Name 4-1 Class Date Additional Vocabulary Support Quadratic Functions and Transformations Concept List Choose the concept from the list below that best represents the item in each box. axis of symmetry parabola translation maximum value parent quadratic function vertex form minimum value quadratic function vertex of the parabola 1. y 5 ax2 1 bx 1 c quadratic function 2. a line that divides a y 3. parabola into two mirror images axis of symmetry x parabola 4. (h, k), where 5. the y-value of the vertex y 5 a(x 2 h)2 1 k vertex of the parabola when the parabola opens up 6. y 5 x2 parent quadratic function minimum value 7. the y-value of the vertex when the parabola opens down 8. y 5 a(x 2 h)2 1 k vertex form 9. a shift of the graph horizontally or vertically translation maximum value Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 1 Name Class 4-1 Date Think About a Plan Quadratic Functions and Transformations Write a quadratic function to represent the areas of all rectangles with a perimeter of 36 ft. Graph the function and describe the rectangle that has the largest area. 1. Write an equation that represents the area of a rectangle with a perimeter of 36 ft. Let x 5 width and y 5 length. 2x 1 2y 5 36 2. Solve your equation for y. y 5 18 2 x 3. Write a quadratic function for the area of the rectangle. z z z z x 18 2 x z R 5 z z ? Q z 2 x2 z 5 z 18x x y A 5 ? y 4. Graph the quadratic function you wrote. 80 5. What point on the graph has a coordinate that 60 represents the largest area? 40 the maximum of the graph: the vertex 6. How can you find the coordinates of this point? What 20 0 are the coordinates? x 0 4 8 12 16 Answers may vary. Sample: Read the x-coordinate of the vertex from the graph and then substitute that value into the quadratic function to get the y-value; (9, 81) 7. Describe the rectangle that has the largest area. What is its area? The rectangle that has the largest area is a square with length 9 ft; 81 ft2 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 2 Name Class 4-1 Date Practice Form G Quadratic Functions and Transformations Graph each function. 1. y 5 3x2 2. f (x) 5 25x2 y 4 3 2 1 5 4. f (x) 5 26x2 ⫺2⫺1 O ⫺1 ⫺2 ⫺3 ⫺4 x ⫺2⫺1 O 1 2 y x 5. ⫺2⫺1 O ⫺1 ⫺2 ⫺3 ⫺4 x 3. y 87 f (x) 5 10x2 4 3 2 1 1 2 ⫺2⫺1 O 8 y 5 3 x2 4 3 2 1 1 2 y x ⫺2⫺1 O 4 6. f (x) 5 5x2 y 4 3 2 1 x 1 2 y ⫺2⫺1 O 1 2 x 1 2 Graph each function. Describe how it was translated from f (x) 5 x2 . 7. f (x) 5 x2 1 4 up 4 units 5 4 3 2 1 8. f (x) 5 (x 2 3)2 right 3 units y x 4 3 2 1 y x O 1 2 3 4 Identify the vertex, axis of symmetry, the maximum or minimum value, and the domain and the range of each function. 9. y 5 (x 2 2)2 1 3 vertex: (2, 3); axis of symmetry: x 5 2; minimum value: 3; domain: all real numbers; range: all real numbers # 3 10. f (x) 5 20.2(x 1 3)2 1 2 vertex: (–3, 2); axis of symmetry: x 5 23; maximum value: 2; domain: all real numbers; range: all real numbers " 2 Graph each function. Identify the axis of symmetry. 11. y 5 (x 1 2)2 2 1 x 5 22 3 2 1 ⫺3⫺2⫺1 O ⫺1 12. y 5 24(x 2 3)2 1 2 x53 y x y 2 1 x O ⫺1 ⫺2 1 1 2 3 4 Write a quadratic function to model each graph. 13. 7 6 5 4 3 2 1 ⫺1 O y 5 (x 2 2)2 1 3 14. y x 6 5 4 3 2 1 ⫺7⫺6⫺5⫺4⫺3⫺2⫺1 O ⫺1 1 2 3 4 5 6 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 3 y x y 5 2(x 1 3)2 2 1 Name Class Date Practice (continued) 4-1 Form G Quadratic Functions and Transformations Describe how to transform the parent function y 5 x2 to the graph of each function below. Graph both functions on the same axes. 15. y 5 3Ax 1 2B 2 4 3 2 1 Translate 2 units to the left; stretch vertically by the factor 3. ⫺3⫺2⫺1 O 1 17. y 5 2 Ax 1 4B 2 2 2 Translate 4 units to the left; shrink vertically by the factor 12 ; translate 2 units down. 16. y 5 2Ax 1 5B 2 1 1 y Translate 5 units to the left; reﬂect across the x-axis; translate 1 unit up. x ⫺6⫺4⫺2 O 18. y 5 20.08Ax 2 0.04B 2 1 1.2 8 6 4 2 Translate 0.04 units to the right; shrink vertically by a factor of 0.08; reﬂect across the x-axis; translate 1.2 units up. 2 x x ⫺6⫺4⫺2 O ⫺2 ⫺4 1 y 8 6 4 2 y 4 2 ⫺4⫺2 O y x 2 4 Write the equation of each parabola in vertex form. 1 20. vertex Q 2, 1 R , point (2, 28) 19. vertex (3, 22), point (2, 3) y 5 5(x 2 3)2 2 2 y 5 24 Q x 2 12 R 2 1 1 21. vertex (24, 224), point (25, 225) y 5 2(x 1 4)2 22. vertex (212.5, 35.5), point (1, 400) y 5 2(x 1 12.5)2 1 35.5 2 24 23. The amount of cloth used to make four curtains is given by the function A 5 24x2 1 40x, where x is the width of one curtain in feet and A is the total area in square feet. Find the width that maximizes the area of the curtains. What is the maximum area? 5 ft; 100 ft2 24. The diagram shows the path of a model rocket launched from 384 ft the ground. It reaches a maximum altitude of 384 ft when it is above a location 16 ft from the launch site. What quadratic function models the height of the rocket? f(x) 5 21.5(x 2 16)2 1 384 Launch 16 ft 25. To make an enclosure for chickens, a rectangular area will be fenced next to a house. Only three sides will need to be fenced. There is 120 ft of fencing material. x a. What quadratic function represents the area of the rectangular enclosure, where x is the distance from the house? A 5 22x2 1 120x b. What dimensions will maximize the area of the enclosure? 30 ft 3 60 ft Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 4 2 32 ft Name Class Date Practice 4-1 Form K Quadratic Functions and Transformations Graph each function. 1. y 5 4x2 y 4 3. y 5 22 x2 8 y 8 y 4 x ⫺8 ⫺4 O ⫺4 1 2. f(x) 5 23x2 ⫺8 ⫺4 O 8 4 x 4 x ⫺8 ⫺4 O ⫺4 8 4 8 ⫺8 ⫺8 Graph each function. How is each graph a translation of f(x) 5 x2 ? 4. f(x) 5 x2 1 4 5. f(x) 5 (x 1 3)2 8 y 8 y 4 ⫺8 ⫺4 O ⫺4 6. f(x) 5 x2 2 2 8 y x 4 4 x ⫺8 ⫺4 O ⫺4 8 ⫺8 4 ⫺8 ⫺4 O ⫺4 8 ⫺8 The vertex moved to (0, 4) 8. f(x) 5 x2 1 6 9. f(x) 5 (x 1 1)2 y 6 4 ⫺8 ⫺4 O ⫺4 8 The vertex moved to (0, 22) y y 4 ⫺8 The vertex moved to (23, 0) 7. f(x) 5 (x 2 5)2 x 4 4 x 4 ⫺8 ⫺4 O ⫺8 The vertex moved to (5, 0) 4 x 8 The vertex moved to (0, 6) 2 ⫺4 ⫺2 O x 2 The vertex moved to (21, 0) What are the vertex, the axis of symmetry, the maximum or minimum value, the domain, and the range of each function? 10. f(x) 5 2(x 2 4)2 1 3 11. f(x) 5 2(x 1 3)2 2 2.5 vertex: (4, 3); axis of sym: x 5 4; min: 3; domain: all real numbers; range: all real numbers L 3 vertex: (23, 22.5); axis of sym: x 5 23; max value: 22.5; domain: all real numbers; range: all real numbers K 22.5 12. f(x) 5 22(x 2 6)2 vertex: (6, 0); axis of sym: x 5 6; max value: 0; domain: all real numbers; range: all real numbers K 0 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 5 Name Class Date Practice (continued) 4-1 Form K Quadratic Functions and Transformations What is the graph of each function? Identify the axis of symmetry. 13. f(x) 5 6(x 2 1)2 2 4 x51 4 y 2 ⫺2 O 14. f(x) 5 2(x 1 5)2 1 2 y x 5 25 2 x 2 15. f(x) 5 (x 1 2)2 2 7 y x 5 22 x 4 ⫺8⫺6 ⫺4 ⫺2 O 4 ⫺8 ⫺4 O ⫺4 ⫺4 ⫺6 ⫺4 x 4 8 ⫺8 What quadratic function models each graph? 16. 17. y 8 6 4 2 ⫺4 ⫺2 O y ⫺6⫺4⫺2 ⫺2 ⫺4 x 18. x O2 ⫺8 2 4 f(x) 5 4(x 2 1)2 1 5 8 6 4 2 ⫺6⫺4⫺2 f(x) 5 234(x 1 3)2 2 2 y x O2 f(x) 5 (x 1 4)2 1 1 19. Error Analysis A classmate said that the vertex of y 5 25(x 1 2)2 2 6 is (2, 6). What mistake did your classmate make? What is the correct vertex? Your classmate forgot to change the sign for h and not to change the sign for k. The correct vertex is (22, 26). 20. Open-Ended Write a quadratic function that has a maximum value. any quadratic function with a negative a value Write the equation of each parabola in vertex form. 21. vertex (23, 7), point (22, 25) f(x) 5 212(x 1 3)2 22. vertex (4, 0), point (26, 23) f(x) 5 20.03(x 2 4)2 17 23. vertex (22, 25), point (26, 0) f(x) 5 5 16 (x 1 2)2 24. vertex (1, 3), point (2, 5) f(x) 5 2(x 2 1)2 1 3 25 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 6 Name Class 4-1 Date Standardized Test Prep Quadratic Functions and Transformations Multiple Choice For Exercises 1−4, choose the correct letter. 1. What is the vertex of the function y 5 3(x 2 7)2 1 4? D (27, 24) (27, 4) (7, 24) (7, 4) 2. Which is the graph of the function f (x) 5 22(x 1 3)2 1 5? F y 6 6 4 4 2 ⫺6 ⫺4 ⫺2 O y ⫺6 ⫺4 ⫺2 O ⫺2 y 2 x ⫺6 ⫺4 ⫺2 O x x 8 y 6 ⫺4 4 ⫺6 2 O ⫺8 x 2 3. Which of the following best describes how to transform y 5 x2 to the graph of y 5 4(x 2 2.5)2 2 3? C Translate 2.5 units left, stretch by a factor of 4, translate 3 units down. Translate 3 units right and 2.5 units down, stretch by a factor of 4. Translate 2.5 units right, stretch by a factor of 4, translate 3 units down. Stretch by a factor of 4, translate 2.5 units left and 3 units down. 4. What is the equation of the parabola with vertex (24, 6) passing through the point (22, 22)? I y 5 22(x 1 4)2 2 6 y 5 2(x 1 4)2 1 6 y 5 2(x 2 4)2 2 6 y 5 22(x 1 4)2 1 6 Short Response 5. A baseball is hit so that its height above ground is given by the equation h 5 216t2 1 96t 1 4, where h is the height in feet and t is the time in seconds after it is hit. Show your work. a. How long does it take the baseball to reach its highest point? b. How high will it go? [2] a. 3 s; b. 148 ft [1] incorrect time to highest point OR incorrect ﬁnal height OR correct time and distance, but no work shown [0] incorrect answers and no work shown OR no answers given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 7 4 6 Name Class 4-1 Date Enrichment Quadratic Functions and Transformations Parabolas in Other Coordinate Systems 1. What are the coordinates of the vertex of the parabola y 5 ax2 ? (0, 0) 2. How do you determine whether the vertex is a maximum or a minimum? If a S 0, vertex is a minimum; if a R 0, vertex is a maximum. 3. What is the equation of the axis of symmetry? x 5 0 Suppose you choose any point (h, k). Through (h, k), draw two lines, one parallel to the x-axis and one parallel to the y-axis. Let the line parallel to the x-axis be called the u-axis, and let the line parallel to the y-axis be called the v-axis. You have now established a new coordinate system—the u-v system. y y ⴝ ax2 u (h, k) (0, 0) 4. In the x-y system, what are the coordinates of the origin of the u-v system? (h, k) 5. In the u-v system, what are the coordinates of the vertex of the parabola y 5 ax2 ? What is the equation of its axis of symmetry? (−h, −k); u 5 2h Suppose point P has coordinates (x, y) in the x-y system and coordinates (u, v) in the u-v system. 6. Write an equation expressing the relationship between u and x. x 5 u 1 h 7. Write an equation expressing the relationship between v and y. y 5 v 1 k 8. Use these relationships to write an equation of the parabola y 5 ax2 in terms of u and v. v 1 k 5 a(u 1 h)2 9. Expand and simplify your equation to express v as a quadratic function of u. v 5 au2 1 2ahu 1 ah2 2 k If we let b 5 2ah and c 5 ah2 2 k, the parabola represented by the quadratic equation v 5 au2 1 bu 1 c in the u-v system is equivalent to the parabola y 5 ax2 in the x-y system. 10. In the u-v system, express the coordinates of the vertex of this parabola in terms of a, b, and c. What is the equation of its axis of symmetry? b b2 b Q 2 2a , c 2 4a R ; u 5 2 2a Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 8 v x Name Class 4-1 Date Reteaching Quadratic Functions and Transformations Parent Quadratic Function The parent quadratic function is y 5 x2 . y Axis of symmetry Substitute 0 for x in the function to get y 5 0. The vertex of the parent quadratic function is (0, 0). xⴝ0 A few points near the vertex are: x y 23 22 21 9 4 1 1 2 3 1 4 9 y ⴝ x2 x Vertex (0, 0) The graph is symmetrical about the line x 5 0 . This line is the axis of symmetry. y Vertex Form of a Quadratic Function y ⴝ a(xⴚh)2ⴙk h)2 The vertex form of a quadratic function is y 5 a(x 2 1 k. The graph of this function is a transformation of the graph of the parent quadratic function y 5 x2. The vertex of the graph is (h, k). If a 5 1, you can graph the function by sliding the graph of the parent function h units along the x-axis and k units along the y-axis. y ⴝ x2 Slide k units x Slide h units Problem What is the graph of y 5 (x 1 3)2 1 2? What are the vertex and axis of symmetry of the function? Step 1 Write the function in vertex form: y 5 1fx 2 (23)g 2 1 2 Step 2 Find the vertex: h 5 23, k 5 2. The vertex is (23, 2). Step 3 Find the axis of symmetry. Since the vertex is (23, 2), the graph is symmetrical about the line x 5 23. The axis of symmetry is x 5 23. Step 4 Because a 5 1, you can graph this function by sliding the graph of the parent function 23 units along the x-axis and 2 units along the y-axis. Plot a few points near the vertex to help you sketch the graph. x y y ⴝ (xⴙ3)2 ⴙ 2 Slide 2 units Slide ⴚ3 units 25 24 23 22 21 6 3 2 3 6 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 9 y y ⴝ x2 x Name Class Reteaching 4-1 Date (continued) Quadratic Functions and Transformations If a 2 1, the graph is a stretch or compression of the parent function by a factor of u a u . 0 , uau , 1 The graph is a vertical compression of the parent function. uau . 1 The graph is a vertical stretch of the parent function y y Vertical y ⴝ ax2 y ⴝ x2 compression y ⴝ ax2 y ⴝ x2 Vertical stretch x x Problem What is the graph of y 5 2(x 1 3)2 1 2? Step 1 Write the function in vertex form: y 5 2fx 2 (23)g 2 1 2 Step 2 Step 3 Step 4 The vertex is (23, 2). The axis of symmetry is x 5 23. Because a 5 2, the graph of this function is a vertical stretch by 2 of the parent function. In addition to sliding the graph of the parent function 3 units left and 2 units up, you must change the shape of the graph. Plot a few points near the vertex to help you sketch the graph. x y y ⴝ 2(xⴙ3)2 ⴙ 2 y Stretch vertically y ⴝ x2 Slide 2 units Slide ⴚ3 units x 25 24 23 22 21 10 4 2 4 10 Exercises Graph each function. Identify the vertex and axis of symmetry. 1. y 5 (x 2 1)2 1 3 (1, 3); y 8 x51 4 2 6 4 2 x ⫺2 O 2 4 6 ⫺2 O y (1, 3); x51 x 2 4 6 3. y 5 (x 1 2)2 1 1 (22, 1); y 8 x 5 22 6 4 x ⫺8 ⫺6 ⫺4 ⫺2 O ⫺2 4. y 5 2(x 2 1)2 1 3 8 6 4 2 2. y 5 (x 1 4)2 2 2 y (24, 22); 6 x 5 24 ⫺6 ⫺4 ⫺2 O 1 5. y 5 2(x 1 4)2 2 2 (24, 22); y 6 x 5 24 4 2 6. y 5 0.9(x 1 2)2 1 1 y 8 6 4 2 x ⫺8 ⫺6 ⫺4 ⫺2 O ⫺2 ⫺6 ⫺4 ⫺2 O Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 10 2 x (22, 1); x 5 22 x 2 Name Class 4-2 Date Additional Vocabulary Support Standard Form of a Quadratic Function Standard Form of a Quadratic The standard form of a quadratic is f (x) 5 ax2 1 bx 1 c, where a 2 0. If a . 0, then the parabola opens upward. If a , 0, then the parabola opens downward. The y-intercept is (0, c). b The x-coordinate of the vertex is 22a . 1. Which function is written in standard form? f (x) 5 4(x 2 5)2 1 1 f (x) 5 3x2 2 x 1 4 2. Which has an x-coordinate of 25? (25, 2) (2, 25) 3. Which quadratic has a y-intercept of 2? f (x) 5 2(x 2 5)2 1 2 f (x) 5 2x2 1 5x 1 2 4. Which parabola opens downward? f (x) 5 223(x 1 1)2 1 5 f (x) 5 4x2 1 1 Vertex Form of a Quadratic The vertex form of a quadratic is f (x) 5 a(x 2 h)2 1 k, where a 2 0. If a . 0, then the parabola opens upward. If a , 0, then the parabola opens downward. The vertex is (h, k). 5. Which function is written in vertex form? f (x) 5 22(x 1 7)2 1 3 f (x) 5 2x2 1 2x 1 6 6. Which quadratic has a vertex of (22, 4)? f (x) 5 x2 2 2x 1 4 f (x) 5 (x 1 2)2 1 4 7. Which parabola opens upward? f (x) 5 25(x 2 7)2 1 1 f (x) 5 14(x 1 5)2 2 9 Write the vertex form of the quadratic written in standard form below. 8. f (x) 5 x2 2 6x 1 12 f(x) 5 (x 2 3)2 1 3 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 11 Name 4-2 Class Date Think About a Plan Standard Form of a Quadratic Function Landscaping A town is planning a playground. It wants to fence in a rectangular space using an existing wall. What is the greatest area it can fence in using 100 ft of donated fencing? Understanding the Problem 1. Write an expression for the width of the playground. Let l be the length of the playground. 100 2 2l 2. Do you know the perimeter of the playground? Explain. No; you do not know how much of the wall will be used. 3. What is the problem asking you to determine? The largest area that can be enclosed by 100 ft of fence on three sides and an unknown length of wall on the fourth side. Planning the Solution 4. Write a quadratic equation to model the area of the playground. A 5 100l 2 2l 2 5. What information can you get from the equation to find the maximum area? Explain. You can ﬁnd the vertex. The x-coordinate of the vertex is the length that gives the maximum area, which is the y-coordinate. Getting an Answer 6. What is the value of l that produces the maximum area? 25 ft 7. What is the greatest area the town can fence in using 100 ft of fencing? 1250 ft2 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 12 Name 4-2 Class Date Practice Form G Standard Form of a Quadratic Function Identify the vertex, the axis of symmetry, the maximum or minimum value, and the range of each parabola. 1. y 5 x2 2 4x 1 1 2. y 5 2x2 1 2x 1 3 Vertex is the maximum: (1, 4); axis of symmetry: x 5 1; range: y K 4 Vertex is the minimum: (2, 23); axis of symmetry: x 5 2; range: y L 23 3. y 5 2x2 2 6x 2 10 4. y 5 3x2 1 18x 1 32 Vertex is the maximum: (23, 21); axis of symmetry: x 5 23; range: y K 21 Vertex is the minimum: (23, 5); axis of symmetry: x 5 23; range: y L 5 5. y 5 2x2 1 3x 2 5 6. y 5 23x2 1 4x Vertex is the minimum: Q 2 34, 2 49 8 R ; axis 3 of symmetry: x 5 2 4 ; range: y L 2 49 8 Vertex is the maximum: Q 23, 43 R ; axis of 2 symmetry: x 5 3 ; range: y K 43 Graph each function. 7. y 5 x2 1 2x 2 5 8. y 5 2x2 1 3x 1 1 y 2 x 642 O 2 4 6 9. y 5 2x2 1 4x 2 4 3 2 1 1 2 3 1 10. y 5 22 x2 2 3x 1 3 y 2 8 6 4 2 2 x 8 2 42 2 4 6 x 1 642 2 4 11. y 5 3x2 2 8x y 12. y 5 23x2 1 18x 2 27 y x x 42 y 2 4 6 8 2 4 y x 2 4 6 8 Write each function in vertex form. 13. y 5 x2 2 8x 1 19 y 5 (x 2 4)2 1 3 14. y 5 x2 2 2x 2 6 y 5 (x 2 1)2 2 7 3 2 9 15. y 5 x2 1 3x y 5 Q x 1 2 R 2 4 16. y 5 2x2 1 x y 5 2 Q x 1 1 R 2 2 1 4 8 17. y 5 2x2 2 12x 1 11 y 5 2(x 2 3)2 2 7 18. y 5 22x2 2 4x 1 6 y 5 22(x 1 1)2 1 8 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 13 Name Class 4-2 Date Practice (continued) Form G Standard Form of a Quadratic Function 19. A small independent motion picture company determines the profit P for producing n DVD copies of a recent release is P 5 20.02n2 1 3.40n 2 16. P is the profit in thousands of dollars and n is in thousands of units. a. How many DVDs should the company produce to maximize the profit? 85,000 b. What will the maximize profit be? $128,500 Sketch each parabola using the given information. 20. vertex (4, 22), y–intercept 6 6 4 2 2 21. vertex (23, 12), point (21, 0) y 12 8 4 x 2 4 6 8 y x 8 642 4 For each function, the vertex of the function’s graph is given. Find the unknown coefficients. 22. y 5 x2 1 bx 1 c; (24, 27) b: 8, c: 9 23. y 5 ax2 2 10x 1 c ; (25, 20) a: 21, c: 25 24. A local nursery sells a large number of ornamental trees every year. The owners have determined the cost per tree C for buying and caring for each tree before it is sold is C 5 0.001n2 2 0.3n 1 50. In this function, C is the cost per tree in dollars and n is the number of trees in stock. a. How many trees will minimize the cost per tree? 150 b. What will the minimum cost per tree be? $27.50 20 ft 25. To line an irrigation ditch, a farmer will use rectangular metal sheets. Each side will be bent x feet from the edge at an angle of 90° to form the trough. If the sheets are 20 ft wide, how far from the edge (x) should the farmer bend them to maximize the area of a cross-section of the trough. 5 ft x For each function, find the y-intercept. 26. y 5 (x 1 3)2 2 5 (0, 4) 27. y 5 22(x 2 2)2 1 6 (0, –2) 28. y 5 2(x 1 1)2 1 9 (0, 8) 1 29. y 5 2(x 1 4)2 2 15 (0, –7) Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 14 x Name Class Date Practice 4-2 Form K Standard Form of a Quadratic Function What are the vertex, the axis of symmetry, the maximum or minimum value, and the range of each parabola? 1. y 5 2x2 1 2x 2 5 vertex: (1, 24); axis of sym: x 5 1; max: 24; range: all real numbers K 24 2. y 5 22x2 2 8x 1 3 vertex: (22, 11); axis of sym: x 5 22; max: 11; range: all real numbers K 11 3. y 5 4x2 2 2x 1 1 vertex: Q 14, 34 R ; axis of sym: x 5 14; min: 34 ; range: all real numbers L 34 What is the graph of each function? 4. y 5 2x2 2 6x 2 11 y 8 4 O 4 x 4 8 5. y 5 5x2 1 10x 1 8 12 y 8 8 12 4 8 4 O 7. y 5 22x2 1 4x 1 3 4 8 4 4 4 x 8 8. y 5 x2 1 4x 1 2 8 4 O 4 x 4 4 8 4 x 8 x 8 y 4 O 8 4 4 4 9. y 5 26x2 2 12x 1 5 y O 11 y 8 8 y 1 6. y 5 2x2 2 3x 1 2 x 8 4 O 4 4 8 Sketch each parabola using the given information. 10. vertex (22, 1), y-intercept 4 y 6 12. vertex (21, 27), point (4, 1) point (25, 1) 8 y 4 4 2 4 2 O 11. vertex (3, 25), x 2 8 4 x 8 4 O 4 8 8 O 4 8 8 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 15 y x 4 8 Name 4-2 Class Date Practice (continued) Form K Standard Form of a Quadratic Function What is the vertex form of the following quadratic functions? 13. y 5 4x2 1 16x 1 19 y 5 4(x 1 2)2 1 3 16. y 5 23x2 2 18x 2 29 y 5 23(x 1 3)2 2 2 14. y 5 2x2 1 2x 1 4 y 5 2(x 2 1)2 1 5 17. y 5 2x2 1 4x 1 9 y 5 2(x 1 1)2 1 7 1 15. y 5 2x2 2 6x 1 15 y 5 12(x 2 6)2 2 3 8 5 2 18. y 5 3x2 2 3x 1 3 y 5 23(x 2 2)2 2 1 19. Reasoning When is it better to have the quadratic function in vertex form instead of standard form? Answers may vary. Sample: When you need to quickly ﬁnd information about the graph of a quadratic function. 20. The Gateway Arch in St. Louis was built in 1965. It is the tallest monument in the United States. The arch can be modeled with the function y 5 20.00635x2 1 4x, where x and y are in feet. a. How high above the ground is the tallest point of the arch? 630 ft b. How far apart are the legs of the arch at their bases? 630 ft 21. The height of a batted ball is modeled by the function h 5 20.01x2 1 1.22x 1 3, where x is the horizontal distance in feet from the point of impact with the bat, and h is the height of the ball in feet. a. What is the maximum height that the ball will reach? about 40 ft b. At what distance from the batter will the ball hit the ground? about 124 ft Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 16 Name Class 4-2 Date Standardized Test Prep Standard Form of a Quadratic Function Multiple Choice For Exercises 1−6, choose the correct letter. 1. What is the vertex of the parabola y 5 x2 1 8x 1 5? B (4, 211) (24, 211) (24, 5) (4, 5) 2. What is the maximum value of the function y 5 23x2 1 12x 2 8? F 4 28 8 2 2 1 3. Which function has the graph shown at the right? D y 5 22x2 2 5x 1 1 y 5 2x2 1 5x 2 1 y 5 22x2 2 5x 2 1 y 5 22x2 1 5x 2 1 2 1 1 2 4. What is the vertex form of the function y 5 3x2 2 12x 1 17? F y 5 3(x 2 2)2 1 5 y 5 3(x 2 2)2 1 17 y 5 3(x 2 2)2 1 11 y 5 3(x 1 2)2 1 5 5. What is the equation of the parabola with vertex (3, 220) and that passes through the point (7, 12)? B y 5 2x2 1 12x 2 2 y 5 2x2 2 12x 2 2 y 5 22x2 1 12x 2 38 y 5 2x2 2 12x 1 38 6. For the function y 5 25x2 2 10x 1 c, the vertex is (21, 8). What is c? H 213 3 23 13 Short Response 7. To increase revenue, a county wants to increase park fees. The overall income will go up, but there will be expenses involved in collecting the fees. For a p% increase in the fees, this cost C will be C 5 0.6p2 2 7.2p 1 48, in thousands of dollars. What percent increase will minimize the cost to the county? Show your work. b 27.2 [2] 6%; the minimum is at the vertex, which is 2 2a 5 2 2(0.6) 5 6. [1] computational error OR work not shown [0] incorrect answer and no work shown OR no answer given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 17 y x 1 2 3 4 Name Class 4-2 Date Enrichment Standard Form of a Quadratic Function The standard form of a quadratic function, y 5 ax2 1 bx 1 c , is useful, but it has the disadvantage that only one of the three constants has a simple geometrical interpretation. 1. Which of the constants in the equation y 5 ax2 1 bx 1 c can be interpreted geometrically? c 2. What is its geometrical interpretation? the y-intercept A more intuitive equation is expressed in terms of constants that have geometrical interpretations. For instance, if I denotes the y-intercept of a parabola whose vertex is (V, W), the equation y 5 I 2 2W (x 2 V)2 1 W describes a parabola with V an axis of symmetry parallel to the y-axis. Using this equation, write the equation of the following parabolas in vertex form. y-intercept 3. 22 4. 1 vertex (1, 4) y 5 26(x 2 1)2 1 4 (23, 8) y 5 2 79 (x 1 3)2 1 8 I The equation y 5 PQ(x 2 P)(x 2 Q) describes a parabola with an axis of symmetry parallel to the y-axis in terms of its y-intercept I and its x-intercepts P and Q, where P and Q are real numbers. Using this equation, write the equation of the following parabolas. y-intercept x-intercepts 5. 24 1 and 5 y 5 2 45 (x 2 1)(x 2 5) 6. 6 24 and 22 y 5 34 (x 1 4)(x 1 2) The equation y 5 I (x 2 P)2 describes a parabola whose axis of symmetry is P2 parallel to the y-axis with y-intercept I and exactly one x-intercept P. 7. Why is this a special case of the previous equation? Answers may vary. Sample: It is the previous equation with P 5 Q. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 18 Name Class Date Reteaching 4-2 Standard Form of a Quadratic Function • The graph of a quadratic function, y 5 ax2 1 bx 1 c, where a 2 0, is a parabola. b . • The axis of symmetry is the line x 5 2 2a b . The y-coordinate of the vertex is • The x-coordinate of the vertex is 2 2a b b R , or the y-value when x 5 2 2a. y 5 f Q 2 2a • The y-intercept is (0, c). Problem What is the graph of y 5 2x2 2 8x 1 5? x52 2(28) b 8 5 5 52 2a 2(2) 4 Find the equation of the axis of symmetry. b 2a x-coordinate of vertex: 2 2 b f Q 22a R 5 f (2) 5 2(2)2 2 8(2) 1 5 Find the y-value when x 5 2. 5 8 2 16 1 5 5 23 y-coordinate of vertex: 23 The vertex is (2, 23). y-intercept: (0, 5) The y-intercept is at (0, c) 5 (0, 5). y 6 5 (0, 5) 4 3 2 1 Because a is positive, the graph opens upward, and the vertex is at the bottom of the graph. Plot the vertex and draw the axis of symmetry. Plot (0, 5) and its corresponding point on the other side of the axis of symmetry. (4 , 5) x O 4 5 2 3 x2 (2, 3) Exercises Graph each parabola. Label the vertex and the axis of symmetry. 1. y 5 23x2 1 6x 2 9 (0, 9) 3. y 5 2x2 2 8x 1 1 2. y 5 2x2 2 8x 2 15 y x1 4 O 4 x 4 (1, 6) 8 (2, 9) 4 y (0, 1) 4 O 4 8 4. y 5 22x2 2 12x 2 7 (4, 1) 4 x 8 (4, 1) 3 y x 9 3 x 4 6 (8, 15) 12 (0, 15) 18 (3, 11) 12 y 6 x = 3 O x 8 4 4 (6, 7) 6 (0, 7) (2, 7) x=2 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 19 Name Class Date Reteaching (continued) 4-2 Standard Form of a Quadratic Function • Standard form of a quadratic function is y 5 ax2 1 bx 1 c. Vertex form of a quadratic function is y 5 a(x 2 h)2 1 k. • For a parabola in vertex form, the coordinates of the vertex are (h, k). Problem What is the vertex form of y 5 3x2 2 24x 1 50? y 5 ax2 1 bx 1 c y 5 3x2 2 24x 1 50 Verify that the equation is in standard form. b 5 224, a 5 3 Find b and a. b x-coordinate 5 22a 5 2224 For an equation in standard form, the x-coordinate of the vertex b can be found by using x 5 2 . 2a Substitute. 54 Simplify. 2(3) y-coordinate 5 3(4)2 2 24(4) 1 50 Substitute 4 into the standard form to ﬁnd the y-coordinate. 52 y 5 3(x 2 4)2 Simplify. 12 Substitute 4 for h and 2 for k into the vertex form. Once the conversion to vertex form is complete, check by multiplying. y 5 3(x2 2 8x 1 16) 1 2 y 5 3x2 2 24x 1 50 The result is the standard form of the equation. Exercises Write each function in vertex form. Check your answers. 5. y 5 x2 2 2x 2 3 y 5 (x 2 1)2 2 4 8. y 5 x2 2 9x y 5 Qx 2 9 2 2R 6. y 5 2x2 1 4x 1 6 2 11. y 5 4x2 1 8x 2 3 y 5 4(x 1 1)2 2 7 y 5 Q x 1 32 R 2 2 49 4 y 5 2(x 2 2)2 1 10 9. y 5 x2 1 x 81 4 7. y 5 x2 1 3x 2 10 y 5 Qx 1 1 2 2R 10. y 5 x2 1 5x 1 4 2 y 5 Q x 1 52 R 2 2 94 1 4 3 12. y 5 4x2 1 9x y 5 34 (x 1 6)2 2 27 13. y 5 22x2 1 2x 1 1 y 5 22 Q x 2 12 R 2 1 32 Write each function in standard form. 14. y 5 (x 2 3)2 1 1 y 5 x2 2 6x 1 10 15. y 5 2(x 2 1)2 2 3 16. y 5 23(x 1 4)2 1 1 y 5 2x2 2 4x 2 1 y 5 23x2 2 24x 2 47 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 20 Name 4-3 Class Date Additional Vocabulary Support Modeling With Quadratic Functions A football player kicks a football and records the height of the ball at different times. When kicked, at 0 seconds, the ball was 2 ft above the ground. One second later the ball was 28 ft above the ground, and 2 seconds after being kicked the ball was 20 ft above the ground. When will the ball hit the ground? You wrote these steps to solve the problem on note cards, but they got mixed up. Substitute the x and y values into the standard form of a quadratic function. Use the quadratic model to determine when the ball hits the ground. Substitute the values of a, b, and c into the standard form of a quadratic function. Solve the system of three linear equations. Use the note cards to write the steps in order. 1. First, substitute the x- and y-values into the standard form of a quadratic function . 2. Second, solve the system of three linear equations . 3. Next, substitute the values of a, b, and c into the standard form of a quadratic function . 4. Finally, use the quadratic model to determine when the ball hits the ground . Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 21 Name Class 4-3 Date Think About a Plan Modeling With Quadratic Functions a. Postal Rates Find a quadratic model for the data. Use 1981 as year 0. Price of First-Class Stamp Year Price (cents) 1981 1991 1995 1999 2001 2006 2007 2008 18 29 32 33 34 39 41 42 b. Describe a reasonable domain and range for your model. (Hint: This is a discrete, real situation.) c. Estimation Estimate when first-class postage was 37 cents. d. Use your model to predict when first-class postage will be 50 cents. Explain why your prediction may not be valid. 1. How can you find the x-coordinates of the data points? Subtract 1981 from each year. 2. What calculator function finds a quadratic model for data? QuadReg z z z z z 0.930 18.586 3. Find a quadratic model for the data. y 5 20.0036 x2 1 x 1 4. What does the domain of your model represent? What set of numbers would be a reasonable domain? Years since 1981; answers may vary. Sample: positive integers 5. What does the range of your model represent? What set of numbers would be a reasonable domain? Cost of ﬁrst-class stamp, in cents; answers may vary. Sample: positive integers 6. How can you find the x-value that produces a given y-value? Answers may vary. Sample: Graph the function on a calculator and use the TRACE function. z 2002 7. Estimate the year when first-class postage was 37 cents. z z 2021 8. Predict the year when first-class postage will be 50 cents. z 9. Why might your prediction not be valid? Answers may vary. Sample: Many factors inﬂuence postal rates, such as inﬂation, fuel costs, and demand for postal services. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 22 z Name Class 4-3 Date Practice Form G Modeling With Quadratic Functions Find an equation in standard form of the parabola passing through the points. 1. (1, 21), (2, 25), (3, 27) y5 x2 2. (1, 24), (2, 23), (3, 24) y 5 2x2 1 4x 2 7 2 7x 1 5 3. (2, 28), (3, 28), (6, 4) x2 y5 4. (21, 212), (2, 26), (4, 212) y 5 2x2 1 3x 2 8 2 5x 2 2 5. (21, 212), (0, 26), (3, 0) 6. (22, 24), (1, 21), (3, 11) y 5 2x2 1 5x 2 6 y 5 x2 1 2x 2 4 7. (21, 26), (0, 0), (2, 6) y5 9. 11. 2x2 8. (23, 2), (1, 26), (4, 9) y 5 x2 2 7 1 5x x f(x) 21 10. y 5 x2 2 x 1 5 x f(x) 7 22 27 1 5 0 1 3 11 2 1 12. y 5 x2 1 5x 2 2 x f(x) 26 22 21 1 4 2 21 2 12 3 9 x f(x) 21 y 5 2x2 1 2x 1 1 y 5 2x2 2 9 13. The table shows the number n of tickets to a school play sold t days after the tickets went on sale, for several days. a. Find a quadratic model for the data. n 5 22t2 1 24t 1 10 b. Use the model to find the number of tickets sold on day 7. 80 c. When was the greatest number of tickets sold? day 6 Day, t Number of Tickets Sold, n 1 32 3 64 4 74 14. The table gives the number of pairs of skis sold in a sporting goods store for several months last year. a. Find a quadratic model for the data, using January as month 1, February as month 2, and so on. s 5 2t2 2 28t 1 108 b. Use the model to predict the number of pairs of skis sold in November. 42 c. In what month were the fewest skis sold? July Month, t Number of Pairs of Skis Sold, s Jan 82 Mar 42 May 18 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 23 Name Class 4-3 Date Practice (continued) Form G Modeling With Quadratic Functions Determine whether a quadratic model exists for each set of values. If so, write the model. 15. f (21) 5 27, f (1) 5 1, f (3) 5 1 16. f (21) 5 13, f (0) 5 6, f (2) 5 28 no y 5 2x2 1 4x 2 2 17. f (2) 5 2, f (24) 5 21, f (22) 5 0 18. f (2) 5 6, f (0) 5 24, f (22) 5 26 no y 5 x2 1 3x 2 4 19. a. Complete the table. It shows the sum of the counting numbers from 1 through n. Number, n 1 2 3 4 5 Sum, s 1 3 6 10 15 b. Write a quadratic model for the data. s 5 12n2 1 12n c. Predict the sum of the first 50 counting numbers. 1275 20. On a suspension bridge, the roadway is hung from cables hanging between support towers. The cable of one bridge is in the shape of the parabola y 5 0.1x2 2 7x 1 150, where y is the height in feet of the cable above the roadway at the distance x feet from a support tower. a. What is the closest the cable comes to the roadway? 27.5 ft b. How far from the support tower does this occur? 35 ft 21. The owner of a small motel has an unusual idea to increase revenue. The motel has 20 rooms. He advertises that each night will cost a base rate of $48 plus $8 times the number of empty rooms that night. For example, if all rooms are occupied, he will have a total income of 20 3 $48 5 $960. But, if three rooms are empty, then his total income will be (20 2 3) 3 ($48 1 $8 ? 3) 5 17 3 $72 5 $1224. a. Write a linear expression to show how many rooms are occupied if n rooms are empty. 20 2 n b. Write a linear expression to show the price paid in dollars per room if n rooms are empty. 48 1 8n c. Multiply the expressions from parts (a) and (b) to obtain a quadratic model for the data. Write the result in standard form. y 5 28n2 1 112n 1 960 d. What will the owner’s total income be if 10 rooms are empty? $1280 e. What is the number of empty rooms that results in the maximum income for the owner? 7 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 24 Name Class 4-3 Date Practice Form K Modeling With Quadratic Functions Find an equation in standard form of the parabola passing through the points. To start, substitute the (x, y) values into y 5 ax2 1 bx 1 c to write a system of equations. 1. (2, 220), (22, 24), (0, 28) 2. (1, 23), (2, 0), (3, 9) y 5 2x2 2 4x 2 8 y 5 3x2 2 6x 1 0 3. (3, 21), (2, 25), (4, 25) 4. (24, 3), (26, 7), (21, 12) y 5 24x2 1 24x 2 37 y 5 x2 1 8x 1 19 5. (2, 1), (1, 21), (4, 27) 6. (21, 2), (22, 7), (0, 7) y 5 22x2 1 8x 2 7 y 5 5x2 1 10x 1 7 7. A player hits a tennis ball across the court and records the height of the ball at different times, as shown in the table. 1 11 a. Find a quadratic model for the data. y 5 22x2 1 x 1 2 b. Use the model to estimate the height of the ball at 4 seconds. 1.5 ft c. What is the ball’s maximum height? 6 ft Time (s) Height (ft) 0 1 2 5.5 6.0 5.5 3 4.0 8. Reasoning Explain why the quadratic model only works up to 4.5 seconds — that height measurements made after 4.5 seconds are not valid. (Remember this is a discrete, real situation.) The ball hits the ground in 4.5 seconds. After it hits the ground, the ball cannot go any lower. 9. The table at the right shows the height of the tides measured at the Santa Monica Municipal Pier in California. Hours are measured from 0.00 at midnight. a. Find a quadratic model for this data using quadratic regression. y 5 0.06x2 2 0.62x 1 4.1 b. Use the model to predict the lowest tide height. 2.5 ft c. When does the lowest tide occur? Time Tide Height (ft) 0.33 3.9 3.30 2.7 11.11 4.6 SOURCE: www.tidesandcurrents.noaa.gov about 5.17 h after midnight Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 25 Name Class 4-3 Date Practice (continued) Form K Modeling With Quadratic Functions 10. The table at the right shows average retail gasoline prices. a. Find a quadratic model for the data using 1976 as year 0, 1986 as year 10, 1996 as year 20, and 2005 as year 29. y 5 0.11x2 1 1.87x 1 61.83 b. Use the model to estimate the average retail gasoline price in 2000. 170.07 cents Year Price per gallon (cents) 1976 61.4 1986 92.7 1996 141.3 2005 208.0 SOURCE: U.S. Dept. of Energy Determine whether a quadratic model exists for each set of values. If so, write the model. 11. f (0) 5 5, f (24) 5 13, f (2) 5 7 12. f (1) 5 1, f (23) 5 219, f (21) 5 29 no yes; y 5 12x2 1 5 13. f (0) 5 0, f (1) 5 2, f (2) 5 4 14. f (25) 5 3, f (22) 5 6, f (0) 5 22 no yes; y 5 2x2 2 6x 2 2 15. The table at the right shows in thousands how many people in the U.S. subscribe to a cellular telephone. a. Find a quadratic model for the data. Let x 5 the number of years since 1985. b. Use the model to estimate the number of subscribers in 1995. c. Describe a reasonable domain and range for this situation. Year U.S Cellular Telephone Subscribership (in thousands) 1985 340 1990 5283 2000 109,478 2004 182,140 SOURCE: CTIA Semi-Annual Wireless Industry a. y 5 597.29x2 2 1724.76x 2 109.82 b. about 42,372 thousand subscribers c. domain: all real numbers L 0 (or from 1985); range: all real numbers L 0 16. Error Analysis In Exercise 15 part (c), your friend said that the range was equal to all real numbers. Why is this incorrect? Because this is a real situation, you cannot have a negative number of subscriptions. Therefore, the range must be greater than or equal to 0. (It cannot be negative.) 17. Reasoning Explain how you know your answer to Exercise 15 part (b) is reasonable. The number of subscribers found in 1995 is reasonable because it is in between the values for 1990 and 2000 from the table. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 26 Name Class 4-3 Date Standardized Test Prep Modeling With Quadratic Functions Multiple Choice For Exercises 1−5, choose the correct letter. 1. Which parabola passes through the points (1, 22), (4, 1), and (5, 22)? B y 5 2x2 1 x 2 3 y 5 x2 2 4x 1 1 y 5 2x2 1 6x 2 7 y 5 x2 2 4x 2 1 2. Which parabola passes through the points in the table at the right? I y5 2x2 2x12 y5 y 5 12x2 2 52x 2 1 2x2 2 4x 2 4 y 5 x2 2 3x 2 2 3. A baseball coach records the height at every second of a ball thrown in x f(x) 21 2 2 24 4 2 the air. Some of the data appears in the table below. Time (s) 0 1 3 Height (ft) 0 64 96 Which equation is a quadratic model for the data? A h 5 216t2 1 80t h 5 232t2 1 80t h 5 248t2 1 112t h 5 216t2 1 64t 4. Use the table in Exercise 3. What is the height of the ball at 2.5 s? H 80 ft 100 ft 88 ft 112 ft 5. Which of the following sets of values cannot be modeled with a quadratic function? C (2, 3), (0, 21), (3, 2) (2, 27), (21, 5), (3, 211) f (2) 5 7, f (21) 5 22, f (0) 5 3 f (2) 5 26, f (0) 5 22, f (21) 5 3 Short Response 6. The accountant for a small company studied the amount spent on advertising and the company’s profit for several years. He made the table below. What is a quadratic model for the data? Show your work. Advertising (Hundreds of Dollars) Profit (Dollars) 1 2 3 269 386 501 [2] y 5 2x2 1 120x 1 150 [1] correct method but computational errors OR correct model but no work shown [0] incorrect answers and no work shown OR no answers given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 27 Name 4-3 Class Date Enrichment Modeling With Quadratic Functions Baseballs in Flight When baseballs are shot out of a cannon, their flight through the air depends on both the angle at which the cannon is set and the initial velocity of the baseball. 6 Height of baseball The equation y 5 0.5x 2 0.01x2 represents the parabolic flight of a certain baseball shot at an angle of 26° with the horizon and at an initial velocity of 25 meters per second. In this equation, y is the height of the baseball, in meters, and x is the horizontal distance traveled, in meters. The graph of the equation is shown to the right. y (10, 4) 4 (40, 4) 2 O 1. Given that the points (10, 4) and (40, 4) lie on the parabola, at what x-coordinate must the vertex lie? x 5 25 (0, 0) 20 x 40 60 Distance traveled 2. Use the equation and your answer to question 1 to find the maximum height of the baseball. 6.25 m 3. Use the point (0, 0) and the location of the vertex to find the total horizontal distance that the baseball will travel. 50 m 4. What is the total horizontal distance that this baseball will travel? 25 m 3 Height of baseball When the angle of the cannon is decreased, the baseball will travel in a different flight. The parabolic flight of the baseball is shown to the right, with the vertex labeled. y 2 (12.5, 1.25) 1 (0, 0) 20 O x 40 60 Distance traveled 5. How far will the baseball travel horizontally before it reaches its maximum height? 25 m 3 Height of baseball Using the same angle, the initial velocity of the baseball is increased to produce the graph of the flight shown to the right. The point shown represents the total horizontal distance the baseball will travel. 2 1 O NOTE: SOURCES: Gustafson, R. David. Concepts of intermediate algebra: an early functions approach. Paciﬁc Grove, Calif.: Brooks/Cole Pub. Co., 1996. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 28 y (0, 0) (50, 0) 20 40 Distance traveled x 60 Name 4-3 Class Date Reteaching Modeling With Quadratic Functions Three non-collinear points, no two of which are in line vertically, are on the graph of exactly one quadratic function. Problem A parabola contains the points (0, 22), (21, 5), and (2, 2). What is the equation of this parabola in standard form? If the parabola y 5 ax2 1 bx 1 c passes through the point (x, y), the coordinates of the point must satisfy the equation of the parabola. Substitute the (x, y) values into y 5 ax2 1 bx 1 c to write a system of equations. First, use the point (0, 22). Use the point (21, 5) next. Finally, use the point (2, 2). y 5 ax2 1 bx 1 c Write the standard form. 22 5 a(0)2 1 b(0) 1 c Substitute. 22 5 c Simplify. 5 5 a(21)2 1 b(21) 1 c Substitute. 55a2b1c Simplify. 2 5 a(2)2 1 b(2) 1 c Substitute. 2 5 4a 1 2b 1 c Simplify. Because c 5 22, the resulting system has two variables. Simplify the equations above. a2b57 4a 1 2b 5 4 Use elimination to solve the system and obtain a 5 3, b 5 24, and c 5 22. Substitute these values into the standard form y 5 ax2 1 bx 1 c. The equation of the parabola that contains the given points is y 5 3x2 2 4x 2 2. Exercises Find an equation in standard form of the parabola passing through the given points. 1. (0, 21), (1, 5), (21, 25) y 5 x2 1 5x 2 1 2. (0, 4), (21, 9), (2, 0) y 5 x2 2 4x 1 4 3. (0, 1), (1, 4), (3, 22) y 5 2x2 1 x 1 1 4. (1, 21), (22, 20), (2, 0) y 5 2x2 2 5x 1 2 5. (21, 25), (0, 21), (2, 1) y 5 2x2 1 3x 2 1 6. (1, 3), (22, 23), (21, 3) y 5 22x2 1 5 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 29 Name Class 4-3 Date Reteaching (continued) Modeling With Quadratic Functions Problem A soccer player kicks a ball off the top of a building. His friend records the height of the ball at each second. Some of her data appears in the table. a. What is a quadratic model for these data? b. Use the model to complete the table. Time (s) Height (ft) 0 112 1 192 2 3 Use the points (0, 112), (1, 192), and (5, 192) to find the quadratic model. Substitute the (t, h) values into h 5 at2 1 bt 1 c to write a system of equations. (0, 112) : 112 5 a(0)2 1 b(0) 1 c c 5 112 (1, 192) : 192 5 a(1)2 1 b(1) 1 c a 1 b 1 c 5 192 (5, 192) : 192 5 a(5)2 1 b(5) 1 c 25a 1 5b 1 c 5 192 4 5 192 6 7 Use c 5 112 and simplify the equations to obtain a system with just two variables. a 1 b 5 80 25a 1 5b 5 80 Use elimination to solve the system. The quadratic model for the data is h 5 216t2 1 96t 1 112 Time (s) Height (ft) Now use this equation to complete the table for the t-values 2, 3, 4, 6, and 7. 0 112 1 192 t 5 2: h 5 216(2)2 1 96(2) 1 112 5 264 1 192 1 112 5 240 2 240 3 256 4 240 t 5 3: h 5 216(3)2 1 96(3) 1 112 5 2144 1 288 1 112 5 256 t 5 4: h 5 216(4)2 1 96(4) 1 112 5 2256 1 384 1 112 5 240 5 192 t 5 6: h 5 216(6)2 1 96(6) 1 112 5 2576 1 576 1 112 5 112 6 112 7 0 t 5 7: h 5 216(7)2 1 96(7) 1 112 5 2784 1 672 1 112 5 0 Exercise 7. The number n of Brand X shoes in stock at the beginning of month t in a store follows a quadratic model. In January (t 5 1), there are 36 pairs of shoes; in March (t 5 3), there are 52 pairs; and in September, there are also 52 pairs. a. What is the quadratic model for the number n of pairs of shoes at the beginning of month t? n 5 2t2 1 12t 1 25 b. How many pairs are in stock in June? 61 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 30 Name Class Date Additional Vocabulary Support 4-4 Factoring Quadratic Expressions Choose the word from the list that best matches each sentence. factoring greatest common perfect square difference of two factor trinomial squares 1. the expression a2 2 b2 difference of two squares 2. rewriting an expression as a product of its factors factoring 3. a trinomial that is the square of a binomial perfect square trinomial 4. a common factor of each term in the expression greatest common factor Choose the word from the list that best matches each sentence. factoring 5. 10 is the greatest common perfect square difference of two factor trinomial squares greatest common factor 6. An example of a 7. When of the expression 20x2 2 50. perfect square trinomial factoring is x2 2 8x 1 16. x2 1 8x 1 15, find numbers with product 15 and sum 8. 8. The difference of two squares will always be a binomial. Multiple Choice 9. Which of the following is a perfect square trinomial? B 2x 2 7 9x2 2 6x 1 1 4x2 2 25 9x2 2 4x 10. Which of the following is a difference of perfect squares? H 2x 2 7 9x2 2 6x 1 1 4x2 2 25 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 31 9x2 2 4x Name 4-4 Class Date Think About a Plan Factoring Quadratic Expressions Agriculture The area in square feet of a rectangular field is x2 2 120x 1 3500. The width, in feet, is x 2 50. What is the length, in feet? Know z z z z length times the width . 1. The area of the field equals the 2 2. The area of the field is x 2 120x 1 3500 ft2 . z z x 2 50 ft. 3. The width of the field is Need 4. To solve the problem I need to: rewrite the expression for the area of the ﬁeld as a product of its factors . Plan z z x 2 50 . 5. One factor is 6. What is the coefficient of the first term of the other factor? 1 How do you know? The coefﬁcient of the trinomial is 1. z 7. What is the sign of the second term of the other factor? negative z How do you know? The second terms of the two factors must have the same sign because the third term of the trinomial is positive. The second term of the ﬁrst factor is negative. z z 70 8. The product of 50 and is 3500. z z z z 70 9. The sum of 50 and is 120. x 2 70 . 10. The other factor is 11. What is the length of the rectangular field, in feet? x 2 70 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 32 Name 4-4 Class Date Practice Form G Factoring Quadratic Expressions Factor each expression. 1. x2 1 11x 1 28 (x 1 7)(x 1 4) 2. x2 1 11x 1 24 (x 1 8)(x 1 3) 3. s2 1 13s 1 42 (s 1 7)(s 1 6) 4. x2 2 10x 1 21 (x 2 7)(x 2 3) 5. y2 2 8y 1 15 ( y 2 5)( y 2 3) 6. x2 2 12x 1 32 (x 2 8)(x 2 4) 7. 2x2 1 9x 2 18 2(x 2 6)(x 2 3) 8. 2w2 1 12w 2 35 2(w 2 7)(w 2 5) 9. 2t2 2 3t 1 54 2(t 1 9)(t 2 6) 10. x2 2 7x 2 60 (x 2 12)(x 1 5) Find the GCF of each expression. Then factor the expression. 11. 6x2 2 9 3(2x2 2 3) 12. 16m2 1 8m 8m(2m 1 1) 13. 2a2 1 22a 1 60 2(a 1 6)(a 1 5) 14. 5x2 1 25x 2 70 5(x 1 7)(x 2 2) 1 1 15. 3x2 1 3x 2 4 31 (x 2 3)(x 1 4) 16. 27x2 1 7x 1 14 27(x 1 1)(x 2 2) Factor each expression. 17. 5x2 2 17x 1 6 (x 2 3)(5x 2 2) 18. 3x2 1 10x 1 8 (x 1 2)(3x 1 4) 19. 2b2 2 9b 2 5 (2b 1 1)(b 2 5) 20. z2 1 12z 1 36 (z 1 6)2 21. 9x2 2 6x 1 1 (3x 2 1)2 22. 4k2 1 12k 1 9 (2k 1 3)2 23. n2 2 49 (n 2 7)(n 1 7) 24. 2x2 2 50 2(x 2 5)(x 1 5) 25. The area of a rectangular field is x2 2 x 2 72 m2 . The length of the field is x 1 8 m. What is the width of the field in meters? x 2 9 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 33 Name 4-4 Class Date Practice (continued) Form G Factoring Quadratic Expressions 26. The product of two integers is w2 2 3w 2 40, where w is a whole number. Write expressions for each of the two integers in terms of w. w 2 8, w 1 5 27. John is j years old. The product of his younger brother’s and older sister’s ages is j2 2 2j 2 15. How old are John’s brother and sister in terms of John’s age? brother: j 2 5; sister: j 1 3 Factor each expression completely. 28. 2x2 1 9x 1 10 (2x 1 5)(x 1 2) 29. 6y2 2 5y 1 1 (2y 2 1)(3y 2 1) 30. 3x2 1 8x 2 3 (x 1 3)(3x 2 1) 31. 4x2 2 7x 2 15 (4x 1 5)(x 2 3) 32. 12t2 1 10t 2 12 2(2t 1 3)(3t 2 2) 33. 210x2 1 x 1 21 2(5x 1 7)(2x 2 3) 34. 24k2 1 2k 1 30 22(2k 1 5)(k 2 3) 1 1 35. 2x2 1 2x 2 10 12 (x 1 5)(x 2 4) 36. x2 2 16x 1 64 (x 2 8)2 37. m2 1 22m 1 121 (m 1 11)2 38. 16x2 2 40x 1 25 (4x 2 5)2 39. 36x2 1 12x 1 1 (6x 1 1)2 40. 22x2 2 32x 2 128 22(x 1 8)2 41. 225p2 1 30p 2 9 2(5p 2 3)2 42. r2 2 144 (r 1 12)(r 2 12) 1 1 1 43. 4x2 2 4 4(x 1 1)(x 2 1) 44. 27s2 1 175 27(s 1 5)(s 2 5) 1 1 45. 225z2 1 1 225(z 1 5)(z 2 5) 46. The radius of the outer circle in the illustration is R. The radius of the inner circle is r. a. Write an expression for the area of the outer circle. πR2 b. Write an expression for the area of the inner circle. πr2 c. Write an expression representing the area of the ring, the shaded region in the illustration. Do not simplify. πR2 2 πr2 d. Factor the expression in part (c). π(R 1 r)(R 2 r) Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 34 R r Name Class 4-4 Date Practice Form K Factoring Quadratic Expressions Factor each expression. 1. x2 1 4x 2 5 (x 2 1)(x 1 5) 2. x2 1 13x 1 42 (x 1 6)(x 1 7) 3. 2x2 2 x 1 12 2(x 1 4)(x 2 3) 4. x2 2 8x 1 16 (x 2 4)(x 2 4) 5. 2x2 1 16x 2 55 2(x 2 11)(x 2 5) 6. x2 1 2x 2 48 (x 2 6)(x 1 8) 7. 2y2 1 17y 2 72 2(y 2 8)(y 2 9) 8. x2 1 7x 1 12 (x 1 4)(x 1 3) 9. x2 2 8x 1 12 (x 2 2)(x 2 6) Find the GCF of each expression. Then factor the expression. 10. 3x2 1 15x 1 12 3(x 1 1)(x 1 4) 11. 29y2 1 6y 23y(3y 2 2) 12. 6x2 1 12x 2 48 6(x 2 2)(x 1 4) 13. 23x2 2 3x 1 60 23(x 2 4)(x 1 5) 14. 2x2 2 10x 2x(x 2 5) 15. 7x2 2 14x 2 56 7(x 1 2)(x 2 4) 16. 10x2 1 100x 10x(x 1 10) 17. 9x2 2 36x 1 27 9(x 2 3)(x 2 1) 18. 25xy2 2 30xy 2 25x 25x(y 1 1)(y 1 5) 19. Writing When you factor a quadratic expression, explain what it means when c , 0 and b . 0. When c R 0, one factor is positive and the other is negative and when b S 0, the factor with the greater absolute value is positive. 20. Error Analysis You factored 2x2 1 10x 2 24 as 2(x 2 6)(x 2 4). Your friend factored it as (x 1 12)(x 2 2). Which of you is correct? What mistake was made? You are correct; your friend forgot to factor out 21. 21. Multiple Choice What is the factored form of 214a2 1 42ab? D a(214a 1 42b) 7(22a2 1 6ab) 22a(7a 2 21b) 214a(a 2 3b) 22. Reasoning The area of a carpet is (x2 2 11x 1 28) ft2 . What are the length and the width of the carpet? (x 2 7) ft and (x 2 4) ft Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 35 Name Class 4-4 Date Practice (continued) Form K Factoring Quadratic Expressions Factor each expression. 23. 2x2 1 7x 1 6 (2x 1 3)(x 1 2) 24. 3x2 2 14x 2 24 (3x 1 4)(x 2 6) 25. 5x2 2 22x 1 21 (5x 2 7)(x 2 3) 26. 4x2 1 18x 1 8 2(x 1 4)(2x 1 1) 27. 2x2 2 8x 1 6 2(x 2 3)(x 2 1) 28. 6x2 1 13x 2 28 (3x 2 4)(2x 1 7) 29. 4x2 2 4x 1 1 30. x2 1 6x 1 9 (2x 2 1)2 (x 1 3)2 31. 4x2 2 16 4(x 2 2)(x 1 2) 32. 9x2 2 4 33. 16x2 2 40x 1 25 (3x 1 2)(3x 2 2) 35. 9x2 2 36x 1 36 9(x 2 2)2 (4x 2 5)2 34. x2 2 25 (x 2 5)(x 1 5) 36. 25x2 2 9 (5x 2 3)(5x 1 3) 37. 4x2 1 24x 1 36 4(x 1 3)2 38. Error Analysis Which of the following examples is factored correctly? Explain. Example 1 The product of two terms with different Example 1 Example 2 signs is negative, so the correct factorization 2 2 of 272 is 27 and 7, not 27 and 27. 4x 2 49 4x 2 49 (2x)2 2 72 (2x 2 7)(2x 1 7) (2x)2 2 72 (2x 2 7)(2x 2 7) 39. You can represent the area of a square tabletop with the expression 16x2 1 24x 1 9. What is the side length of the tabletop in terms of x? 4x 1 3 units Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 36 Name Class 4-4 Date Standardized Test Prep Factoring Quadratic Expressions Multiple Choice For Exercises 1−6, choose the correct letter. 1. What is the complete factorization of 2x2 1 x 2 15? B (x 2 5)(2x 1 3) (x 2 3)(2x 1 5) (x 1 3)(2x 2 5) (x 1 5)(2x 2 3) 2. What is the complete factorization of 2x2 1 3x 1 28? I (x 2 4)(x 2 7) 2(x 1 4)(x 1 7) 2(x 2 4)(x 1 7) 2(x 2 7)(x 1 4) 3. What is the complete factorization of 6x2 1 9x 2 6? A 3(2x 2 1)(x 1 2) 3(x 2 2)(2x 1 1) (3x 1 2)(2x 2 3) 3(x 2 2)(2x 2 1) 4. What is the complete factorization of 16x2 2 56x 1 49? G (4x 2 7)(4x 1 7) (4x 1 7)2 (4x 2 7)2 16(x 2 7)2 5. What is the complete factorization of 5x2 2 20? C (5x 2 4)(x 1 5) 5(x 1 2)(x 2 2) 5(x 1 4)(x 2 4) 5(x 2 2)2 6. What is the complete factorization of x2 2 14x 1 24? I (x 2 8)(x 2 3) (x 1 2)(x 2 12) (x 2 4)(x 2 6) (x 2 12)(x 2 2) Short Response 7. The area in square meters of a rectangular parking lot is x2 2 95x 1 2100. The width in meters is x 2 60. What is the length of the parking lot in meters? Show your work. [2] x 2 35 [1] correct method but an error in ﬁnding the length OR correct length but no work shown [0] incorrect answer and no work shown OR no answer given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 37 Name Class 4-4 Date Enrichment Factoring Quadratic Expressions To factor a quadratic expression of the form ax2 1 bx 1 c, break the middle term of the expression into two terms and use common factors to complete the factoring. 2x2 2 3x 2 5 2x2 1 2x 2 5x 2 5 Rewrite 23x as 2x 2 5x. 2x(x 1 1) 2 5(x 1 1) Factor the ﬁrst two terms. Then factor the third and fourth terms. (2x 2 5)(x 1 1) Rewrite the expression using the Distributive Property. This same method can be used to factor polynomials with more than three terms. 1. Rewrite x3 1 3x2 1 4x 1 12 by finding a common factor for the first two terms and another for the last two terms. x2(x 1 3) 1 4(x 1 3) 2. Factor x3 1 3x2 1 4x 1 12 using the Distributive Property. (x2 1 4)(x 1 3) Use this method to factor the polynomials below. 3. 3x2 1 xy 2 12x 2 4y (x 2 4)(3x 1 y) 4. a3 2 2a2 1 5a 2 10 (a2 1 5)(a 2 2) 5. x4 1 2x3 2 2x 2 4 (x3 2 2)(x 1 2) 6. b3 1 3b2 2 2b 2 6 (b2 2 2)(b 1 3) 7. m3 1 4m2 2 9m 2 36 (m 1 4)(m 2 3)(m 1 3) 8. c4 2 c2d2 1 c2d 2 d3 (c2 1 d )(c 2 d )(c 1 d ) Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 38 Name Class 4-4 Date Reteaching Factoring Quadratic Expressions Problem What is 6x2 2 5x 2 4 in factored form? a 5 6, b 5 25, and c 5 24 Find a, b, and c; they are the coefﬁcients of each term. ac 5 224 and b 5 25 We are looking for factors with product ac and sum b. Factors of 224 1, 224 21, 24 2, 212 22, 12 3, 28 Sum of factors 223 23 210 10 25 23, 8 4, 26 24, 6 5 22 2 The factors 3 and 28 are the combination whose sum is 25. 6x2 1 3x 2 8x 2 4 Rewrite the middle term using the factors you found. 3x(2x 1 1) 2 4(2x 1 1) Find common factors by grouping the terms in pairs. (3x 2 4)(2x 1 1) Rewrite using the Distributive Property. 3 Check 3 (3x 2 4)(2x 1 1) You can check your answer by multiplying the factors together. 6x2 1 3x 2 8x 2 4 6x2 2 5x 2 4 Remember that not all quadratic expressions are factorable. Exercises Factor each expression. 1. x2 1 6x 1 8 (x 1 4)(x 1 2) 2. x2 2 4x 1 3 (x 2 3)(x 2 1) 3. 2x2 2 6x 1 4 2(x 2 2)(x 2 1) 4. 2x2 2 11x 1 5 (2x 2 1)(x 2 5) 5. 2x2 2 7x 2 4 (2x 1 1)(x 2 4) 6. 4x2 1 16x 1 15 (2x 1 5)(2x 1 3) 7. x2 2 5x 2 14 (x 1 2)(x 2 7) 8. 7x2 2 19x 2 6 (7x 1 2)(x 2 3) 9. x2 2 x 2 72 (x 2 9)(x 1 8) 10. 2x2 1 9x 1 7 (2x 1 7)(x 1 1) 11. x2 1 12x 1 32 (x 1 4)(x 1 8) 12. 4x2 2 28x 1 49 (2x 2 7)(2x 2 7) 13. x2 2 3x 2 10 (x 2 5)(x 1 2) 14. 2x2 1 9x 1 4 (2x 1 1)(x 1 4) 15. 9x2 2 6x 1 1 (3x 2 1)(3x 2 1) 16. x2 2 10x 1 9 (x 2 1)(x 2 9) 17. x2 1 4x 2 12 (x 1 6)(x 2 2) 18. x2 1 7x 1 10 (x 1 5)(x 1 2) 19. x2 2 8x 1 12 (x 2 6)(x 2 2) 20. 2x2 2 5x 2 3 (2x 1 1)(x 2 3) 21. x2 2 6x 1 5 (x 2 1)(x 2 5) 22. 3x2 1 2x 2 8 (3x 2 4)(x 1 2) Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 39 Name Class 4-4 Date Reteaching (continued) Factoring Quadratic Expressions • a2 1 2ab 1 b2 5 (a 1 b)2 Factoring perfect square trinomials a2 2 2ab 1 b2 5 (a 2 b)2 • a2 2 b2 5 (a 1 b)(a 2 b) Factoring a difference of two squares Problem What is 25x2 2 20x 1 4 in factored form? There are three terms. Therefore, the expression may be a perfect square trinomial. a2 5 25x2 and b2 5 4 Find a2 and b2. a 5 5x and b 5 2 Take square roots to ﬁnd a and b. Check that the choice of a and b gives the correct middle term. 2ab 5 2 ? 5x ? 2 5 20x Write the factored form. a2 2 2ab 1 b2 5 (a 2 b)2 25x2 2 20x 1 4 5 (5x 2 2)2 (5x 2 2)2 Check (5x 2 2)(5x 2 2) 25x2 2 10x 2 10x 1 4 25x2 2 20x 1 4 You can check your answer by multiplying the factors together. Rewrite the square in expanded form. Distribute. Simplify. Exercises Factor each expression. 23. x2 2 12x 1 36 24. x2 1 30x 1 225 (x 2 6)2 25. 9x2 2 12x 1 4 (x 1 15)2 (3x 2 2)2 26. x2 2 64 (x 1 8)(x 2 8) 27. 9x2 2 42x 1 49 29. 27x2 2 12 3(3x 1 2)(3x 2 2) 30. 49x2 1 42x 1 9 32. 9x2 2 16 (3x 1 4)(3x 2 4) 33. 8x2 2 18 2(2x 1 3)(2x 2 3) 34. 81x2 1 126x 1 49 35. 125x2 2 100x 1 20 36. 2x2 1 196 37. 216x2 2 24x 2 9 5(5x 2 2)2 (3x 2 7)2 28. 25x2 2 1 (5x 1 1)(5x 2 1) 31. 16x2 2 32x 1 16 (7x 1 3)2 16(x 2 1)2 (9x 1 7)2 2(x 1 14)(x 2 14) 2(4x 1 3)2 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 40 Name Class 4-5 Date Additional Vocabulary Support Quadratic Equations Problem What are the solutions of the quadratic equation 2x2 2 4x 5 6? Explain Work Justify 2x2 2 4x 5 6 First, write the equation. Second, subtract 6 from each side to set equal to 0. 2x2 2 4x 2 6 5 0 Next, factor out the GCF, 2. 2(x2 2 2x 2 3) 5 0 Then, factor the trinomial. 2(x 2 3)(x 1 1) 5 0 Then, use the Zero-Product Property. Finally, solve for x. x 2 3 5 0 or x 1 1 5 0 x 5 3 or x 5 21 Original equation Subtraction Property of Equality Distributive Property Factor the quadratic expression Zero-Product Property Addition Property of Equality Additio Solution 3 or 21 Exercise What are the solutions of the quadratic equation 3x2 2 6x 5 23? Explain Work First, write the equation. Justify 3x2 2 6x 5 23 Original equation Second, add 3 to each side to set equal to 0. 3x2 2 6x 1 3 5 0 Addition Property of Equality Next, factor out the GCF, 3. 3(x2 2 2x 1 1) 5 0 Distributive Property Then, factor the trinomial. 3(x 2 1)(x 2 1) 5 0 Then, use the Zero-Product Property. x 2 1 5 0 or x 2 1 5 0 Factor the quadratic expression Zero-Product Property Finally, solve for x. Addition Property of Equality x51 Solution 1 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 41 Name 4-5 Class Date Think About a Plan Quadratic Equations Landscaping Suppose you have an outdoor pool measuring 25 ft by 10 ft. You want to add a cement walkway around the pool. If the walkway will be 1 ft thick and you have 304 ft3 of cement, how wide should the walkway be? Understanding the Problem x x 25 ft walkway in feet. 2. If you lay the pieces of walkway end to end, what is the total length of the walkway? 4x 1 70 3. What is the thickness of the walkway? 1 ft 4. What is the problem asking you to determine? The width of a walkway that is (4x 1 70) ft long and 1 ft thick, that has volume 304 ft3. Planning the Solution 5. Write a quadratic equation to model the volume of the walkway. V 5 4x2 1 70x 6. What method can you use to find the solutions of your quadratic equation? Answers may vary. Sample: Substitute 304 for V in the equation. Subtract 304 from each side to set it equal to zero. Graph y 5 4x2 1 70x 2 304 on a graphing calculator and use the ZERO option in the CALC feature to solve for x. Getting an Answer 7. How many solutions of your quadratic equation do you need to find? Explain. One; only one solution is positive, and a negative solution does not make sense. 8. How wide should the walkway be? about 3.6 ft Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 42 x 10 ft 1. Draw a diagram of the pool and the walkway. Let x 5 the width of the x Name Class 4-5 Date Practice Form G Quadratic Equations Solve each equation by factoring. Check your answers. 1. x2 2 2x 2 24 5 0 24, 6 2. 3x2 5 x 1 4 21, 43 3. x2 2 6x 1 9 5 0 3 4. 3x2 1 45 5 24x 3, 5 3 5. 4x2 1 6x 5 0 22, 0 6. 7x2 5 21x 0, 3 7. (x 1 2)2 5 49 29, 5 8. x 1 3 5 24x2 21, 3 3 8 Solve each equation using tables. Give each answer to at most two decimal places. 9. 5x2 1 7x 2 6 5 0 22, 0.6 10. x2 2 2x 5 1 20.41, 2.41 11. 2x2 2 x 5 5 21.35, 1.85 12. x2 2 4x 1 2 5 0 0.59, 3.41 13. 3x2 1 7x 5 1 22.47, 0.14 14. 2x2 2 3x 5 15 22.09, 3.59 Solve each equation by graphing. Give each answer to at most two decimal places. 15. 10x2 5 4 2 3x 20.8, 0.5 16. 3x2 1 2x 5 2 21.22, 0.55 17. 4x2 2 x 5 6 21.11, 1.36 18. 4x2 1 3x 5 6 2 2x 22, 0.75 19. x2 1 4 5 6x 0.76, 5.24 1 20. 5 2 x 5 2x2 24.32, 2.32 21. A woman drops a front door key to her husband from their apartment window several stories above the ground. The function h 5 216t2 1 64 gives the height h of the key in feet, t seconds after she releases it. a. How long does it take the key to reach the ground? 2 s b. What are the reasonable domain and range for the function h? 0 K t K 2; 0 K h K 64 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 43 Name 4-5 Class Date Practice (continued) Form G Quadratic Equations 22. The function C 5 75x 1 2600 gives the cost, in dollars, for a small company to manufacture x items. The function R 5 225x 2 x2 gives the revenue, also in dollars, for selling x items. How many items should the company produce so that the cost and revenue are equal? 20 or 130 23. The function a 5 2.4t 2 0.1t2 gives the amount a, in micromilligrams (mmg), of a drug in a patient’s bloodstream t hours after being ingested in tablet form. When is the amount of the drug equal to 8 mmg? (Hint: Multiply the equation you write by 10 before solving.) 4 h; 20 h 24. You use a rectangular piece of cardboard measuring 20 in. by 30 in. to x 30 ft construct a box. You cut squares with sides x in. from each corner of the piece of cardboard and then fold up the sides to form the bottom. a. Write a function A representing the area of the base of the box in terms of x. A 5 (30 2 2x)(20 2 2x) 5 4x2 2 100x 1 600 b. What is a reasonable domain for the function A? 0 R x R 10 c. Write an equation if the area of the base must be 416 in.2. 4x2 2 100x 1 600 5 416 d. Solve the equation in part (c) for values of x in the reasonable domain. 2 e. What are the dimensions of the base of the box? 26 in. by 16 in. Solve each equation by factoring, using tables, or by graphing. If necessary, round your answer to the nearest hundredth. 25. 9x2 5 49 2 7, 7 3 3 2 27. 4x 1 1 5 8x 0.13, 1.87 26. x2 1 10x 1 17 5 0 27.83, 22.17 29. 4(x2 2 x) 5 19 21.74, 2.74 30. 25x2 1 20x 1 4 5 0 2 25 31. 3x2 5 4x 1 32 2 8, 4 3 32. x2 2 5x 2 12 5 0 21.77, 6.77 28. 5x2 2 2x 2 7 5 0 21, 75 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 44 x x 20 ft x Name 4-5 Class Date Practice Form K Quadratic Equations Solve each equation by factoring. Check your answers. To start, factor the quadratic expression. 1. x2 2 x 2 30 5 0 x 5 25, x 5 6 2. x2 2 10x 5 221 x 5 3, x 5 7 3. x2 5 210x 2 9 x 5 21, x 5 29 4. x2 2 5x 5 0 x 5 0, x 5 5 5. 10x 2 24 5 x2 x 5 4, x 5 6 6. x2 5 212x x 5 0, x 5 212 Solve each equation using tables. Give each answer to at most two decimal places. To start, enter the equation as Y1. Make a table and look for where the y-values change sign. 7. x2 1 x 5 12 x 5 3, x 5 24 8. 10x2 1 26x 1 16 5 0 x 5 21, x 5 21.6 10. 2x2 2 13x 1 18 5 0 x 5 2, x 5 4.5 11. 2x2 5 10x x 5 5, x 5 0 9. 2x2 1 11x 5 6 x 5 26, x 5 0.5 12. 0.5x2 2 8 5 0 x 5 24, x 5 4 Write a quadratic equation with the given solutions. 13. 4 and 25 x2 1 x 2 20 5 0 14. 26 and 0 x2 1 6x 5 0 15. 3 and 8 x2 2 11x 1 24 5 0 16. Writing Explain when you would prefer to use factoring to solve a quadratic equation and when you would prefer to use tables. When a 5 1 and c factors easily into integers, it is better to solve by factoring. When c doesn’t factor easily, it is better to solve using tables. 17. A parabolic jogging path intersects both ends of a street. The path has the equation x2 2 25x 5 0. If one end of the street is considered to be x 5 0 and the street lies on the x-axis, where else does the path intersect the street? at x 5 25 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 45 Name Class 4-5 Date Practice (continued) Form K Quadratic Equations Solve each equation by graphing. Give each answer to at most two decimal places. 18. 2x2 2 x 2 10 5 0 x 5 22 and x 5 2.5 21. 4x2 2 5x 2 26 5 0 x 5 3.25 and x 5 22 19. 6x2 2 13x 5 28 x 5 21.33 and x 5 3.5 22. 6x2 2 23x 5 18 x 5 4.5 and x 5 20.67 20. 4x2 1 27x 5 12 x N 0.42 and x N 27.17 23. 4x2 2 9x 1 5 5 0 x 5 1.25 and x 5 1 24. The students in Mr. Wilson’s Physics class are making golf ball catapults. The flight of group A’s ball is modeled by the equation y 5 20.014x2 1 0.68x, where x is the ball’s distance from the catapult. The units are in feet. a. How far did the ball fly? about 48.6 feet b. How high above the ground did the ball fly? about 8.3 feet c. What is a reasonable domain and range for this function? Answers may vary. Sample: domain: 0 K x K 48.6; range: 0 K y K 8.3 25. A rectangular pool is 20 ft wide and 50 ft long. The pool is surrounded by a walkway. The walkway is the same width all the way around the pool. The total area of the walkway is 456 square ft. How wide is the walkway? 3 ft 26. Reasoning The equation used to solve Exercise 25 has two solutions. Why is only one solution used to answer the question? One of the solutions is negative, and the walkway cannot have a negative width. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 46 Name Class Date Standardized Test Prep 4-5 Quadratic Equations Gridded Response Solve each exercise and enter your answer in the grid provided. 1. What is the positive solution of the equation x2 5 2x 1 35? Solve by factoring. 2. What is the positive solution of the equation 5x2 1 2x 2 16 5 0? Solve by factoring. 3. What is the positive solution of the equation x2 2 3x 5 1? Solve by using a table or by graphing. If necessary, round your answer to the nearest hundredth. 4. What is the positive solution of the equation 3x2 2 5x 2 7 5 0? Solve by using a table or by graphing. If necessary, round your answer to the nearest hundredth. 1 5. What is the positive solution of the equation 2x2 2 3x 5 5? Solve by using a table or by graphing. If necessary, round your answer to the nearest hundredth. Answers 1. – 2. 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 – 1 .6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 3. 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 – 3 . 30 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 4. 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 – 2 .57 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 5. 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 47 – 7 .3 6 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Name Class 4-5 Date Enrichment Quadratic Equations Fencing Costs Solve each of the following problems by using a quadratic equation. 1. The area of a rectangular field is 1200 m2 . Two parallel sides are fenced with aluminum at $20/m. The remaining two sides are fenced with steel at $10/m. The total cost of the fencing is $2200. a. What is the length of each side fenced with aluminum? 40 m, or 15 m b. What is the length of each side fenced with steel? 30 m, or 80 m, respectively 2. The perimeter of a rectangular field is 140 yards. The land sells for $10>yd2 and the total cost of the land is $12,000. What are the dimensions of the field? 30 yd by 40 yd 3. The area of a rectangular field is 875 m2. Two adjacent sides of the field are fenced with wood costing $5/m. The remaining two sides are fenced with steel costing $10/m. The total cost of the fencing is $900. What are the dimensions of the field? 25 m by 35 m 4. The area of a field shaped like a right triangle is 600 m2. The legs of the field are fenced with steel at $10/m, while the hypotenuse is fenced with aluminum at $20/m. The perimeter of the field is 120 m. The total cost of the fencing is $1700. a. What is the length of each side fenced with steel? 30 m, 40 m b. What is the length of the side fenced with aluminum? 50 m 5. The area of a field shaped like a right triangle is 750 yd2. One leg of the field is fenced with wood costing $5/yd. The remainder of the perimeter of the field is fenced with steel costing $10/yd. The perimeter of the field is 150 yd. The total cost of the fencing is $1200. a. What is the length of the leg fenced with wood? 60 yd b. What is the length of the leg fenced with steel? 25 yd 6. The area of a rectangular field is 1000 yd2 . Two parallel sides are fenced with aluminum at $15/yd. One of the remaining sides is fenced with steel at $10/yd, and all but 10 yd of the remaining side is fenced with wood costing $5/yd. The remaining 10 yd are left unfenced. The total cost of the fencing is $1525. a. What is the length of the side fenced with steel? 25 yd or 80 yd b. What is the length of each side fenced with aluminum? 40 yd, or 1212 yd, respectively. 7. Two rectangular fields with identical shapes and areas are to be fenced side by side. The total area enclosed is 1200 yd2 . The shared side is fenced with wood costing $5/yd. The remaining perimeter of the two joined fields is fenced with aluminum at $15/yd. The total cost of the fencing is $2250. a. What is the length of the shared side? 30 yd, or 3427 yd b. What is the length of each side fenced with aluminum? 30 yd and 20 yd, or 3427 yd and 1712 yd, respectively Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 48 Name Class Date Reteaching 4-5 Quadratic Equations There are several ways to solve quadratic equations. If you can factor the quadratic expression in a quadratic equation written in standard form, you can use the Zero-Product Property. If ab 5 0 then a 5 0 or b 5 0. Problem What are the solutions of the quadratic equation 2x2 1 x 5 15? 2x2 1 x 5 15 2x2 1 x 2 15 5 0 (2x 2 5)(x 1 3) 5 0 2x 2 5 5 0 or x 1 3 5 0 2x 5 5 or x 5 23 5 x52 or Write the equation. Rewrite in standard form, ax2 1 bx 1 c 5 0. Factor the quadratic expression (the nonzero side). Use the Zero-Product Property. Solve for x. x 5 23 Check the solutions: 2 5 5 5 x 5 2: 2 Q 2 R 1 Q 2 R 0 15 25 5 2 1 2 0 15 x 5 23: 2(23)2 1 (23) 0 15 15 5 15 15 5 15 18 2 3 0 15 Both solutions check. The solutions are x 5 52 and x 5 23. Exercises Solve each equation by factoring. Check your answers. 1. x2 2 10x 1 16 5 0 2, 8 2. x2 1 2x 5 63 29, 7 3. x2 1 9x 5 22 211, 2 4. x2 2 24x 1 144 5 0 12 5. 2x2 5 7x 1 4 2 12, 4 6. 2x2 5 25x 1 12 24, 32 7. x2 2 7x 5 212 3, 4 8. 2x2 1 10x 5 0 25, 0 9. x2 1 x 5 2 22, 1 11. x2 5 25x 2 6 23, 22 12. x2 1 x 5 20 25, 4 10. 3x2 2 5x 1 2 5 0 23, 1 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 49 Name Class 4-5 Date Reteaching (continued) Quadratic Equations Some quadratic equations are difficult or impossible to solve by factoring. You can use a graphing calculator to find the points where the graph of a function intersects the x-axis. At these points f (x) 5 0, so x is a zero of the function. The values r1 and r2 are the zeros of the function y 5 (x 2 r1)(x 2 r2). The graph of the function intersects the x-axis at x 5 r1 , or (r1, 0), and x 5 r2 , or (r2, 0). Problem What are the solutions of the quadratic equation 3x2 5 2x 1 7? Step 1 Rewrite the equation in standard form, ax2 1 bx 1 c 5 0. 3x2 2 2x 2 7 5 0 Step 2 Enter the equation as Y1 in your calculator. Step 3 Graph Y1. Choose the standard window and see if the zeros of the function Y1 are visible on the screen. If they are not visible, zoom out and determine a better viewing window. In this case, the zeros are visible in the standard window. Step 4 Use the ZERO option in the CALC feature. For the first zero, choose bounds of 22 and 21 and a guess of 21.5. The screen display gives the first zero as x 5 21.230139. Plot1 Plot2 Plot3 \Y153X2 2 2X 2 7 \Y25 \Y35 \Y45 \Y55 \Y65 \Y75 Zero X 5 21.230139 Y 5 0 Similarly, the screen display gives the second zero as x 5 1.8968053. The solutions to two decimal places are x 5 21.23 and x 5 1.90. Zero X 5 1.8968053 Y 5 0 Exercises Solve the equation by graphing. Give each answer to at most two decimal places. 13. x2 5 5 22.24, 2.24 14. x2 5 5x 1 1 20.19, 5.19 15. x2 1 7x 5 3 27.41, 0.41 16. x2 1 x 5 5 22.79, 1.79 17. x2 1 3x 1 1 5 0 22.62, –0.38 18. x2 5 2x 1 4 21.24, 3.24 19. 3x2 2 5x 1 9 5 8 0.23, 1.43 20. 4 5 2x2 1 3x 22.35, 0.85 21. x2 2 6x 5 27 1.59, 4.41 22. 2x2 5 8x 1 8 26.83, 21.17 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 50 Name Class 4-6 Date Additional Vocabulary Support Completing the Square Problem Solve 2x2 1 12x 2 2 5 0 by completing the square. Justify your steps. 2x2 1 12x 2 2 5 0 Write the original equation. 2x2 1 12x 5 2 Rewrite so the variable terms are on one side of the equation and the constants are on the other side. 2x2 12x 2 2 1 2 52 Divide each side by 2 so the coefﬁcient of x2 will be 1. x2 1 6x 5 1 Simplify. 2 6 2 b 2 Q 2 R 5 Q 2 R 5 (3) 5 9 2 Find Q b2 R . x2 1 6x 1 9 5 1 1 9 Add 9 to both sides. (x 1 3)2 5 10 Factor the trinomial. x 1 3 5 4 !10 x 5 23 4 !10 Find square roots. Solve for x. Exercise Solve 3x2 2 24x 2 9 5 0 by completing the square. Justify your steps. Write the original equation 3x2 2 24x 2 9 5 0 . Rewrite so the variable terms are on one side of the 3x2 2 24x 5 9 equation and the constants are on the other side 3x2 24x 9 3 2 3 53 . Divide each side by 3 so the coefﬁcient of x2 will be 1 . Simplify x2 2 8x 5 3 . b 2 Find Q 2 R . x2 2 8x 1 16 5 3 1 16 Add 16 to both sides . (x 2 4)2 5 19 Factor the trinomial . Find square roots . Solve for x . b 2 28 2 Q 2 R 5 Q 2 R 5 (24)2 5 16 x 2 4 5 4 !19 x 5 4 4 !19 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 51 Name Class 4-6 Date Think About a Plan Completing the Square Geometry The table shows some possible dimensions of rectangles Width Length with a perimeter of 100 units. Copy and complete the table. 1 49 a. Plot the points (width, area). Find a model for the data set. 2 48 b. What is another point in the data set? Use it to verify your 47 3 model. 46 4 c. What is a reasonable domain for this function? Explain. 45 5 d. Find the maximum possible area. What are its dimensions? e. Find an equation for area in terms of width without using y the table. Do you get the same equation as in 200 part (a)? Explain. Area 49 96 141 184 225 150 100 1. What points should you plot? Plot the points on the graph. 50 (1, 49), (2, 96), (3, 141), (4, 184), (5, 225) x O 1 2 3 4 5 2. Use your graphing calculator to find a model for the data set. A 5 2x2 1 50x 3. What is another point in the data set? Use it to verify your model. ? Answers may vary. Sample: (6, 264); 264 5 2(6)2 1 50(6); 264 5 264 4. What does the domain of your function represent? the width of the rectangle 5. The domain must be greater than 0 and less than 50 . 6. A reasonable domain is: all real numbers greater than 0 and less than 50 7. Write the vertex form of your function. A 5 2(x 2 25)2 1 625 8. The maximum possible area is 625 units2 . The dimensions of this rectangle are 25 units 9. If the width of the rectangle is x, then the length is Area 5 length times width 5 by . 25 units 50 2 x (50 2 x) ? . x 5 2x2 1 50x 10. Is the equation in Exercise 9 the same as your model in Exercise 2? Explain. Yes; the functions are the same. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 52 Name 4-6 Class Date Practice Form G Completing the Square Solve each equation by finding square roots. 1. 3x2 5 75 w5 2. 5x2 2 45 5 0 w3 3. 4x2 2 49 5 0 w72 4. 6x2 5 216 w6 5. 2x2 5 14 w!7 6. 3x2 2 96 5 0 w4 !2 7. A box is 4 in. high. Its length is 1.5 times its width. The volume of the box is 1350 in.2. What are the width and length of the box? 15 in.; 22.5 in. Solve each equation. 8. x2 1 12x 1 36 5 25 211, 21 9. x2 2 10x 1 25 5 144 27, 17 49 13 1 10. x2 1 6x 1 9 5 4 2 2 , 2 11. x2 2 22x 1 121 5 225 24, 26 12. 16x2 1 8x 1 1 5 16 254, 34 13. 25x2 2 30x 1 9 5 81 265, 12 5 Complete the square. 14. x2 1 22x 1 121 15. x2 2 30x 1 225 16. x2 1 5x 1 25 4 1 1 17. x2 2 2 x 1 16 18. 25x2 1 10x 1 1 19. 4x2 2 12x 1 9 Solve each quadratic equation by completing the square. 20. x2 1 10x 2 1 5 0 25 6 !26 21. x2 1 2x 2 7 5 0 21 6 2 !2 22. 2x2 1 6x 1 10 5 0 3 6 !19 23. x2 1 5x 5 3x 1 11 21 6 2 !3 24. 3x2 1 4x 5 2x2 1 3 22 6 !7 2 6 !7 3 25. x2 2 2x 2 4 5 0 2 26. 20.2x2 1 0.4x 1 0.8 5 0 1 6 !5 27. 4x2 1 20x 1 1 5 0 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 53 25 6 2 !6 2 Name Class 4-6 Date Practice (continued) Form G Completing the Square Rewrite each equation in vertex form. 28. y 5 x2 2 6x 1 4 y 5 (x 2 3)2 2 5 29. y 5 x2 1 14x 1 50 y 5 (x 1 7)2 1 1 30. y 5 3x2 1 8x 1 2 y 5 3 Q x 1 43 R 2 2 10 3 31. y 5 22x2 1 6x 2 2 y 5 22 Q x 2 32 R 2 1 52 Find the value of k that would make the left side of each equation a perfect square trinomial. 32. x2 1 kx 1 196 5 0 28 33. 64x2 2 kx 1 1 5 0 16 34. x2 2 kx 1 16 5 0 8 35. 4x2 2 kx 1 9 5 0 12 36. 16x2 1 kx 1 9 5 0 24 1 1 37. 4x2 2 kx 1 25 5 0 15 38. The quadratic function d 5 2t2 1 4t 1 33 models the depth of water in a flood channel after a rainstorm. The time in hours after it stops raining is t and d is the depth of the water in feet. a. Solve the equation 2t2 1 4t 1 33 5 0. 2 6 !37 b. Approximate the positive solution found in part (a) to two decimal places. 8.08 c. Interpret the answer to part (b) in terms of the problem. The water level will be 0 ft after about 8 h. 39. While in orbit, a space scientist measures the pressure inside a container as it is being heated and then cooled. She records the information and discovers the pressure p, in pounds per square inch, is related to the time t in minutes after the experiment begins according to the equation p 5 20.2t2 1 1.6t . a. Complete the square in the expression 20.2t2 1 1.6t . 20.2t2 1 1.6t 2 3.2 b. Rewrite the equation for p in vertex form. p 5 20.2(t 2 4)2 1 3.2 c. What is a reasonable domain for this function? Explain. 0 K t K 8; other values of t result in d. When does the maximum pressure occur? What is the maximum pressure? negative values for p. 4 min; 3.2 lbs/in.2 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 54 Name Class 4-6 Date Practice Form K Completing the Square Solve each equation by finding square roots. To start, remember to isolate x2. 1. x2 2 9 5 0 2. x2 1 4 5 20 3. x2 1 15 5 16 x2 5 9 x2 5 16 x2 5 1 x 5 w3 x 5 w4 x 5 w1 4. 2x2 2 64 5 0 x 5 w4 !2 5. 4x2 2 100 5 0 x 5 w5 6. 5x2 2 25 5 0 x 5 w!5 7. You are painting a large wall mural. The wall length is 3 times the height. The area of the wall is 300 ft2. a. What are the dimensions of the wall? height 5 10 ft; length 5 30 ft b. If each can of paint covers 22 ft2, will 12 cans be enough to cover the wall? No; 12 cans will only cover 264 ft2. 8. The lengths of the sides of a carpet have the ratio of 4.4 to 1. The area of the carpet is 1154.7 ft2. What are the dimensions of the carpet? 16.2 ft, 71.3 ft 9. A packing box is 4 ft deep. One side of the box is 1.5 times longer than the other. The volume of the box is 24 ft3. What are the dimensions of the box? 2 ft, 3 ft Solve each equation. To start, factor the perfect square trinomial. 10. x2 2 14x 1 49 5 81 11. x2 1 6x 1 9 5 1 12. 9x2 2 12x 1 4 5 49 (x 2 7)2 5 81 (x 1 3)2 5 1 (3x 2 2)2 5 49 x 5 16, x 5 22 x 5 22, x 5 24 x 5 3, x 5 253 13. 4x2 1 36x 1 81 5 16 5 x 5 213 2 , x 5 22 14. x2 1 2x 1 1 5 36 x 5 27, x 5 5 15. x2 2 16x 1 64 5 9 x 5 11, x 5 5 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 55 Name Class 4-6 Date Practice (continued) Form K Completing the Square Complete the following squares. z 16. x2 1 8x 1 16 z z 100 z 18. x2 2 14x 1 z 49 z z 289 z 21. x2 2 46x 1 z 529 z 20 2 Q2R 5 8 2 Q 2 R 5 42 5 16 19. x2 2 24x 1 z 17. x2 1 20x 1 144 z 20. x2 1 34x 1 Solve the following equations by completing the square. 22. x2 2 8x 2 5 5 0 x2 2 8x 5 5 x2 2 8x 1 16 5 5 1 16 (x 2 4)2 5 21 23. x2 1 12x 1 9 5 0 x2 1 12x 5 29 x2 1 12x 1 36 5 29 1 36 24. x2 2 10x 5 211 x 5 5 6 "14 x 5 26 6 3 !3 x 2 4 5 4 !21 z 21 x 5 4 6 " z 25. 2x2 1 11x 2 23 5 2x 1 3 26. x2 2 18x 1 64 5 0 x 5 23 6 "22 x 5 9 6 "17 27. 3x2 2 42x 1 78 5 0 x 5 7 6 "23 Write the following equations in vertex form. 28. y 5 x2 1 10x 2 9 y 5 (x 1 5)2 2 34 29. y 5 x2 2 18x 1 13 y 5 (x 2 9)2 2 68 30. y 5 x2 1 32x 2 8 y 5 (x 1 16)2 2 264 Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 56 Name Class 4-6 Date Standardized Test Prep Completing the Square Multiple Choice For Exercises 1−6, choose the correct letter. 1. What are the solutions of the equation 36x2 2 12x 1 1 5 4? D 216 , 16 4, 8 212 , 16 216 , 12 2. What are the solutions of the equation 2x2 1 16x 1 28 5 0? G 24 4 !30 24 4 !2 4 4 !2 4 4 !30 9 24 3. Which value completes the square for x2 2 3x? A 9 4 3 2 9 49 4. Which value for k would make the left side of x2 1 kx 1 64 5 0 a perfect square trinomial? H 7 2 7 7 4 7 8 5. What are the solutions of the equation x2 5 8x 2 1? C 24 4 !17 24 4 !15 4 4 !15 4 4 !17 6. Which equation is the vertex form of y 5 23x2 1 12x 2 7? G y 5 23(x 2 2)2 2 5 y 5 23(x 1 2)2 2 5 y 5 23(x 2 2)2 1 5 y 5 23(x 1 2)2 1 5 Short Response 7. The equation p 5 2x2 1 8x 1 5 gives the price p, in dollars, for a product when x million units are produced. a. What are the solutions of the equation 2x2 1 8x 1 5 5 0? b. What is the positive solution to part (a) rounded to two decimal places? What does this solution mean in terms of this problem? [2] a. 4 6 !21 b. 8.58; the price will be $0 when 8.58 million units are produced. [1] incorrect solution OR incorrect interpretation [0] no answers given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 57 Name Class 4-6 Date Enrichment Completing the Square You can approximate square roots with an interesting quadratic relationship. 2 b 2 1. Multiply Q a 1 2a R . a2 1 b 1 b 2 4a 2. What happens to the value of a fraction as the denominator gets larger? As the denominator gets larger, the fraction becomes smaller. 2 If a is much larger than b, the value of the fraction b 2 is very small and has little effect in the expression write a2 4a b2 1 b 1 2 . So when a is much larger than b, you can 4a b 2 Q a 1 2a R < a2 1 b b a 1 2a < "a2 1 b If you can write a number as a2 1 b where a . b, then an approximate value of b its square root is a 1 2a . For example, to approximate !17 , let a 5 4 and b 5 1. b 1 < 4.125 !17 5 "42 1 1 so !17 < a 1 2a 5 4 1 2(4) Evaluate this square root on a calculator. !17 < 4.123 Use the formula to approximate each square root. Then find each square root using your calculator. Round to the nearest thousandth. 3. !26 5.100; 5.099 4. !404 20.100; 20.100 5. !174 13.192; 13.191 6. !1773 42.107; 42.107 7. !963 31.032; 31.032 8. !83 9.111; 9.110 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 58 Name Class Date Reteaching 4-6 Completing the Square Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a binomial. 2 2 Start with the expression x2 1 bx. Add Q b2 R . Now the expression is x2 1 bx 1 Q b2 R , 2 2 which can be factored into the square of a binomial: x2 1 bx 1 Q b2 R 5 Q x 1 b2 R . To complete the square for an expression ax2 1 abx, first factor out a. Then find the value that completes the square for the factored expression. Problem What value completes the square for 22x2 1 10x? Think Write Write the expression in the form a(x2 1 bx). 22x2 1 10x 5 22(x2 2 5x) Find b2 . 5 b 25 2 5 2 5 22 2 2 Add Q b2 R to the inner expression to complete the square. 22 cx2 2 5x 1 a22 b d 5 22ax2 2 5x 1 4 b Factor the perfect square trinomial. 22 Q x 2 2 R Find the value that completes the square. 22 Q 4 R 5 2 2 5 5 2 25 25 Exercises What value completes the square for each expression? 1. x2 1 2x 1 2. x2 2 24x 144 4. x2 2 20x 100 5. x2 1 5x 7. 2x2 2 24x 72 8. 3x2 1 12x 12 10. 5x2 1 80x 320 11. 27x2 1 14x 27 25 4 3. x2 1 12x 36 81 6. x2 2 9x 4 9. 2x2 1 6x 29 75 12. 23x2 2 15x 2 4 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 59 25 Name Class 4-6 Date Reteaching (continued) Completing the Square You can easily graph a quadratic function if you first write it in vertex form. Complete the square to change a function in standard form into a function in vertex form. Problem What is y 5 x2 2 6x 1 14 in vertex form? Think Write Write an expression using the terms that contain x. x2 2 6x Find b2 . 26 b 2 5 2 5 23 2 Add Q b2 R to the expression to complete the square. x2 2 6x 1 (23)2 5 x2 2 6x 1 9 Subtract 9 from the expression so that the equation is unchanged. y 5 x2 2 6x 1 9 1 14 2 9 Factor the perfect square trinomial. y 5 (x 2 3)2 1 14 2 9 Add the remaining constant terms. y 5 (x 2 3)2 1 5 Exercises Rewrite each equation in vertex form. 13. y 5 x2 1 4x 1 3 (x 1 2)2 2 1 14. y 5 x2 2 6x 1 13 (x 2 3)2 1 4 15. y 5 2x2 1 4x 2 10 2(x 2 2)2 2 6 16. y 5 x2 2 2x 2 3 (x 2 1)2 2 4 17. y 5 x2 1 8x 1 13 (x 1 4)2 2 3 18. y 5 2x2 2 6x 2 4 2(x 1 3)2 1 5 19. y 5 2x2 1 10x 2 18 2(x 2 5)2 1 7 20. y 5 x2 1 2x 2 8 (x 1 1)2 2 9 21. y 5 2x2 1 4x 2 3 2(x 1 1)2 2 5 22. y 5 3x2 2 12x 1 8 3(x 2 2)2 2 4 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 60 Name Class Date Additional Vocabulary Support 4-7 The Quadratic Formula The column on the left shows the steps used to solve a problem using the quadratic formula. Use the column on the left to answer each question in the column on the right. Problem 1. Read the title of the problem. What Solve by using the quadratic formula. process are you going to use to solve the problem? 4x2 1 x 5 2 Answers may vary. Sample: Solve by using the quadratic formula. Write in standard form. 2. What is the standard form of a quadratic equation? 4x2 1 x 2 2 5 0 ax2 1 bx 1 c 5 0 Find the values of a, b, and c. 3. Explain how you know which value is a. Answers may vary. Sample: It is the a 5 4, b 5 1, c 5 22 leading coefﬁcient. Substitute the values into the quadratic formula. x5 4. Write the quadratic formula. x5 2(1) 4 "(1)2 2 4(4)(22) 2b 6 "b2 2 4ac 2a 2(4) 5. Explain what the symbol 4 means. Simplify. Answers may vary. Sample: The symbol x5 5 21 4 "1 2 (232) means there are two answers, one found 8 by adding and the other by subtracting. 21 4 "33 8 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 61 Name 4-7 Class Date Think About a Plan The Quadratic Formula Sports A diver dives from a 10 m springboard. The equation f (t) 5 24.9t2 1 4t 1 10 models her height above the pool at time t. At what time does she enter the water? Understanding the Problem 1. What does the function represent? the diver’s height above the water at time t in seconds 2. What is the problem asking you to determine? at what time the diver’s height above the water equals 0 3. Do you need to use the height of the springboard to solve the problem? Explain. No; you don’t use the height of the springboard. You only need to ﬁnd the zeros of the function. Planning the Solution 4. What are three possible methods for solving this problem? Graph the equation, factor the equation; use the Quadratic Formula. 5. If a solution exists, which method will give an exact solution? Explain. The Quadratic Formula; you can only estimate from a graph, and not all equations have expressions that are factorable. Getting an Answer 6. Is there more than one reasonable solution to the problem? Explain. No; the equation has two zeros, but one of them is negative. Time moves forward and cannot be negative. 7. At what time does the diver enter the water? The diver enters the water about 1.89 s after starting her dive. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 62 Name 4-7 Class Date Practice Form G The Quadratic Formula Solve each equation using the Quadratic Formula. 1. x2 2 8x 1 15 5 0 3, 5 2. x2 1 12x 1 35 5 0 27, 25 3. 3x2 1 5x 5 2 22, 13 4. 2x2 1 3 5 7x 12, 3 5. x2 1 16 5 8x 4 6. x2 5 4x 2 1 2 6 "3 7. x(2x 2 5) 5 12 232, 4 8. 23x2 2 8x 1 16 5 0 24, 43 10. x2 1 10x 1 22 5 0 25 6 "3 9. x2 1 4x 5 3 22 6 "7 3 11. 4x(x 1 1) 5 7 21 6 2"2 12. x(2x 2 3) 5 9 22, 3 2 13. The principal at a high school is planning a concert to raise money for the music programs. He determines the profit p from ticket sales depends on the price t of a ticket according to the equation p 5 2200t2 1 3600t 2 6400. All amounts are in dollars. If the goal is to raise $8500, what is the smallest amount the school should charge for a ticket to the concert? $6.45 14. The equation y 5 x2 2 12x 1 45 models the number of books y sold in a bookstore x days after an award-winning author appeared at an autographsigning reception. What was the first day that at least 100 copies of the book were sold? day 16 15. The height of the tide measured at a seaside community varies according to the number of hours t after midnight. If the height h, in feet, is currently given by the equation h 5 212t2 1 6t 2 9, when will the tide first be at 6 ft? about 3.55 h past midnight or about 3:33 A.M. 16. The height h, in feet, of a model rocket t seconds after launch is given by h 5 256t 2 16t2 . As the rocket descends, it deploys a recovery parachute when it reaches 200 ft above the ground. At what time does the parachute deploy? about 15.2 s Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 63 Name 4-7 Class Practice Date Form G (continued) The Quadratic Formula Evaluate the discriminant for each equation. Determine the number of real solutions. 17. x2 1 5x 1 8 5 0 27; none 18. x2 2 5x 1 4 5 0 9; two 19. 29x2 1 12x 2 4 5 0 0; one 20. 23x2 1 5x 2 4 5 0 223; none 21. 4x2 1 4x 5 21 0; one 22. 6x2 5 x 1 2 49; two 23. 5x 1 1 5 3x2 37; two 24. 4x2 2 x 1 3 5 0 247; none 25. 4x2 1 36x 1 81 5 0 0; one 26. 5x2 5 3x 2 2 231; none 27. 16x2 2 56x 1 49 5 0 0; one 28. 4x2 2 16x 1 11 5 0 80; two 29. In Exercise 16, the height of the rocket was given by h 5 256t 2 16t2 . Use the discriminant to answer the following questions. a. Will the rocket reach an altitude of 1000 ft? yes b. Will the rocket reach an altitude of 1024 ft? yes c. Will the rocket reach an altitude of 1048 ft? no 30. The number n of people using the elevator in an office building every hour is given by n 5 t2 2 10t 1 40. In this equation, t is the number of hours after the building opens in the morning, 0 # t # 12. Will the number of people using the elevator ever be less than 15 in any one hour? Use the discriminant to answer. no Solve each equation using any method. When necessary, round real solutions to the nearest hundredth. 3 31. 4x2 1 x 2 3 5 0 21, 4 32. 5x2 2 6x 2 2 5 0 20.27, 1.47 33. x2 2 5x 2 9 5 0 21.41, 6.41 1 1 34. 15x2 2 2x 2 1 5 0 25, 3 3 35. 2x2 5 5x 2 3 1, 2 36. 4x2 1 3x 5 5 21.55, 0.80 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 64 Name Class 4-7 Date Practice Form K The Quadratic Formula Solve each equation using the Quadratic Formula. To start, find the values of a, b, and c. Substitute those values into the Quadratic Formula. When necessary round real solutions to the nearest hundredth. 1. x2 2 4x 1 3 5 0 2. 2x2 1 3x 2 4 5 0 3. 8x2 2 2x 2 5 5 0 a 5 1, b 5 24, c 5 3 a 5 2, b 5 3, c 5 24 a 5 8, b 5 22, c 5 25 2(24) 4 "(24)2 2 (4)(1)(3) 2(1) x 5 3, x 5 1 2(3) 4 "(3)2 2 (4)(2)(24) 2(2) x 5 0.85, x 5 22.35 2(22) 4 "(22)2 2 (4)(8)(25) 2(8) x 5 0.93, x 5 20.68 4. x2 1 3x 5 3 5. 4x2 1 3 5 9x x 5 0.79, x 5 23.79 x 5 0.41, x 5 1.84 6. 2x 2 5 5 2x2 x 5 1.45, x 5 23.45 7. Your school sells yearbooks every spring. The total profit p made depends on the amount x the school charges for each yearbook. The profit is modeled by the equation p 5 22x2 1 70x 1 520. What is the smallest amount in dollars the school can charge for a yearbook and make a profit of at least $1000? To start, substitute 1000 for p in the equation. 1000 5 22x2 1 70x 1 520 Then, write the equation in standard form. 2x2 2 70x 1 480 5 0 $9.36 8. Engineers can use the formula d 5 0.05s2 1 1.1s to estimate the minimum stopping distance d in feet for a vehicle traveling s miles per hour. a. If a car can stop after 65 feet, what is the fastest it could have been traveling when the driver put on the brakes? 26.7 miles per hour b. Reasoning Explain how you knew which of the two solutions from the Quadratic Formula to use. (Hint: Remember this is a real situation.) The other solution was negative. You cannot drive negative miles per hour. 9. Reasoning Explain why a quadratic equation has no real solutions if the discriminant is less than zero. The discriminant is the value under the square root sign. If it is less than zero, you are trying to take the square root of a negative number. The square root of a negative number is not a real number. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 65 Name 4-7 Class Date Practice (continued) Form K The Quadratic Formula Evaluate the discriminant for each equation. Determine the number of real solutions. 10. 212x2 1 5x 1 2 5 0 11. x2 2 x 1 6 5 0 12. 2x 2 5 5 2x2 (5)2 2 4(212)(2) (21)2 2 4(1)(6) (2)2 2 4(1)(25) 121, 2 real solutions 223, 0 real solutions 24, 2 real solutions 13. 4x2 1 7 5 9x 231, 0 real solutions 14. x2 2 4x 5 24 0, 1 real solution 15. 3x 1 6 5 26x2 2135, 0 real solutions Solve each equation using any method. When necessary, round real solutions to the nearest hundredth. 16. 7x2 1 3x 5 12 17. x2 1 6x 2 7 5 0 18. 5x 5 23x2 1 2 7x2 1 3x 2 12 5 0 (x 1 7)(x 2 1) 5 0 23x2 2 5x 1 2 5 0 23 4 "(3)2 2 4(7)(212) 2(7) x 5 1.11, x 5 21.54 x 5 27, x 5 1 5 4 "(5)2 2 4(23)(2) 2(23) x 5 0.33, x 5 22 19. 212x 1 7 5 5 2 2x2 x 5 5.83, x 5 0.17 20. 9x2 2 6x 2 4 5 25 x 5 0.33 21. 2x 2 24 5 2x2 x 5 4, x 5 26 Without graphing, determine how many x-intercepts each function has. 22. y 5 2x2 2 3x 1 5 23. y 5 2x2 2 4x 1 1 24. y 5 x2 1 3x 1 3 (3)2 2 4(2)(5) (24)2 2 4(2)(1) (3)2 2 4(1)(3) 0 x-intercepts 2 x-intercepts 0 x-intercepts 25. y 5 9x2 2 12x 1 7 0 x-intercepts 26. y 5 25x2 1 8x 2 3 2 x-intercepts 27. y 5 x2 1 16x 1 64 1 x-intercept Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 66 Name Class 4-7 Date Standardized Test Prep The Quadratic Formula Multiple Choice For Exercises 1−6, choose the correct letter. 1. What is the solution of 3x2 1 2x 2 5 5 0? Use the Quadratic Formula. A 253 , 1 21, 53 213 , 5 25, 13 2. What is the solution of 2x2 2 8x 1 3 5 0? Use the Quadratic Formula. H 24 4 !22 2 24 4 !10 2 4 4 !10 2 4 4 !22 2 3. What is the solution of x2 2 5x 5 5? Use the Quadratic Formula. D 25, 1 5 4 !5 2 21, 5 5 4 3 !5 2 4. What is the solution of x2 5 6x 2 3? Use the Quadratic Formula. I 23 4 !6 3 23 3 4 !6 5. What is the discriminant of the equation 3x2 2 7x 1 1 5 0? B 61 !37 37 219 6. What is the discriminant of the equation 4x2 1 28x 1 49 5 0? H 25472 0 2756 1568 Extended Response 7. The equation d 5 n2 2 12n 1 43 models the number of defective items d produced in a manufacturing process when there are n workers in a restricted area. Use the discriminant to answer the following questions. Show your work. a. Will the number of defective items ever be 10? b. Will the number of defective items ever be 7? c. Will the number of defective items ever be 5? [4] a. yes; b. yes; c. no; the answers are based on the value of the discriminant for each case: (a) 12, (b) 0, and (c) –8. [3] appropriate method with one computational error [2] appropriate method with several computational errors [1] correct answers but no work shown [0] incorrect answers and no work shown OR no answers given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 67 Name 4-7 Class Date Enrichment The Quadratic Formula The value of the discriminant of a quadratic equation tells you how many solutions the quadratic equation will have and the type of solutions they will be. The discriminant can also tell you if the quadratic equation is factorable. Factor each of the following quadratic equations, if possible. Then determine the value of the discriminant for each equation. 1. 6x2 2 11x 1 3 5 0 (3x 2 1)(2x 2 3); 49 2. 4x2 2 x 2 7 5 0 not factorable; 113 3. x2 1 7x 1 10 5 0 (x 1 5)(x 1 2); 9 4. 16x2 2 1 5 0 (4x 2 1)(4x 1 1); 64 5. x2 1 x 2 5 5 0 not factorable; 21 6. 2x2 1 5x 2 1 5 0 not factorable; 33 7. x2 2 11x 1 30 5 0 (x 2 5)(x 2 6); 1 8. x2 1 9x 1 2 5 0 not factorable; 73 9. x2 1 5x 2 8 5 0 not factorable; 57 10. 3x2 1 8x 1 4 5 0 (3x 1 2)(x 1 2); 16 11. x2 2 7x 2 5 5 0 not factorable; 69 12. x2 1 4x 1 3 5 0 (x 1 3)(x 1 1); 4 Copy and complete the table using your previous answers. Discriminants of quadratics that are factorable 49 9 64 1 16 4 Discriminants of quadratics that are not factorable 113 21 33 73 57 69 13. What types of numbers did you get for the values of the discriminants for the quadratics that were factorable? The numbers are all perfect squares. 14. Writing How can the value of the discriminant tell you if a quadratic equation is factorable? Answers may vary. Sample: If the discriminant is a perfect square, the equation will be factorable. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 68 Name Class Date Reteaching 4-7 The Quadratic Formula You can solve some quadratic equations by factoring or completing the square. You can solve any quadratic equation ax2 1 bx 1 c 5 0 by using the Quadratic Formula: x5 2b 4 "b2 2 4ac 2a Notice the 4 symbol in the formula. Whenever b2 2 4ac is not zero, the Quadratic Formula will result in two solutions. Problem What are the solutions for 2x2 1 3x 5 4? Use the Quadratic Formula. x5 5 2x2 1 3x 2 4 5 0 Write the equation in standard form: ax2 1 bx 1 c 5 0 a 5 2 ; b 5 3 ; c 5 24 a is the coefﬁcient of x2, b is the coefﬁcient of x, c is the constant term. 2b 4 "b2 2 4ac Write the Quadratic Formula. 2a 2(3) 4 "(3)2 2 4(2)(24) 2(2) Substitute 2 for a, 3 for b, and −4 for c. 5 23 4 !41 4 Simplify. 5 23 1 !41 23 2 !41 or 4 4 Write the solutions separately. Check your results on your calculator. Replace x in the original equation with 23 1 !41 23 2 !41 and . Both values 4 4 for x give a result of 4. The solutions check. (23 2 V (41)) / 4 X 22.350781059 2X2 1 3X 4 (23 1 V (41)) / 4 X .8507810594 2X2 1 3X 4 Exercises What are the solutions for each equation? Use the Quadratic Formula. 1. 2x2 1 7x 2 3 5 0 3. 2x2 5 4x 1 3 7 1 !37 2 2 1 !10 2 or or 7 2 !37 2 2. x2 1 6x 5 10 23 1 !19 or 23 2 !19 2 2 !10 2 9 4. 4x2 1 81 5 36x 2 5. 2x2 1 1 5 5 2 7x 24 or 12 6. 6x2 2 10x 1 3 5 0 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 69 5 1 !7 6 or 5 2 !7 6 Name Class 4-7 Date Reteaching (continued) The Quadratic Formula There are three possible outcomes when you take the square root of a real number n: . 0 S two real values (one positive and one negative) n • 5 0 S one real value (0) , 0 S no real values 2b 4 "b2 2 4ac . The value under the Now consider the quadratic formula: x 5 2a radical symbol determines the number of real solutions that exist for the equation ax2 1 bx 1 c 5 0: b2 . 0 S two real solutions 2 4ac • 5 0 S one real solution , 0 S no real solutions The value under the radical, b2 2 4ac, is called the discriminant. Problem What is the number of real solutions of 23x2 1 7x 5 2? 23x2 1 7x 5 2 23x2 1 7x 2 2 5 0 Write in standard form. a 5 23 , b 5 7 , c 5 22 b2 Find the values of a, b, and c. 2 4ac Write the discriminant. (7)2 2 4(23)(22) Substitute for a, b, and c. 49 2 24 Simplify. 25 The discriminant, 25, is positive. The equation has two real roots. Exercises What is the value of the discriminant and what is the number of real solutions for each equation? 7. x2 1 x 2 42 5 0 169; two 8. 2x2 1 13x 2 40 5 0 9; two 9. x2 1 2x 1 5 5 0 –16; none 1 10. x2 5 18x 2 81 0; one 11. 2x2 1 7x 1 44 5 0 225; two 12. 4x2 2 5x 1 25 5 0 0; one 13. 2x2 1 7 5 5x –31; none 14. 4x2 1 25x 5 21 961; two 15. x2 1 5 5 3x –11; none 1 16. 9x2 5 4x 2 36 0; one 1 17. 2x2 1 2x 1 3 5 0 –2; none 1 18. 6x2 5 2x 1 18 16; two Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 70 Name 4-8 Class Date Additional Vocabulary Support Complex Numbers Complete the vocabulary chart by filling in the missing information. Word or Word Phrase Deﬁnition Picture or Example imaginary unit The imaginary unit i is a complex number whose square is 21. i 5 !21 pure imaginary number A pure imaginary number is of the form a 1 bi, where a 5 0 and b 2 0. 2. any number of the form a 1 bi where a and b are real numbers complex number complex number plane In the complex number plane, the point (a, b) represents the complex number a 1 bi. To graph, locate the real part on the horizontal axis and the imaginary part on the vertical axis. 4. Answers may vary. Sample: The absolute value of a complex number is its distance from the origin in the complex number plane. absolute value of a complex number complex conjugates number pairs of the form a 1 bi and a 2 bi 1. Answers may vary. Sample: 5i 7 2 4i 3. 4i 2i 42 4i Real 2 4 2 3i u 8 1 6i u 5 "82 1 62 5 10 5. Answers may vary. Sample: 23 1 4i and 23 2 4i Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 71 Imaginary Name Class 4-8 Date Think About a Plan Complex Numbers A student wrote the numbers 1, 5, 1 1 3i, and 4 1 3i to represent the vertices of a quadrilateral in the complex number plane. What type of quadrilateral has these vertices? Know 1. The vertices of the quadrilateral are: 1, 5, 1 1 3i, and 4 1 3i . 2. You can write the vertices in the form a 1 bi as: 1 1 0i, 5 1 0i, 1 1 3i, and 4 1 3i . Need 3. To solve the problem I need to: graph the numbers in the complex plane Plan 4. How do you find the coordinates that represent each complex number? For the complex number a 1 bi, the real part a is the horizontal coordinate and the imaginary part b is the vertical coordinate. 5. What are the points you need to graph? (1, 0), (5, 0), (1, 3), (4, 3) 6. Graph your points in the complex plane. Connect the points with straight lines to form a quadrilateral. Imaginary axis 4i 2i Real axis O 2 4 6 2i 7. What type of quadrilateral did you draw? Explain how you know. Trapezoid; the top and bottom sides are parallel, but the left and right sides are not parallel. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 72 . Name 4-8 Class Date Practice Form G Complex Numbers Simplify each number by using the imaginary number i. 1. !249 7i 2. !2144 12i 3. !27 i"7 4. !210 i"10 5. !28 2i"2 6. !248 4i"3 ⴚ4 ⴙ 8i Plot each complex number and find its absolute value. 8. 6 2 4i 2"13 7. 23i 3 4i 2i 9. 24 1 8i 4"5 4 2 2i Simplify each expression. ⴚ3i Imaginary axis Real axis 2 4 6 ⴚ 4i 10. (22 1 3i) 1 (5 2 2i) 3 1 i 11. (26 1 7i) 1 (6 2 7i) 0 12. (4 2 2i) 2 (21 1 3i) 5 2 5i 13. (25 1 3i) 2 (28 1 2i) 3 1 i 14. (4 2 3i)(25 1 4i) 28 1 31i 15. (2 2 i)(23 1 6i) 15i 16. (5 2 3i)(5 1 3i) 34 17. (21 1 3i)2 28 2 6i 18. (4 2 i)2 15 2 8i 19. (22i)(5i)(2i) 210i 20. A6 2 !216B 1 A24 1 !225B 2 1 i 21. A22 1 !29B 1 A21 2 !236B 23 2 3i 22. A25 1 !24B 2 A3 2 !216B 28 1 6i 23. A7 2 !21B 2 !281 7 2 10i 24. 3i(2 1 2i) 26 1 6i 25. 2(3 2 7i) 2 i(24 1 5i) 11 2 10i 26. A2 1 !24B A21 1 !29B 28 1 4i 27. A5 1 !21B A2 2 !236B 16 2 28i Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 73 Name 4-8 Class Date Practice (continued) Form G Complex Numbers Write each quotient as a complex number. 28. 5 1 2i 1 5 2 2 4i 4i 29. 3 2 2i 1 30. 4 2 3i 18 25 1 25 i 3i 3 6 2 5i 22 1 i 5 7 14 31. 5 2 2i 35 29 1 29 i Find the factors of each expression. Check your answer. 32. x2 1 36 ( x 1 6i)(x 2 6i) 33. 2x2 1 8 2( x 1 2i)(x 2 2i) 34. 5x2 1 5 5( x 1 i)(x 2 i) 1 35. x2 1 9 ( x 1 1 i)(x 2 1 i) 3 3 36. 16x2 1 25 (4 x 1 5i)(4 x 2 5i) 37. 24x2 2 49 2(2 x 1 7i)(2 x 2 7i) Find all solutions to each quadratic equation. 38. x2 1 2x 1 5 5 0 21 6 2i 39. 2x2 1 2x 2 10 5 0 1 6 3i "3 "31 3 41. 24x2 1 6x 2 3 5 0 4 6 4 i 3 40. 2x2 2 3x 1 5 5 0 4 6 4 i "14 "13 1 42. 3x2 1 2x 1 5 5 0 23 6 3 i 1 43. 2x2 2 2x 1 7 5 0 2 6 2 i 44. a. Name the complex number represented by each point on the graph at the right. A: 2 1 3i; B: 24 1 2i; C: 23 2 3i; D: 24i A: 22 2 3i; b. Find the additive inverse of each number. B: 4 2 2i; C: 3 1 3i; D: 4i c. Find the complex conjugate of each number. A: 2 2 3i; d. Find the absolute value of each number. B: 24 2 2i; A: !13; B: 2 !5; C: 3 !2; D: 4 C: 23 1 3i; D: 4i Imaginary axis B Prentice Hall Gold Algebra 2 • Teaching Resources A 2i 4 2 2i Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 74 4i C 4i D Real axis 2 4 Name Class Date Practice 4-8 Form K Complex Numbers Simplify each number by using the imaginary number i. 1. !2100 2. !22 i!2 !21 ? 100 3. !248 4i!3 4. !236 6i !21 ? !100 10i Plot each complex number and find its absolute value. 5. 5i 6. 3 1 2i 8i imaginary axis 4i 5i 4i real axis 8 4 O 4i 4 8 4 O 4i 8 8i 7. 7 2 1i 8i imaginary axis 3 2i 4i 8 8 4 O 4i real axis 4 8i 4 9i real axis 4 7 1i !97 5 !2 (9 1 2) 1 (6i 2 i) 10. (212i) 2 (3 1 3i) 23 2 15i 11. (22i)(5 1 4i) 8 2 10i 11 1 5i Write each quotient as a complex number. 12. 5 1 4i 7i 5 1 4i 27i 7i Q 27i R 4 7 21 1 5i 13. 3 2 2i 2 2 6i 14. 2 2 3i 22 13 21 1 i 6 2 13 i 2 57i Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 75 4i 8i Simplify each expression. 9. (9 1 6i) 1 (2 2 i) 8i imaginary axis 8 4 O 4i 8 8i !13 5 8. 24 1 9i 8i imaginary axis real axis 4 8 Name Class Date Practice (continued) 4-8 Form K Complex Numbers Find the factors of each expression. Check your answer. 15. 2x2 1 32 2(x2 1 16) 16. x2 1 4 ( x 1 2i)(x 2 2i) 17. 3x2 1 3 3( x 1 i)(x 2 i) 19. 9x2 1 49 (3 x 1 7i)(3 x 2 7i) 20. 4x2 1 25 (2 x 1 5i)(2 x 2 5i) 2(x2 1 42) 2( x 1 4i)(x 2 4i) 18. x2 1 64 ( x 1 8i)(x 2 8i) Find all solutions to each quadratic equation. 21. 2x2 2 3x 1 7 5 0 3 4 "(23)2 2 4(2)(7) 2(2) 3 4 !9 2 56 x5 4 x5 x5 22. 4x2 2 5x 1 6 5 0 23. x(x 2 3) 1 3 5 0 !3 x 5 32 6 2 i 5 4 "(25)2 2 4(4)(6) x5 2(4) !71 x 5 58 6 8 i 3 4 !247 4 3 x544 !47 4 i 24. Error Analysis Robert solved the equation 2x2 1 16 5 0. His solution was x 5 4 !28 i. What errors did Robert make? What is the correct solution? Robert made two errors. He left a negative number under the radical sign, and he did not simplify !8. The correct solution is x 5 w2i!2. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 76 Name Class Date Standardized Test Prep 4-8 Complex Numbers Multiple Choice For Exercises 1−8, choose the correct letter. 1. What is the simplified form of (28 1 5i) 1 (3 2 2i)? B 214 1 31i 25 1 3i 211 1 7i 5 2 3i 2. What is the simplified form of (11 2 6i) 2 (24 1 12i)? I 7 1 6i 7 2 18i 15 1 6i 15 2 18i 3. What is the simplified form of (5 1 !236) 2 (24 2 !249)? D 9 2 13i 9 1 85i 12i 9 1 13i 15 15i 4. What is the simplified form of (25i)(23i)? G 215i 215 5. What is the simplified form of (23 1 2i)(1 2 4i)? C 22 2 2i 211 2 10i 5 1 14i 23 2 8i 55 2 48i 55 1 48i 5 3 4 2 2i 7 11 10 2 10i 6. What is the simplified form of (8 2 3i)2 ? H 73 16 2 6i 5 1 3i 7. What is 4 2 2i written as a complex number? A 7 13 11 1 10 1 10i 10 1 10i 8. What is the factored form of the expression 4x2 1 36? G 4(x 1 3i)2 4(x 1 6i)2 4(x 1 3i)(x 2 3i) 4(x 1 6i)(x 2 6i) Short Response 9. What are the solutions of 2x2 1 3x 1 6 5 0? Show your work. !39 [2] 234 6 4 i; [1] quadratic formula properly used, but some computational errors OR correct solutions without work shown [0] incorrect answers and no work shown OR no answers given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 77 Name 4-8 Class Date Enrichment Complex Numbers Mathematical quantities like distance, height, and area are represented by a single real number that indicates the size or magnitude. Other quantities used in science such as force, velocity, and acceleration have both magnitude and direction. These quantities are called vectors. On a graph vectors are represented by a directed line segment or arrow. Vector v, v , is shown on the complex number plane below. Algebraically, vectors are represented by complex numbers. The vector shown below is represented by k3, 5l because the arrow ends at the point represented by the complex number 3 1 5i. W Imaginary v Real The magnitude of the vector is the absolute value of the complex number. Write the complex number represented by each vector. Then determine the magnitude of the vector. 1. k22, 7l 22 1 7i; !53 2. k5, 1l 5 1 i; !26 3. k3, 24l 3 2 4i; 5 4. k210, 15l 210 1 15i; 5 !13 5. k213, 221l 213 2 21i; !610 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 78 Name Class Date Reteaching 4-8 Complex Numbers • A complex number consists of a real part and an imaginary part. It is written in the form a 1 bi, where a and b are real numbers. • i 5 !21 and i2 5 (!21)(!21) 5 21 • When adding or subtracting complex numbers, combine the real parts and then combine the imaginary parts. • When multiplying complex numbers, use the Distributive Property or FOIL. Problem What is (3 2 i) 1 (2 1 3i)? (3 2 i) 1 (2 1 3i) 5 3 2 i 1 2 1 3i Circle real parts. Put a square around imaginary parts. 5 (3 1 2) 1 (21 1 3)i Combine. 5 5 1 2i Simplify. Problem What is the product (7 2 3i)(24 1 9i)? Use FOIL to multiply: (7 2 3i)(24 1 9i) 5 7(24) 1 7(9i) 1 (23i)(24) 1 (23i)(9i) (7 2 3i)(24 1 9i) 5 228 1 63i 1 12i 2 27i2 First 5 7(24) 5 228 1 75i 2 27i2 Outer 5 7(9i) You can simplify the expression by substituting 21 for i2. Inner 5 (23i)(24) (7 2 3i)(24 1 9i) 5 228 1 75i 2 27(21) Last 5 (23i)(9i) 5 21 1 75i Exercises Simplify each expression. 1. 2i 1 (24 2 2i) 24 2. (3 1 i)(2 1 i) 5 1 5i 3. (4 1 3i)(1 1 2i) 22 1 11i 4. 3i(1 2 2i) 6 1 3i 5. 3i(4 2 i) 3 1 12i 6. 3 2 (22 1 3i) 1 (25 1 i) 22i 7. 4i(6 2 2i) 8 1 24i 8. (5 1 6i) 1 (22 1 4i) 3 1 10i 9. 9(11 1 5i) 99 1 45i Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 79 Name Class Date Reteaching (continued) 4-8 Complex Numbers • The complex conjugate of a complex number a 1 bi is the complex number a 2 bi. • (a 1 bi)(a 2 bi) 5 a2 1 b2 • To divide complex numbers, use complex conjugates to simplify the denominator. Problem 4 1 5i What is the quotient 2 2 i ? 4 1 5i 22i 5 4 1 5i 21i 5 22i ?21i 5 5 5 5 8 1 4i 1 10i 1 5i2 (2 2 i)(2 1 i) 8 1 4i 1 10i 1 5i2 22 1 12 8 1 14i 1 5(21) 4 1 1 3 1 14i 5 3 5 5 1 14 5i The complex conjugate of 2 2 i is 2 1 i. Multiply both numerator and denominator by 2 1 i. Use FOIL to multiply the numerators. Simplify the denominator. (a 1 bi)(a 2 bi) 5 a2 1 b2 Substitute 21 for i2. Simplify. Write as a complex number a 1 bi. Exercises Find the complex conjugate of each complex number. 10. 1 2 2i 1 1 2i 11. 3 1 5i 3 2 5i 12. i 2i 13. 3 2 i 3 1 i 14. 2 1 3i 2 2 3i 15. 25 2 2i 25 1 2i Write each quotient as a complex number. 3i 16. 1 2 2i 265 1 35i 6 9 17. 3 1 5i 17 2 15 17 i 18. 2 1 2i 2 2 2i i 2 1 5i 1 19. 3 2 i 10 1 17 10 i 24 2 i 10 20. 2 1 3i 211 13 1 13 i 61i 7 21. 25 2 2i 232 29 1 29 i Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 80 Name Class 4-9 Date Additional Vocabulary Support Quadratic Systems Use the list below to complete the Venn diagram. solve using substitution one solution is possible where two lines intersect solve using elimination two solutions are possible where two parabolas intersect Quadratic System of Equations solve using substitution Linear System of Equations where two parabolas intersect where two lines intersect solve using elimination two solutions are possible one solution is possible Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 81 Name 4-9 Class Date Think About a Plan Quadratic Systems Business A company’s weekly revenue R is given by the formula R 5 2p2 1 30p, where p is the price of the company’s product. The company is considering hiring a distributor, which will cost the company 4p 1 25 per week. a. Use a system of equations to find the values of the price p for which the company will still remain profitable if they hire this distributor. b. Which value of p will maximize the profit after including the distributor cost? 1. What does it mean for the company to be profitable? It means that the weekly revenue is greater than the distributor cost. z z R 5 2p2 1 30p . 2. The weekly revenue is represented by the function z z D 5 4p 1 25 3. The distributor cost D is represented by the function . R/D 4. Solve this system of equations by graphing. 200 150 5. For what values of p will the company remain profitable? 1 R p R 25 6. How can you find the new weekly revenue of the company 100 50 0 if they hire the distributor? p 0 10 20 30 40 Subtract the distributor cost function from the weekly revenue function. 7. Write the new weekly revenue function and graph it. 5 2p2 1 26p 2 25z z R R 200 150 8. How can you find the value of p that will maximize the profit? 100 Find the maximum of the graph of the new weekly revenue 50 function. 0 p 0 9. What value of p will maximize the profit? 13 10. What is the maximum profit? 144 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 82 10 20 30 40 Name Class 4-9 Date Practice Form G Quadratic Systems Solve each system by graphing. Check your answers. 1. e y 5 2x2 1 3x 1 2 (0, 2) y 5 3x 1 2 2. e y 5 x2 1 2x 2 3 (22, 23), (2, 5) y 5 2x 1 1 3. e y 5 22x2 1 4x 1 3 (21, 23), (2, 3) y 5 2x 2 1 4. e y 5 2x2 2 5x (21, 7), (2, 22) y 5 23x 1 4 Solve each system by substitution. Check your answers. 5. e y 5 x2 1 5x 2 2 (22, 28), (0, 22) y 5 3x 2 2 6. e y 5 2x2 1 x 1 12 (25, 218), (4, 0) y 5 2x 2 8 7. e y 5 x2 2 2x 2 3 (0, 23), (4, 5) y 5 2x 2 3 8. e y 5 2x2 2 5x 1 6 (2, 4) y 5 3x 2 2 9. e y 5 2x2 1 2x 1 18 (27, 245), (4, 10) y 5 5x 2 10 10. e y 5 x2 2 2x 2 2 (23, 13), (2, 22) y 5 23x 1 4 x1y55 (22, 7), (1, 4) y 1 1 5 3x2 1 2x 12. e x 1 y 5 x2 2 6 (22, 0), (2, 24) x1y1250 14. e y 1 4 5 x2 2 3x (1, 26), (5, 6) y 1 9 5 3x 11. e x5y25 13. e 2 (22, 3), (1, 6) x 1 2x 5 y 2 3 15. e x2 1 y 2 10 5 0 (23, 1), (4, 26) x1y1250 16. e 2 (27, 14), (1, 6) x 2 y 5 25x 17. e y 1 5x 5 x2 2 3 (2, 29), (6, 3) y 2 3x 5 215 18. e x1y57 y 2 2x 5 2x2 2 4 (1, 23), (3, 27) y 1 2x 5 21 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 83 Name Class 4-9 Date Practice (continued) Form G Quadratic Systems Solve each system. 19. e y 5 2x2 1 2x 2 3 (21, 26), (0, 23) y 5 x2 1 4x 2 3 20. e y 5 x2 1 2x 2 3 (23, 0), (1, 0) y 5 2x2 2 2x 1 3 21. e y 5 2x2 1 x 2 5 (21, 24), (0, 25) y 5 2x2 2 2x 2 5 22. e y 5 2x2 1 x 1 2 (21, 0), (3, 24) y 5 x2 2 3x 2 4 23. e y 5 x2 1 1 (22, 5), (2, 5) y 5 2x2 2 3 24. e y 5 2x2 2 4 (25, 46), (1, 22) y 5 x2 2 4x 1 1 26. e y y . x2 2 4x 6 y , 2x2 1 6 (⫺1,5) Solve each system by graphing. 25. e y y , x2 1 5 (23, 14) (3, 14) 12 y . 2x2 2 4 8 4 ⫺6 ⫺2 27. e y . x2 2 x y , x2 1 3 (⫺3, 12) ⫺6 x 2 y 28. e 8 O x 6 (3,⫺3) ⫺6 6 y y # 4x2 1 4x y $ x2 1 4x 8 4 x ⫺6 2 O ⫺6 6 (0, 0) 6 x 29. In business, a break-even point is the point (x, y) at which the graphs of the revenue and cost functions intersect. For one manufacturing company, the revenue from producing x items is given by the function y 5 2x 1 12 and the cost of producing x items is given by y 5 2x2 1 10x 1 5. Find all break-even points. (1, 14), (7, 26) 30. Two skaters are practicing at the same time on the same rink. One skater follows the path y 5 22x 1 32, while the other skater follows the curve y 5 22x2 1 18x. Find all points where they might collide if they are not careful. (2, 28), (8, 16) Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 84 Name Class 4-9 Date Practice Form K Quadratic Systems Solve each system by graphing. Check your answers. y 5 3x2 1 2x 2 5 1. e y 5 2x 2 2 y 4 ⫺8 ⫺4 (⫺1, ⫺4) y 5 212 x2 2 x 1 3 2. e y 5 3x 1 14 y (21, 24), (1, 0) ⫺8 ⫺4 O ⫺4 y 5 2x2 1 3x 2 1 y 5 2x 2 1 8 y no solution 4 (1, 0) x 4 3. e 4 x 4 8 ⫺8 ⫺4 ⫺4 (0, ⫺1) (0, 21), (4, 25) x 4 8 (4, ⫺5) ⫺8 Solve each system by substitution. Check your answers. 4. e y 5 22x2 1 4x 2 1 y 5 2x 2 5 5. e 2x 2 5 5 22x2 1 4x 2 1 y 5 x2 1 2 y 5 2x 1 4 (22, 6) and (1, 3) 6. e y 5 6x2 1 5x 1 3 y 5 2x 1 3 (0, 3) and (21, 4) (2, 21) and (21, 27) Solve each system using your graphing calculator. 7. e y 5 x2 2 2x 1 4 y 5 2x2 1 2x 1 4 8. e (0, 4) and (2, 4) y 5 x2 1 3x 2 2 y 5 x2 1 5x 1 4 9. e y 5 2x2 1 7x 2 2 y 5 2x2 1 3x 1 2 (1, 4) (23, 22) 10. Reasoning What is the least number of solutions a quadratic system can have? Explain what that means. A quadratic system can have no solutions. This means the graphs of the equations do not intersect. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 85 Name Class 4-9 Date Practice (continued) Form K Quadratic Systems 11. You work at a restaurant whose weekly profit is given by the formula P 5 2c2 1 14c 1 800, where c is the average price of the food, in dollars. The manager wants to add delivery service, which will cost the restaurant D 5 5c 1 300 per week. a. Use a graphing calculator to find the highest average price c the restaurant can sell its food at and still make a profit if they add delivery. $27.31 b. What will the weekly profit P be if the restaurant sells its food at this average price and doesn’t offer delivery? $436.50 c. Reasoning Even though these equations have two solutions, why is only one solution useful? (Hint: Remember this is a real situation.) The other solution is negative. You cannot have negative price. Solve each system. When necessary, round solutions to the nearest hundredth. 12. e y 5 2x2 1 5x 2 2 y 5 3x 2 1 (1, 2) 15. e y 5 x2 2 4x 1 5 y 5 23x 1 5 14. e (0, 5) and (1, 2) y 5 x2 1 7x 2 2 y 5 4x 2 5 no solutions 13. e 16. e y 5 25x2 2 x 1 3 y 5 2x 2 2 y 5 12x2 1 5x 2 32 y5x29 (23, 212) and (25, 214) 17. e (21, 21) and (1, 23) y 5 2x2 2 x 2 4 y5x12 (2.3, 4.3) and (21.3, 0.7) y 5 2x2 2 x 1 3 has y 5 23x 1 3 no solutions. Her work is below. What mistake did she make? What is the solution of this system? 18. Error Analysis A classmate said that the quadratic system e 2x2 2 x 1 3 5 23x 1 3 2x2 1 2x 1 6 5 0 x2 1 x 1 3 5 0 21 4 "12 2 (4)(1)(3) 2(1) 21 4 !211 2 In the second line of her solution, she added 3 to the left side of the equation and subtracted 3 from the right side instead of adding 3 to both sides; (0, 3) and (21, 6). Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 86 Name Class Date Standardized Test Prep 4-9 Quadratic Systems Multiple Choice For Exercises 1−4, choose the correct letter. y 5 x2 2 4x 1 5 1. What is the solution of the system? e D y 5 22x 1 8 (0, 5), (2, 1) (22, 17), (1, 6) (21, 10), (4, 5) (21, 10), (3, 2) 2. What is the solution of the system? e y 5 2x2 2 2x 1 4 G y 5 2x 1 2 (22, 4), (2, 0) (23, 5), (1, 1) (22, 4), (1, 1) (23, 1), (2, 0) 3. What is the solution of the system? e y 5 x2 2 4x 1 3 A y 5 22x 1 6 (21, 8), (3, 0) (21, 8), (4, 3) (22, 10), (3, 0) (0, 3), (4, 22) 4. What is the solution of the system? e y 5 2x2 1 4x 1 5 G y 5 x2 2 2x 2 3 (22, 27), (2, 9) (21, 0), (3, 8) (21, 0), (4, 5) (1, 24), (4, 5) Short Response 5. What is the solution of the system? Solve by graphing. e y , 2x2 1 3x y . x2 2 x 2 6 8 y [2] 4 ⫺6 x 6 ⫺6 [1] an error in graphing one of the functions OR the wrong region is shaded [0] no answer given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 87 Name Class 4-9 Date Enrichment Quadratic Systems y Write a system of inequalities for the graph at the right. 8 B 1. Write the equation for Graph A in the 6 form y 5 a(x 2 h)2 1 k. 4 y 5 (x 2 3)2 1 1 A 2 2. Rewrite the equation for Graph A as an inequality. x ⫺8 ⫺6 ⫺4 ⫺2 O ⫺2 2 4 6 8 C Explain how you decided which inequality to use. 3)2 y K (x 2 1 1; answers may vary. Sample: The region below the graph is shaded and the boundary is included. 3. Write an inequality for Graph B. Explain how you decided which inequality to use. y L 2»x 1 1… 2 4; answers may vary. Sample: The region above the graph is shaded and the boundary is included. 4. Write an inequality for Graph C. Explain how you decided which inequality to use. y S 212 (x 1 4)2 1 4.5; answers may vary. Sample: The region above the graph is shaded and the boundary line is not included. All three inequalities together describe the system of inequalities for this graph. Write the system of inequalities for each graph. y 5. 6. 8 8 y 6 4 2 2 ⫺8 ⫺6 ⫺4 ⫺2 O ⫺2 ᎐9 x 2 4 6 8 y R (x 1 1)2 2 4, y L (x 2 5)2 1 5 ᎐6 ᎐4 ᎐2 ᎐2 ᎐4 ᎐6 ᎐8 x 2 4 6 9 y K (x 1 4)2 2 8, y S x2 2 7, y K 2»x… 1 7 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 88 Name Class 4-9 Date Reteaching Quadratic Systems You used graphing and substitution to solve systems of linear equations. You can use these same methods to solve systems involving quadratic equations. Problem What is the solution of the system of equations? e y 5 2x 2 3 x2 2 2x 2 8 5 2x 2 3 x2 2 4x 2 5 5 0 (x 1 1)(x 2 5) 5 0 x 1 1 5 0 or x 2 5 5 0 x 5 21 or x 5 5 y 5 x2 2 2x 2 8 y 5 2x 2 3 Write one equation. Substitute x2 2 2x 2 8 for y in the linear equation. Write in standard form. Factor the quadratic expression. Use the Zero-Product Property. Solve for x. Because the solutions to the system of equations are ordered pairs of the form (x, y), solve for y by substituting each value of x into the linear equation. You can use either equation, but the linear equation is easier. x 5 21: x 5 5: y 5 2x 2 3 5 2(21) 2 3 5 25 y 5 2x 2 3 5 2(5) 2 3 5 7 S S (21, 25) (5, 7) The solutions are (21, 25) and (5, 7). Check these by graphing the system and identifying the points of intersection. Exercises Solve each system. 1. e y 5 x2 1 3x 2 5 (21, 27), (1, 21) y 5 3x 2 4 2. e y 5 2x2 1 5x 2 1 (1, 3), (5, 21) y 5 2x 1 4 3. e y 5 2x2 2 x 2 5 (21, 22), (3, 10) y 5 3x 1 1 4. e y 5 x2 1 3x 2 7 (25, 3), (1, 23) y 5 2x 2 2 5. e y 5 2x2 2 5x 1 1 (1, 22), (4, 13) y 5 5x 2 7 6. e y 5 2x2 2 2x 1 3 (24, 25), (1, 0) y5x21 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 89 Name Class 4-9 Date Reteaching (continued) Quadratic Systems To solve a system of linear inequalities, you graph each inequality and find the region where the graphs overlap. You can also use this technique to solve a system of quadratic inequalities. Problem What is the solution of this system of inequalities? e Step 1 y , 2x2 1 4x y . x2 2 2x 2 8 Graph the equation y 5 2x2 1 4x. Use a dashed boundary line because the points on the curve are not part of the solution. Choose a point on one side of the curve and check if it satisfies the inequality. y , 2x2 1 4x ? 0 , 2(2)2 1 4(2) 0,4 Check the point (2, 0). The inequality is true. ᎐8 ᎐6 ᎐4 ᎐2 ᎐8᎐6᎐4᎐2 ᎐4 y x 2 4 6 8 Points below the curve satisfy the inequality, so shade that region. Step 2 y Graph the equation y 5 x2 2 2x 2 8. Use a dashed boundary line ᎐8 ᎐6 because the points on the curve are not part of the solution. Choose a ᎐4 point on one side of the curve and check if it satisfies the inequality. ᎐2 x 2 y . x 2 2x 2 8 ᎐8᎐6᎐4᎐2 2 4 6 8 ? 0 . (2)2 2 2(2) 2 8 0 . 28 ᎐4 ᎐6 ᎐8 Check the point (2, 0). The inequality is true. Points above the curve satisfy the inequality, so shade that region. Step 3 ᎐2 ᎐4 ᎐6 ᎐8 The solution to the system of both inequalities is the set of points satisfying both inequalities. In other words, the solution is the region where the graphs overlap. The region contains no boundary points. ᎐8᎐6᎐4᎐2 ᎐4 ᎐6 Exercises Solve each system by graphing. 7. e y , 2x2 1 6 y . x2 2 2 8 y 6 2 ᎐8 ᎐4 9. e y , 2x2 2 2x 1 8 y . 2x2 1 4 2 ᎐8 ᎐4 x ᎐4 ᎐6 ᎐8 y . x2 2 x 2 2 y , 2x2 2 x 1 6 8 y 2 ᎐8 ᎐4 468 ᎐4 ᎐6 ᎐8 6 8. e y 10. e x y . x2 2 6x y , x2 2 6x 1 7 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 90 ᎐4 ᎐6 ᎐8 6 y 2 ᎐8 ᎐4 468 x 468 ᎐4 ᎐6 ᎐8 x 2 68 y x 2 4 6 8 Name Class Date Chapter 4 Quiz 1 Form G Lessons 4-1 through 4-4 Do you know HOW? Graph each function. y 1. y 5 2x2 2 8x 1 3 4 ⫺6 O 8 y 2. y 5 2( x 1 3)2 1 4 4 x ⫺4 O ⫺4 6 ⫺6 4 x 8 ⫺8 3. What are the vertex, axis of symmetry, maximum or minimum value, domain, and range of the function y 5 2x2 2 4x 1 3? vertex (22, 7); axis of symmetry x 5 22; maximum value 7; domain: all real numbers; range: all real numbers K 7 4. Rewrite the equation y 5 4x2 2 4x 2 5 in vertex form. Name the vertex and the axis of symmetry. y 5 4 Q x 2 12 R 2 2 6; vertex Q 12, 26 R ; axis of symmetry x 5 12 Write each expression in factored form. 5. 20 2 t 2 t2 (5 1 t)(4 2 t) or 2(t 1 5)(t 2 4) 6. 25w2 1 30w 1 9 (5w 1 3)2 Find an equation in standard form of the parabola passing through the following points. 7. (0, 24), (1, 1), (2, 8) 8. (21, 1), (0, 5), (2, 7) y 5 x2 1 4x 2 4 y 5 2x2 1 3x 1 5 Write the equation in vertex form for each parabola. y 9. ⫺4 O 10. 2 x y 5 (x 1 3) 2 9 ⫺4 4 2 ⫺2 O ⫺8 y y 5 22(x 2 2)2 1 4 x 2 4 6 ⫺4 Do you UNDERSTAND? 11. Compare and Contrast The x-coordinate of the vertices of the parabolas y1 5 x2 2 3x 1 1 and y2 5 22x2 1 6x 2 2 are the same. Explain why this is so and explain how the y-coordinates of the vertices are related. The values of a and b in the two 6 23 standard forms are proportional, x 5 2 2(22) 5 2 2(1) . Because y2 5 22y1, the y-coordinate of the second parabola is 22 times the y-coordinate of the ﬁrst parabola. 12. Error Analysis A student says that the graph of the parabola y 5 x2 1 1001 is “one thousand times larger” than the parabola y 5 x2 1 1. Explain why this is not correct. “One thousand times larger” would be y 5 1000x2 1 1000. The graph is only shifted 1000 units upward. Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 91 Name Class Date Chapter 4 Quiz 2 Form G Lessons 4-5 through 4-9 Do you know HOW? Complete the square. 1. x2 1 9x 1 81 4 2. 4x2 2 12x 1 9 Evaluate the discriminant of each equation. Determine how many real solutions each equation has. 3. 2x2 2 4x 1 5 5 0 224; 0 4. 6x2 2 11x 2 10 5 0 361; 2 Simplify each expression. 5. (22 1 4i) 1 (7 2 3i) 5 1 i 6. (3 2 5i) 2 (21 1 7i) 4 2 12i 7. (21 1 2i)(3 1 10i) 223 2 4i Solve each equation using the Quadratic Formula. 9. 9x2 1 6x 1 1 5 25 22, 43 8. x2 1 4x 5 6 22 6 !10 10. 2x2 2 x 2 5 5 0 1 6 !41 4 !23 11. x2 1 3x 1 8 5 0 2 32 6 2 i Solve each system of equations. 12. e y 5 2x2 2 2x 1 8 (25, 27), (1, 5) y 5 2x 1 3 13. e y 5 2x2 2 3x 2 1 (21, 4), (6, 53) y 5 x2 1 2x 1 5 Do you UNDERSTAND? 14. Compare and Contrast Compare multiplying two binomials in x with multiplying two complex numbers. They are similar but a product of imaginary numbers may be simpliﬁed using i2 5 21, while x2-terms cannot be further simpliﬁed. 15. a. Open Ended Write a system of two quadratic equations that has exactly one ordered pair as the solution. Answer may vary. Sample: y 5 x2 and y 5 2x2 b. Open Ended Write a system of two quadratic equations that has exactly two ordered pairs as the solution. Answer may vary. Sample: y 5 x2 and y 5 2x2 1 1 c. Reasoning Is it possible for a system of two quadratic equations to have exactly three ordered pairs for the solution? If so, give an example. no Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 92 Name Class Date Chapter 4 Test Form G Do you know HOW? 1. Write the equation of the parabola in standard form. Find the coordinates of the points on the other side of the axis of symmetry corresponding to P and Q. Label these points P9 and Q9, respectively. y 5 2x2 2 6x 2 5; Pr(22, 3), Qr(21, 0) 4 P y 2 Q x ⫺4 ⫺2 O Sketch a graph of the quadratic function with the given vertex and through the given point. 2. vertex (3, 4); point (5, 8) y 8 4 8 y 4 3. vertex (23, 22); point (1, 2) ⫺6 4 8x O Graph each quadratic function. Name the axis of symmetry and the coordinates of the vertex. 4. y 5 x2 1 5 5. y 5 x2 2 4x 2 3 y 8 4 (0, 5) x⫽0 ⫺4 O 4 x 6. y 5 2x2 1 7x 2 2 y 8 4 O y ⫺4 ⫺4 ⫺8 1 7. y 5 2x2 2 6 1 (3 1 , 10 4) 2 1 x ⫽3 2 y 4 O ⫺4⫺4 4 8 x x 68 (2,⫺7) x⫽2 x⫽0 4 x (0, ⫺6) Simplify each expression. 8. (3 1 i) 2 (7 1 6i) 24 2 5i 9. (3 2 4i)(5 1 2i) 23 2 14i 11. 3 !225 1 4 4 1 15i 10. (24 2 9i) 1 (5 2 7i) 1 2 16i Solve each quadratic equation. 12. x2 2 16 5 0 4, 24 13. 2x2 2 3x 2 11 5 0 14. x2 1 3x 2 10 5 0 2, 25 15. 3x2 1 48 5 0 w 4i 3 6 !97 4 16. Anthony has 10 ft of framing and wants to use it to make the largest rectangular picture frame possible. Find the maximum area that can be enclosed by his frame. 614 ft 2 Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 93 ⫺4 x 4 Name Class Date Chapter 4 Test (continued) Form G Write each function in vertex form. Sketch the graph of the function and label its vertex. 17. y 5 x2 1 4x 2 7 y 5 (x 1 2)2 2 11 18. y 5 2x2 1 4x 2 1 y 5 2(x 2 2)2 1 3 y ⫺8 ⫺4 O ⫺4 6 4 2 2x y ⫺6⫺4⫺2O 2 ⫺4 ⫺8 (2, 3) x 6 (ⴚ2, ⴚ11) 1 20. y 5 2x2 2 5x 1 12 y 5 12 (x 2 5)2 2 12 19. y 5 3x2 1 18x y 5 3(x 1 3)2 2 27 y ⫺8 O ⫺8 8 y 10 8 6 4 2 O x ⫺2 2 4 5, ⴚ 1 2 x ⫺16 (ⴚ3, ⴚ27) Evaluate the discriminant of each equation. Determine how many real solutions each equation has. 21. x2 1 5x 1 6 5 0 1; 2 real solutions 22. 3x2 2 4x 1 3 5 0 220; no real solutions 23. 22x2 2 5x 1 4 5 0 57; 2 real solutions 24. 16x2 2 8x 1 1 5 0 0; 1 real solution Solve each system. 25. e y 5 2x2 1 5x 1 1 (21, 25), (4, 5) y 5 2x 2 3 26. e y 5 x2 2 x 1 2 (24, 22), (2, 4) y 5 2x2 1 x 2 6 Solve the following systems of inequalities by graphing. 27. e y , x2 1 2x 2 3 y . x2 2 9 6 y 28. e y . x2 1 3x 2 4 y , 2x2 2 x 1 2 8 y 4 x O ⫺6 6 ⫺6 ⫺6 O ⫺4 x 6 ⫺8 Do you UNDERSTAND? 29. Reasoning Suppose a parabola has a vertex in Quadrant IV and a , 0 in its equation y 5 ax2 1 bx 1 c. How many real solutions will the equation ax2 1 bx 1 c 5 0 have? none 30. Open-Ended Write a complex number with an absolute value between 3 and 8. Answers may vary. Sample: 3 2 4i Prentice Hall Gold Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 94 Name Class Date Chapter 4 Quiz 1 Form K Lessons 4–1 through 4–4 Do you know HOW? Graph each function. Identify the axis of symmetry. 1. y 5 (x 2 3)2 1 1 x 5 3 2. y 5 (x 1 5)2 2 5 x 5 25 y y 6 4 4 O x O ⫺10 ⫺4 ⫺4 2 2 4 6 x ⫺8 Identify the vertex, the axis of symmetry, the maximum or minimum value, and the range of each parabola. 3. y 5 2x2 2 8x 2 19 vertex 5 (24, 23); axis of symmetry x 5 24; maximum 5 23; range 5 all real numbers K 23 4. y 5 x2 2 2x 2 3 vertex 5 (1, 2 4); axis of symmetry x 5 1; minimum 5 2 4; range 5 all real numbers L 4 Find an equation in standard form of the parabola passing through the points. 5. (24, 3), (21, 26), (3, 10) y 5 x2 1 2x 2 5 6. (21, 9), (0, 4), (2, 6) y 5 2x2 2 3x 1 4 Factor each expression. 7. 4x2 1 2x 2x(2x 1 1) 8. t2 1 2t 2 15 (t 1 5)(t 2 3) 9. r 2 1 14r 1 48 (r 1 6)(r 1 8) 10. 8y 2 2 2y 2 1 (4y 1 1)(2y 2 1) Do you UNDERSTAND? 11. Writing Explain how the sign of a in the equation y 5 a(x 2 h)2 1 k tells you whether the parabola has a minimum or a maximum value. If a S 0, the parabola has a minimum value. If a R 0, the parabola has a maximum value. 12. The area of a rectangular garden in square feet is x2 2 5x 2 300. The width is (x – 20) feet. What is the length of the garden? (x 1 15) ft Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 95 Name Class Date Chapter 4 Quiz 2 Form K Lessons 4-5 through 4-9 Do you know HOW? Solve each equation by factoring. Check your answers. 1. x2 2 5x 5 0 x 5 0 and x 5 5 2. x2 1 3x 2 4 5 0 x 5 1 and x 5 24 3. x2 1 10x 5 221 x 5 23 and x 5 27 Solve each quadratic equation by completing the square. 4. x2 1 10x 1 24 5 0 26, 24 5. x2 2 2x 2 2 5 0 1 1 "3, 1 2 "3 6. x2 1 8x 1 11 5 0 24 1 "5, 24 2 "5 8. (2 1 5i)(6 2 i) 17 1 28i 9. Simplify each expression. 7. (8 1 2i) 1 (3 2 4i) 11 2 2i 4 1 3i 9i 1 4 2 3 9i Solve each system by substitution. Check your answers. 10. e y 5 x2 2 2x 2 1 y5x21 (0, 21) and (3, 2) 11. e y 5 23x2 1 x 1 4 y 5 22x 2 2 12. e (21, 0) and (2, 26) y 5 x2 1 4x 2 6 y5x22 (1, 21) and (24, 26) Do you UNDERSTAND? 13. Open-Ended Write a quadratic equation with a discriminant less than 0. What does this tell you about the solution of the equation? Any quadratic equation where b2 2 4ac R 0. When the discriminant is less than zero, the solution is two complex numbers. 14. Writing Explain how you would use the Quadratic Formula to solve 3x2 1 5 5 x 1 9. First, rewrite the equation in ax2 1 bx 1 c 5 0 form. Then determine the values of a, b, and c. Use these in the Quadratic Formula x 5 2b 6 "b2 2 4ac . 2a Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 96 Name Class Date Chapter 4 Test Form K Do you know HOW? Identify the vertex, the axis of symmetry, the maximum or minimum value, and the domain and the range of each function. 1. y 5 3(x 2 2)2 1 6 vertex 5 (2, 6); axis of symmetry x 5 2; minimum 5 6; domain 5 all real numbers; range 5 all real numbers L 6 2. y 5 2(x 1 4)2 23 vertex 5 (24, 23); axis of symmetry x 5 2 4; maximum 5 23; domain 5 all real numbers; range = all real numbers K 23] y 3. y 5 2 x2 2 2x 1 3 4. y 5 3x2 2 4x 1 1 y 6 4 4 O ⫺10 ⫺4 ⫺4 2 Factor each expression. ⫺4 ⫺2 O ⫺8 2 x 5. 4c2 1 4c 1 1 6. g2 2 49 (2c 1 1)2 (g 1 7)(g 27) Use a graphing calculator to solve each equation. Give each answer to at most two decimal places. 7. 5x2 1 9x 1 4 5 0 x 5 2 1 and x 5 2 0.8 8. 23x2 2 2x 1 7 5 0 x 5 1.2 and x 5 21.9 Complete the square. 9. x2 1 14x 1 j 49 10. x2 2 18x 1 j 81 Evaluate the discriminant for each equation. Determine the number of real solutions. 11. 5x2 1 x 1 6 5 0 2119; 0 real solutions 12. 23x2 2 4x 1 1 5 0 28; 2 real solutions Plot each complex number and find its absolute value. 13. 7 2 2i !53 8i imaginary axis 4i 14. 8i 8 8i imaginary axis 4i real axis ⫺8 ⫺4 O ⫺4i 4 ⫺8 ⫺4 O ⫺4i 7 ⫺ 2i ⫺8i real axis 4 ⫺8i Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 97 x Name Class Date Chapter 4 Test (continued) Form K Find all solutions to each quadratic equation. 15. y 5 2x2 2 2x 1 5 3 1 3 1 1 i, 2 i 2 2 2 2 16. y 5 3x2 1 2x 1 4 2 Solve each system by graphing. 17. y . x2 d 6 y 18. y . x2 1 3x 4 y , 2 x2 1 4 d 2 4 2O 2 "11 "11 1 1 1 i, 2 2 i 3 3 3 3 x 2 6 4 y . x2 2 2 2 4 4 y 2O 2 x 2 Do you UNDERSTAND? 19. The parabolic path of a hit tennis ball can be modeled by the table at the right. The top of the net is at (4, 10). a. Find a quadratic model for the data. y 5 20.5x2 1 3x 1 4.5 b. Will the ball go over the net? If not, will it hit the net on the way up or the way down? No; it will hit the net on the way down. 4 x y 1 2 7 7 8.5 1 20. Writing Explain the relationship between the x-intercepts of quadratic function and the zeros of a quadratic function. They are the same thing because the x-intercepts are the x-coordinates where the quadratic function equals zero. 21. The period of a pendulum is the time the pendulum takes to swing back and forth. The function l 5 0.81t2 relates the length l in feet of a pendulum to the period t. a. If a pendulum is 30 ft long, what is the period of the pendulum in seconds? t 5 6.1 s b. Reasoning Why does only one of the solutions work for this problem? The other solution is negative and you cannot have negative time. Prentice Hall Foundations Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 98 Name Class Date Chapter 4 Performance Tasks Task 1 Write your own quadratic function in the standard form y 5 ax2 1 bx 1 c, such that a, b, and c do not equal zero. a. Describe at least two methods that can be used to determine the graph of your function. b. Write your quadratic function in vertex form. c. Find the maximum or minimum value. Explain how you can determine this value. d. Determine the zeros of your function. Give an algebraic reason for the existence or nonexistence of real-valued zeros of your quadratic function. [4] Student describes at least two possible methods that clearly demonstrate an in-depth understanding of the mathematical principles involved. Student writes the function in vertex form correctly. Student ﬁnds the maximum or minimum value and zeros. [3] Student describes at least one possible graphing method that demonstrates understanding of the mathematical principles involved. Further detail or more clarity is needed. Student writes the function in vertex form correctly. Student ﬁnds the maximum or minimum value and zeros. [2] Student does not describe possible graphing methods in sufﬁcient detail to demonstrate understanding of the concepts. Student does not write the function in vertex form correctly. Student ﬁnds the maximum or minimum value and zeros. [1] Student does not write on equation in the correct form. Student does not ﬁnd all values. [0] Response is missing or inappropriate. Task 2 Give complete answers. a. Indicate whether the graph of the equation y 5 x2 2 4x 1 1 opens up, down, left, or right. Explain how you know. opens up; The coefﬁcient of x2 is positive. b. Give the coordinates of the vertex and the equation of the axis of symmetry. (2, 23); x 5 2 c. Sketch the graph of the equation. 2 O ⫺2 y x 2 4 [4] Student correctly identiﬁes the direction in which the parabola opens. Explanation indicates an understanding of the direction. Student identiﬁes the coordinates of the vertex correctly. Equation of the axis of symmetry is written with no errors. Graphs are drawn neatly and labeled accurately. [3] Student correctly identiﬁes the direction in which the parabola opens. Explanation could be more detailed. Student identiﬁes the coordinates of the vertex. Equation of the axis of symmetry has minor errors. Graphs could be more neatly drawn and labeled. [2] Student does not correctly identify the direction in which the parabola opens. Explanation has minor errors. Student identiﬁes the vertex and axis of symmetry with minor errors. Graphs have missing information or minor errors. [1] Student does not correctly identify the direction in which the parabola opens. Explanation is not sufﬁcient or contains major errors. Student does not identify the coordinates of the vertex correctly. Equation of the axis of symmetry is written with signiﬁcant errors. Graphs are not neatly drawn and labeled. [0] Response is missing or inappropriate. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 99 Name Class Date Chapter 4 Performance Tasks (continued) Task 3 Give complete answers. a. Explain when to use each method for solving quadratic equations. Give an example equation for each. • factoring • completing the square • the Quadratic Formula • graphing b. Solve the equation 4x2 2 4x 1 7x 2 7 5 2(x 2 1) by whichever method seems easiest. Explain why you decided to use the method you did. 1, 2 54 c. Give an example of when you would use a quadratic model to model data. Give an example of when you would use a linear model to model data. [4] Student explanation of each method is sufﬁciently detailed to indicate a thorough understanding of the method. Example equations are appropriate. Equation is solved correctly and explanation of method chosen is detailed. Examples of linear and quadratic models are clear and indicate that student understands when each is appropriate. [3] Student explanation of each method is detailed enough to indicate an understanding of each method. Example equations are appropriate. Equation is solved with only minor errors and explanation of method chosen is provided but could have more detail. Examples of linear and quadratic models are clear and indicate that student understands when each is appropriate. [2] Student explanation of each method is not enough to indicate an understanding of each method. Example equations are appropriate. Equation is not solved correctly because of major errors. Explanation of method chosen is provided but does not provide sufﬁcient detail. Examples of linear and quadratic models are not clear and no explanation is included. [1] Student explanation lacks major information. Example equations are not appropriate. Equation is not solved correctly. Explanation of method chosen is missing or has major errors. Examples of linear and quadratic models are missing or no explanation is included. [0] Response is missing or inappropriate. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 100 Name Class Date Chapter 4 Cumulative Review Multiple Choice For Exercises 1−13, choose the correct letter. 3 2 1. Which line is perpendicular to the graph of y 5 22x 1 3 ? C 3 3 y 5 2x 1 6 y 5 3x 1 23 y 5 23x 1 2 6 y 5 3x 1 23 2. What system describes this graph? I 3 2 1 ⫺4 ⫺3 y x O ⫺1 ⫺1 ⫺2 ⫺3 1 2 y$x • x $ 22 x#0 y.x • x . 22 x,0 y5x •x $ 2 x#0 y.x • x $ 22 x#0 3. Which of these is the standard form of y 5 8x 1 12? D y 2 8x 2 12 5 0 28x 1 y 5 12 y 2 8x 5 12 8x 2 y 5 212 4. What is the solution of the system? e y 5 2x2 1 3x 2 3 H y 5 2x 2 5 (0, 23), (2, 21) (21, 27), (2, 21) (21, 27), (3, 23) (21, 210), (3, 2) 5. Which number is irrational? C 27 !144 9 !2 20.5 6. Which point lies on the graph of 2x 2 y 1 z 5 0? I (0, 4, 28) (0, 2, 4) (12, 26, 6) (0, 25, 25) 7. Which of these is the solution of 27x . 4x 1 33? D x.3 x,3 x . 23 x , 23 7 1 3i 3 1 3i 8. Simplify (5 1 6i) 1 (2 2 3i). H 4 7 2 3i 9. At which vertex is the objective function C 5 3x 2 4y maximized? B (9, 0) (22, 220) (25, 22) Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 101 (0, 29) Name Class Date Chapter 4 Cumulative Review (continued) 3x 1 y 1 z 5 7 10. Which ordered triple is a solution to • x 1 3y 2 z 5 13 ? F y 5 2x 2 1 (2, 3, 22) (2, 23, 220) (22, 25, 18) (22, 3, 10) 5 or 25 4 or 21 11. Which is the solution to |x 2 3| 2 1 5 4? A 8 or 22 28 or 2 12. Which number completes the square of the expression 2x2 2 3x 1 ? G 9 8 9 9 16 9 22 13. Which expression is (3 2 2i)(24 1 i) simplified? D 21 2 i 7 2 3i 212 1 2i 210 1 11i Short Response 8 y 14. Find the discriminant of each quadratic equation. a. x2 b. 2x2 1 4x 1 9 5 0 220 y , 2x2 1 x 1 6 15. Solve by graphing. e y . x2 1 x 2 2 1 3x 5 0 9 4 ⫺6 ⫺4 x 6 ⫺8 16. Open-Ended Write a quadratic function in vertex form. Write the equation for the axis of symmetry of the parabola. Answers may vary. Sample: y 5 3(x 2 3)2 1 4, axis of symmetry x 5 3 17. Solve for x: ax 1 by 5 c. x 5 c 2a by x1y2z52 18. Solve the system: • 2x 1 y 1 z 5 3 Q 2, 212, 212 R 3x 1 y 1 z 5 5 Extended Response 19. Jack’s Bowling Alley charges $1.50 to rent shoes and $4.50 for each game bowled. Jill’s Bowling Alley charges $2.50 to rent shoes and $4 for each game bowled. How many games must be bowled in order to make the cost of bowling at Jack’s the same as the cost of bowling at Jill’s? [4] Jack’s Alley: c 5 1.50 1 4.5g; Jill’s Alley: c 5 2.50 1 4g; solve 1.50 1 4.5g 5 2.50 1 4g (OR equivalent equation); 2 games [3] appropriate methods, but with one computational error [2] incorrect equation solved correctly or correct equation solved incorrectly [1] correct number of games, without work shown 20. The length of a rectangle is 5 inches more than its width. The area of the rectangle is 14 square inches. What are the dimensions of the rectangle? [4] l 5 5 1 w , A 5 l ? w 5 14, solve (5 1 w) ? w 5 14 (OR equivalent equation); width: 2 in., length: 7 in. [3] appropriate methods, but with one computational error [2] incorrect equation solved correctly or correct equation solved incorrectly [1] correct width and length, without work shown Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 102 TEACHER INSTRUCTIONS Chapter 4 Project Teacher Notes: On Target About the Project The Chapter Project gives students an opportunity to use quadratic equations in real-life situations. Students use tables and graphs to study the paths of arrows. Students also make a three-dimensional target or a moving video target. Students display the target and their other findings. Introducing the Project • Encourage students to keep all project-related materials in a separate folder. • Ask students to show the path of an arrow if it is aimed horizontally. Ask them how the path changes if the arrow is aimed upward and to name the shape of this path. • Have students look at Activity 4. Encourage them to start this part of the project now so they will have time to create good targets. Activity 1: Graphing Students graph possible parabolic paths for arrows shot while standing or while seated in a wheelchair. Then, they identify the similarities of and differences between their graphs. Activity 2: Analyzing Students find a parabolic model for the path of an arrow. Activity 3: Modeling Students graph data relating the weight of an arrow to its spine, the distance the center of the arrow bends when a constant weight is attached. Then, they decide whether a linear model or a quadratic model is a better fit for the data. Activity 4: Researching Students research new archery styles using three-dimensional targets or moving video targets, and then create their own targets. Finishing the Project You may wish to plan a project day on which students share their completed projects. Encourage students to explain their processes as well as their results. Ask students to review their project work and update their folders. • Have students review their methods for making their graphs, for writing equations to model the graphs, and for creating the targets for the project. • Ask groups to share their insights that resulted from completing the project, such as shortcuts they found for graphing, for modeling, or for making their targets. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 103 Name Class Date Chapter 4 Project: On Target Beginning the Chapter Project Archery has its roots in prehistoric times. Cave drawings in Spain, France, and North Africa show hunters using bows and arrows. In the Far East, people made bows by gluing wood, bone, and animal tendons together. Early Native Americans also used bows and arrows. Archery has also become a recreational sport. In 1988, at the Olympic Games in Barcelona, it was an archer who lit the Olympic flame. In this project, you will research topics such as how archers choose their arrows and how technology has changed the sport. You may want to finish your project by making a display or other presentation. List of Materials • Calculator • Graph paper • Measuring tape Activities Activity 1: Graphing An archer releases an arrow at shoulder height. • Measure the distance from the floor to your shoulder when you are standing. Suppose you release an arrow and it hits the target at a point 5 ft above the ground. Sketch a possible parabolic path of your arrow’s flight using this information. Archery is one of only a few sports in which athletes using wheelchairs can compete with other athletes. • Measure the distance from the floor to your shoulder while you are sitting in a chair. Sketch the possible path of an arrow released by someone using a wheelchair. • Describe the similarities and differences between your two sketches. 4 cm 4 cm 4 cm 4 cm Check students’ work. Activity 2: Analyzing An archer releases an arrow from a shoulder height of 1.39 m. When the arrow hits the target 18 m away, it hits point A. When the target is removed, the arrow lands 45 m away. Find the maximum height of the arrow along its parabolic path. 1.41 m Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 104 8 cm A Name Class Date Chapter 4 Project: On Target (continued) Activity 3: Modeling Archers need to use arrows that do not bend easily. The table shows how the weight of an arrow affects its spine, or the distance the center of the arrow bends when a certain constant weight is attached. Graph the data in the table to find a linear and a quadratic model for the data. Use the regression feature on your calculator to find each model. Which model is a better fit? Explain. Weight (in grams) 140 150 170 175 205 Weight (in inches) 1.4 1.25 0.93 0.78 0.43 linear: y 5 20.0152x 1 3.513 quadratic: y 5 0.00006x2 2 0.0348x 1 5.172 Answers may vary. Sample: The quadratic model is a better ﬁt. The graph of the quadratic function that models the data is closer to more of the points than is the graph of the linear function that models the data. Activity 4: Researching Research the new styles of archery that use three-dimensional targets or moving video targets. Create one of these targets using readily-available materials or a computer program. Check students’ work. Finishing the Project The activities should help you to complete your project. Present your project for this chapter as a visual display, a demonstration or, if equipment is available, as a videotape. Reﬂect and Revise Present your information to a small group of classmates. Decide if your work is complete, clear, and convincing. If needed, make changes to improve your presentation. Extending the Project Interview an archer. Find techniques archers use to increase the range and accuracy of a shot. Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 105 Name Class Date Chapter 4 Project Manager: On Target Getting Started Read the project. As you work on the project, you will need a calculator, a measuring tape, materials on which you can record your calculations, and materials to make accurate and attractive graphs. Keep all of your work for the project in a folder. Checklist ☐ Activity 1: sketching parabolic paths ☐ Activity 2: finding equations ☐ Activity 3: modeling using regression ☐ Activity 4: researching archery ☐ presentation Suggestions ☐ Measure standing and sitting height to your shoulder. ☐ Use given information to write the equation of a parabola. Find the vertex of the parabola. ☐ Determine whether there is an archery club in your area. ☐ How might a videotaped presentation be more useful in studying parabolic paths of arrows than a real-life demonstration? What other objects in sports or real-life situations follow parabolic paths? Scoring Rubric 4 Calculations are correct. Graphs are neat, accurate, and labeled correctly, and they clearly show the differences between the situations. Explanations are thorough and well thought out. The target is designed well and neatly made. The display is well organized. 3 Calculations and explanations are mostly correct, with some minor errors. Graphs are neat and mostly accurate with minor errors in scale. The target is designed adequately, but is not neatly made. The display presents clear information, but is not well organized. 2 Calculations contain both minor and major errors. Graphs are not accurate. Explanations lack detail. The target is poorly designed. 1 Major concepts are misunderstood. Project satisfies few of the requirements and shows poor organization and effort. 0 Major elements of the project are incomplete or missing. Your Evaluation of Project Evaluate your work, based on the Scoring Rubric. Teacher’s Evaluation of Project Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 106

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