Graphing Calculator EL-9650/9600c/9450/9400 Handbook Vol. 1 Algebra EL-9650 EL-9450 Contents 1. Linear Equations 1-1 1-2 Slope and Intercept of Linear Equations Parallel and Perpendicular Lines 2. Quadratic Equations 2-1 2-2 Slope and Intercept of Quadratic Equations Shifting a Graph of Quadratic Equations 3. Literal Equations 3-1 3-2 3-3 Solving a Literal Equation Using the Equation Method (Amortization) Solving a Literal Equation Using the Graphic Method (Volume of a Cylinder) Solving a Literal Equation Using Newton’s Method (Area of a Trapezoid) 4. Polynomials 4-1 4-2 Graphing Polynomials and Tracing to Find the Roots Graphing Polynomials and Jumping to Find the Roots 5. A System of Equations 5-1 Solving a System of Equations by Graphing or Tool Feature 6. Matrix Solutions 6-1 6-2 Entering and Multiplying Matrices Solving a System of Linear Equations Using Matrices 7. Inequalities 7-1 7-2 7-3 7-4 Solving Inequalities Solving Double Inequalities System of Two-Variable Inequalities Graphing Solution Region of Inequalities 8. Absolute Value Functions, Equations, Inequalities 8-1 8-2 8-3 8-4 8-5 Slope and Intercept of Absolute Value Functions Shifting a graph of Absolute Value Functions Solving Absolute Value Equations Solving Absolute Value Inequalities Evaluating Absolute Value Functions 9. Rational Functions 9-1 9-2 Graphing Rational Functions Solving Rational Function Inequalities 10. Conic Sections 10-1 10-2 10-3 10-4 Graphing Parabolas Graphing Circles Graphing Ellipses Graphing Hyperbolas Read this first 1. Always read “Before Starting” The key operations of the set up condition are written in “Before Starting” in each section. It is essential to follow the instructions in order to display the screens as they appear in the handbook. 2. Set Up Condition As key operations for this handbook are conducted from the initial condition, reset all memories to the initial condition beforehand. 2nd F OPTION E 2 CL Note: Since all memories will be deleted, it is advised to use the CE-LK1P PC link kit (sold separately) to back up any programmes not to be erased, or to return the settings to the initial condition (cf. 3. Initial Settings below) and to erase the data of the function to be used. • To delete a single data, press 2nd F OPTION C and select data to be deleted from the menu. • Other keys to delete data: to erase equations and remove error displays CL : to cancel previous function 2nd F QUIT : 3. Initial settings Initial settings are as follows: ✩ Set up ( 2nd F SET UP ): ✩ Format ( 2nd F FORMAT ): ✩ Stat Plot ( 2nd F STAT PLOT E ): ✩ Shade ( 2nd F DRAW G ): ✩ Zoom ( ZOOM A ): ✩ Period ( 2nd F FINANCE C ): Rad, FloatPt, 9, Rect, Decimal(Real), Equation RectCoord, OFF, OFF, Connect, Sequen 2. PlotOFF 2. INITIAL 5. Default 1. PmtEnd Note: ✩ returns to the default setting in the following operation. ( 2nd F OPTION E 1 ENTER ) 4. Using the keys Press 2nd F to use secondary functions (in yellow). To select “sin-1”: 2nd F sin ➔ Displayed as follows: Press ALPHA to use the alphabet keys (in blue). To select A: ALPHA sin ➔ Displayed as follows: 2nd F sin-1 ALPHA A 5. Notes • Some features are provided only on the EL-9650/9600c and not on the EL-9450/9400. (Substitution, Solver, Matrix, Tool etc.) • As this handbook is only an example of how to use the EL-9650/9600c and 9450/9400, please refer to the manual for further details. Using this Handbook This handbook was produced for practical application of the SHARP EL-9650/9600c and EL-9450/9400 Graphing Calculator based on exercise examples received from teachers actively engaged in teaching. It can be used with minimal preparation in a variety of situations such as classroom presentations, and also as a self-study reference book. Introduction Explanation of the section Example Example of a problem to be solved in the section Important notes to read before operating the calculator A quadratic equation of y in terms of x can be expressed by the standard form y = a (x -h)2+ k, where a is the coefficient of the second degree term ( y = ax 2 + bx + c) and ( h, k) is the vertex of the parabola formed by the quadratic equation. An equation where the largest exponent on the independent variable x is 2 is considered a quadratic equation. In graphing quadratic equations on the calculator, let the x- variable be represented by the horizontal axis and let y be represented by the vertical axis. The graph can be adjusted by varying the coefficients a, h, and k. Example EL-9650/9600c Graphing Calculator Graph various quadratic equations and check the relation between the graphs and the values of coefficients of the equations. 1. Graph y = x 2. Graph y = x 3. Graph y = x 4. Graph y = x 2 and y = (x-2)2. 2 and y = x 2+2. 2 and y = 2x 2. 2 and y = -2x 2. Step & Key Operation Display 2-1 Change the equation in Y2 to y = x2+2. Y= 2nd F * 2 0 SUB ENTER There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. -2 View both graphs. 2 As the Substitution feature is only available on the EL-9650/9600c, this section does not apply to the EL-9450/9400. GRAPH Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-1 Enter the equation y = x2 for Y1. 1-2 Enter the equation y = (x-2) 2 for Y2 Y= X/θ/T/n Step & Key Operation ALPHA 2nd F ( 1-3 0 C Notice that the addition of 2 moves the basic y =x2 graph up two units and the addition of -2 moves the basic graph down two units on the y-axis. This demonstrates the fact that adding k (>0) within the standard form y = a (x h)2 + K will move the basic graph up K units and placing k (<0) will move the basic graph down K units on the y axis. x2 3-1 1 EZ Notes *Use either pen touch or cursor to operate. Change the equation in Y2 to y = 2x2. Y= using Sub feature. A clear step-by-step guide to solving the problems Explains the process of each step in the key operations Slope and Intercept of Quadratic Equations ENTER Before Starting Notes EL-9650/9600c Graphing Calculator ENTER SUB 1 ENTER ) ENTER * 1 * 0 3-2 * ENTER ENTER 2 GRAPH Display SUB 2 ENTER View both graphs. * GRAPH ENTER View both graphs. 2nd F ENTER Notice that the addition of -2 within the quadratic operation moves the basic y =x2 graph two unitsin(adding Change right the equation Y2 to 2 moves y = -2x2.it left two units) on the x-axis. This shows that placing an h (>0) within the standard 2 form y = a (x - h) right (-) graph Y=+ k will move 2nd F the 2 SUB basic * h units and placing an h (<0) will move it left h units on the x-axis. ENTER Notice that the multiplication of 2 pinches or closes the basic y=x2 graph. This demonstrates the fact that multiplying an a (> 1) in the standard form y = a (x - h) 2 + k will pinch or close the basic graph. 4-1 4-2 View both graphs. Notice that the multiplication of -2 pinches or closes the basic y =x2 graph and flips it (reflects it) across the x-axis. This demonstrates the fact that multiplying an a (<-1) in the standard form y = a (x - h) 2 + k will pinch or close the basic graph and flip it (reflect it) across the x-axis. 2-1 GRAPH Illustrations of the calculator screen for each step The EL-9650/9600c/9450/9400 allows various quadratic equations to be graphed easily. Also the characteristics of quadratic equations can be visually shown through the relationship between the changes of coefficient values and their graphs, using the Substitution feature. Merits of Using the EL-9650/9600c/9450/9400 2-1 Highlights the main functions of the calculator relevant to the section • When you see the sign on the key: * means same series of key strokes can be done with screen touch on the EL-9650/9600c. * ( : for the corresponding key; : for the corresponding keys underlined.) * * Key operations may also be carried out with the cursor (not shown). • Different key appearance for the EL-9450/9400: for example X/ /T/n ➔ X / T We would like to express our deepest gratitude to all the teachers whose cooperation we received in editing this book. We aim to produce a handbook which is more replete and useful to everyone, so any comments or ideas on exercises will be welcomed. (Use the attached blank sheet to create and contribute your own mathematical problems.) Thanks to Dr. David P. Lawrence at Southwestern Oklahoma State University for the use of his teaching resource book (Applying Pre-Algebra/Algebra using the SHARP EL-9650/9600c Graphing Calculator). Other books available: Graphing Calculator EL-9450/9400 TEACHERS’ GUIDE EL-9650/9600c/9450/9400 Graphing Calculator Slope and Intercept of Linear Equations A linear equation of y in terms of x can be expressed by the slope-intercept form y = mx+b, where m is the slope and b is the y - intercept. We call this equation a linear equation since its graph is a straight line. Equations where the exponents on the x and y are 1 (implied) are considered linear equations. In graphing linear equations on the calculator, we will let the x variable be represented by the horizontal axis and let y be represented by the vertical axis. Example Draw graphs of two equations by changing the slope or the y- intercept. 1. Graph the equations y = x and y = 2x. 2. Graph the equations y = x and y = 12 x. 3. Graph the equations y = x and y = - x. 4. Graph the equations y = x and y = x + 2. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. 1-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Enter the equation y = x for Y1 and y = 2x for Y2. Y= 1-2 Notes X/ /T/n ENTER * 2 X/ /T/n View both graphs. The equation Y1 = x is displayed first, followed by the equation Y2 = 2x. Notice how Y2 becomes steeper or climbs faster. Increase the size of the slope (m>1) to make the line steeper. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-1 Enter the equation y = 12 x for Y2. Y= 1 2-2 * a/b 2 CL * View both graphs. GRAPH X/ /T/n Notice how Y2 becomes less steep or climbs slower. Decrease the size of the slope (0<m<1) to make the line less steep. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 1-1 EL-9650/9600c/9450/9400 Graphing Calculator 3-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Enter the equation y = - x for Y2. Y= 3-2 Notes * CL (-) X/ /T/n View both graphs. Notice how Y2 decreases (going down from left to right) instead of increasing (going up from left to right). Negative slopes (m<0) make the line decrease or go down from left to right. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 4-1 Enter the equation y = x + 2 for Y2. Y= 4-2 * CL View both graphs. GRAPH X/ /T/n + 2 Adding 2 will shift the y = x graph upwards. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Making a graph is easy, and quick comparison of several graphs will help students understand the characteristics of linear equations. 1-1 EL-9650/9600c/9450/9400 Graphing Calculator Parallel and Perpendicular Lines Parallel and perpendicular lines can be drawn by changing the slope of the linear equation and the y intercept. A linear equation of y in terms of x can be expressed by the slopeintercept form y = mx + b, where m is the slope and b is the y-intercept. Parallel lines have an equal slope with different y-intercepts. Perpendicular lines have 1 ). These characteristics can be slopes that are negative reciprocals of each other (m = - m verified by graphing these lines. Example Graph parallel lines and perpendicular lines. 1. Graph the equations y = 3x + 1 and y = 3x + 2. 2. Graph the equations y = 3x - 1 and y = - 31 x + 1. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM 1-1 * ( ENTER ALPHA Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) * ) 7 * Notes Enter the equations y = 3x + 1 for Y1 and y = 3x + 2 for Y2. Y= 3 1-2 C 3 X/ /T/n X/ /T/n + + 1 ENTER * 2 View the graphs. These lines have an equal slope but different y-intercepts. They are called parallel, and will not intersect. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-1 Enter the equations y = 3x - 1 for Y1 and y = - 1 x + 1 for Y2. 3 Y= CL 3 CL (-) 1 + 1 X/ /T/n a/b — 3 1 ENTER * * X/ /T/n 1-2 EL-9650/9600c/9450/9400 Graphing Calculator 2-2 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) View the graphs. GRAPH Notes These lines have slopes that are negative reciprocals of 1 ). They are each other (m = - m called perpendicular. Note that these intersecting lines form four equal angles. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The Graphing Calculators can be used to draw parallel or perpendicular lines while learning the slope or y-intercept of linear equations. 1-2 EL-9650/9600c Graphing Calculator Slope and Intercept of Quadratic Equations A quadratic equation of y in terms of x can be expressed by the standard form y = a (x - h) 2+ k, where a is the coefficient of the second degree term (y = ax 2 + bx + c) and (h, k) is the vertex of the parabola formed by the quadratic equation. An equation where the largest exponent on the independent variable x is 2 is considered a quadratic equation. In graphing quadratic equations on the calculator, let the x-variable be represented by the horizontal axis and let y be represented by the vertical axis. The graph can be adjusted by varying the coefficients a, h, and k. Example Graph various quadratic equations and check the relation between the graphs and the values of coefficients of the equations. 1. Graph y = x 2. Graph y = x 3. Graph y = x 4. Graph y = x 2 and y = (x - 2) 2. and y = x 2 + 2. and y = 2x 2. and y = - 2x 2. 2 2 2 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-1 Enter the equation y = x 2 for Y1. Y= 1-2 X/ /T/n x2 Enter the equation y = (x - 2) 2 for Y2 using Sub feature. 1 EZ ALPHA C 2nd F SUB 1 ENTER ) ( 1-3 0 ENTER ENTER * 1 ENTER View both graphs. GRAPH * ENTER 2 * ENTER Notice that the addition of -2 within the quadratic operation moves the basic y = x 2 graph right two units (adding 2 moves it left two units) on the x-axis. This shows that placing an h (>0) within the standard form y = a (x - h) 2 + k will move the basic graph right h units and placing an h (<0) will move it left h units on the x-axis. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-1 EL-9650/9600c Graphing Calculator Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 2-1 Change the equation in Y2 to y = x 2+2. Y= * 2 ENTER 2-2 2nd F 0 SUB ENTER View both graphs. Notice that the addition of 2 moves the basic y = x 2 graph up two units and the addition of - 2 moves the basic graph down two units on the y-axis. This demonstrates the fact that adding k (>0) within the standard form y = a (x h) 2 + k will move the basic graph up k units and placing k (<0) will move the basic graph down k units on the y-axis. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Change the equation in Y2 to y = 2x 2. Y= * 0 3-2 2nd F SUB 2 ENTER ENTER Notice that the multiplication of 2 pinches or closes the basic y = x 2 graph. This demonstrates the fact that multiplying an a (> 1) in the standard form y = a (x - h) 2 + k will pinch or close the basic graph. View both graphs. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 4-1 Change the equation in Y2 to y = - 2x 2. Y= * 2nd F SUB (-) 2 ENTER 4-2 View both graphs. GRAPH Notice that the multiplication of -2 pinches or closes the basic y =x 2 graph and flips it (reflects it) across the x-axis. This demonstrates the fact that multiplying an a (<-1) in the standard form y = a (x - h) 2 + k will pinch or close the basic graph and flip it (reflect it) across the x-axis. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The EL-9650/9600c allows various quadratic equations to be graphed easily. Also the characteristics of quadratic equations can be visually shown through the relationship between the changes of coefficient values and their graphs, using the Substitution feature. 2-1 EL-9650/9600c/9450/9400 Graphing Calculator Shifting a Graph of Quadratic Equations A quadratic equation of y in terms of x can be expressed by the standard form y = a (x - h) 2 + k, where a is the coefficient of the second degree term (y = ax 2 + bx + c) and (h, k) is the vertex of the parabola formed by the quadratic equation. An equation where the largest exponent on the independent variable x is 2 is considered a quadratic equation. In graphing quadratic equations on the calculator, let the x-variable be represented by the horizontal axis and let y be represented by the vertical axis. The relation of an equation and its graph can be seen by moving the graph and checking the coefficients of the equation. Example Move or pinch a graph of quadratic equation y = x 2 to verify the relation between the coefficients of the equation and the graph. 1. Shift the graph y = x upward by 2. 2. Shift the graph y = x to the right by 3. 3. Pinch the slope of the graph y = x . 2 2 2 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. 1-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Access Shift feature and select the equation y = x 2. 2nd F SHIFT/CHANGE 1 1-2 A * * Move the graph y = x 2 upward by 2. ENTER 1-3 Notes * Save the new graph and observe the changes in the graph and the equation. ENTER ALPHA Notice that upward movement of the basic y = x 2 graph by 2 units in the direction of the yaxis means addition of 2 to the y-intercept. This demonstrates that upward movement of the graph by k units means adding a k (>0) in the standard form y = a (x - h) 2 + k. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-2 EL-9650/9600c/9450/9400 Graphing Calculator 2-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Move the graph y = x 2 to the right by 3. CL 2-2 Notes (three times) ENTER * Save the new graph and observe the changes in the graph and the equation ENTER ALPHA Notice that movement of the basic y = x2 graph to the right by 3 units in the direction of the x-axis is equivalent to the addition of 3 to the x -intercept. This demonstrates that movement of the graph to the right means adding an h (>0) in the standard form y = a (x - h) 2 + k and movement to the left means subtracting an h (<0). ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Access Change feature and select the equation y = x 2. 2nd F SHIFT/CHANGE 1 3-2 B * * Pinch the slope of the graph. ENTER 3-3 Save the new graph and observe the changes in the graph and the equation. ENTER ALPHA Notice that pinching or closing the basic y = x 2 graph is equivalent to increasing an a (>1) within the standard form y = a (x - h) 2 + k and broadening the graph is equivalent to decreasing an a (<1). ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The Shift/Change feature of the EL-9650/9600c/9450/9400 allows visual understanding of how graph changes affect the form of quadratic equations. 2-2 EL-9650/9600c Graphing Calculator Solving a Literal Equation Using the Equation Method (Amortization) The Solver mode is used to solve one unknown variable by inputting known variables, by three methods: Equation, Newton’s, and Graphic. The Equation method is used when an exact solution can be found by simple substitution. Example Solve an amortization formula. The solution from various values for known variables can be easily found by giving values to the known variables using the Equation method in the Solver mode. The formula : P = L I -N 1-(1+ 12 ) I / 12 -1 P= monthly payment L= loan amount I= interest rate N=number of months 1. Find the monthly payment on a $15,000 car loan, made at 9% interest over four 2. 3. years (48 months) using the Equation method. Save the formula as “AMORT”. Find amount of loan possible at 7% interest over 60 months with a $300 payment, using the saved formula. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-1 Access the Solver feature. This screen will appear a few seconds after “SOLVER” is displayed. 2nd F SOLVER 1-2 Select the Equation method for solving. A 2nd F SOLVER 1 1-3 * * Enter the amortization formula. 2nd F SOLVER P = a/b 1 — ( ALPHA ab ALPHA ab 1 (-) I (-) a/b 1 ALPHA N ÷ 1 L ALPHA ( 2 + 1 ) * * 2 * ) 1 3-1 EL-9650/9600c Graphing Calculator Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-4 Enter the values L=15,000, I=0.09, N=48. ENTER ENTER 8 1-5 1 * • * 0 5 9 0 0 ENTER * 0 4 ENTER The monthly payment (P) is $373.28. Solve for the payment(P). * ( CL 2nd F EXE ) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-1 Save this formula. 2nd F SOLVER 2-2 C * ENTER * Give the formula the name AMORT. A M O R T ENTER ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Recall the amortization formula. B 2nd F SOLVER 0 3-2 3-3 1 * * Enter the values: P = 300, I = 0.01, N = 60 ENTER 3 0 • 1 ENTER 0 0 ENTER * 6 Solve for the loan (L). * 2nd F 1 0 ENTER * ENTER The amount of loan (L) is $17550.28. EXE ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ With the Equation Editor, the EL-9650/9600c displays equations, even complicated ones, as they appear in the textbook in easy to understand format. Also it is easy to find the solution for unknown variables by recalling a stored equation and giving values to the known variables in the Solver mode when using the EL-9650/9600c. 3-1 EL-9650/9600c Graphing Calculator Solving a Literal Equation Using the Graphic Method (Volume of a Cylinder) The Solver mode is used to solve one unknown variable by inputting known variables. There are three methods: Equation, Newton’s, and Graphic. The Equation method is used when an exact solution can be found by simple substitution. Newton’s method implements an iterative approach to find the solution once a starting point is given. When a starting point is unavailable or multiple solutions are expected, use the Graphic method. This method plots the left and right sides of the equation and then locates the intersection(s). Example Use the Graphic method to find the radius of a cylinder giving the range of the unknown variable. The formula : V = πr 2h ( V = volume r = radius h = height) 1. Find the radius of a cylinder with a volume of 30in 3 2. 3. and a height of 10in, using the Graphic method. Save the formula as “V CYL”. Find the radius of a cylinder with a volume of 200in 3 and a height of 15in, using the saved formula. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-1 Access the Solver feature. This screen will appear a few seconds after “SOLVER” is displayed. 2nd F SOLVER 1-2 Select the Graphic method for solving. 2nd F SOLVER 3 1-3 * * Enter the formula V = πr 2h. V ALPHA R 1-4 A x2 = ALPHA ALPHA 2nd F π ALPHA H Enter the values: V = 30, H = 10. Solve for the radius (R). ENTER 0 3 ENTER 0 ENTER * * 2nd F * 1 EXE 3-2 EL-9650/9600c Graphing Calculator Step & Key Operation Notes Display *Use either pen touch or cursor to operate. 1-5 Set the variable range from 0 to 2. 0 ENTER 2 * ENTER The graphic solver will prompt with a variable range for solving. 30 3 = <3 π 10π r =1 ➞ r 2 = 12 = 1 <3 r =2 ➞ r 2 = 22 = 4 >3 r2= Use the larger of the values to be safe. 1-6 The solver feature will graph the left side of the equation (volume, y = 30), then the right side of the equation (y = 10r 2), and finally will calculate the intersection of the two graphs to find the solution. The radius is 0.98 in. Solve. 2nd F ( EXE CL ) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2 Save this formula. Give the formula the name “V CYL”. C 2nd F SOLVER V SPACE C ENTER * Y L * ENTER ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Recall the formula. Enter the values: V = 200, H = 15. 2nd F SOLVER 3-2 B ENTER 2 1 ENTER 5 0 0 * 0 1 * ENTER 0 ENTER Solve the radius setting the variable range from 0 to 4. * ENTER 2nd F 2nd F EXE EXE 0 ENTER 4 200 14 = < 14 π 15π 2 r = 3 ➞ r = 32 = 9 < 14 r = 4 ➞ r 2 = 42 = 16 > 14 r2= Use 4, the larger of the values, to be safe. The answer is : r = 2.06 ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ One very useful feature of the calculator is its ability to store and recall equations. The solution from various values for known variables can be easily obtained by recalling an equation which has been stored and giving values to the known variables. The Graphic method gives a visual solution by drawing a graph. 3-2 EL-9650/9600c Graphing Calculator Solving a Literal Equation Using Newton's Method (Area of a Trapezoid) The Solver mode is used to solve one unknown variable by inputting known variables. There are three methods: Equation, Newton’s, and Graphic. The Newton’s method can be used for more complicated equations. This method implements an iterative approach to find the solution once a starting point is given. Example Find the height of a trapezoid from the formula for calculating the area of a trapezoid using Newton’s method. 1 The formula : A= h(b+c) (A = area h = height b = top face c = bottom face) 2 1. Find the height of a trapezoid with an area of 25in 2 2. 3. and bases of length 5in and 7in using Newton's method. (Set the starting point to 1.) Save the formula as “A TRAP”. Find the height of a trapezoid with an area of 50in2 with bases of 8in and 10in using the saved formula. (Set the starting point to 1.) There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-1 Access the Solver feature. This screen will appear a few seconds after “SOLVER” is displayed. 2nd F SOLVER 1-2 Select Newton's method for solving. A 2nd F SOLVER 2 1-3 * Enter the formula A = 21 h(b+c). ALPHA A ALPHA = 1 ALPHA H ( ALPHA B a/b + 2 * ALPHA ) C 1-4 * Enter the values: A = 25, B = 5, C = 7 ENTER * 2 5 5 ENTER ENTER * 7 * ENTER 3-3 EL-9650/9600c Graphing Calculator Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-5 Solve for the height and enter a starting point of 1. 2nd F * 1-6 1 EXE ENTER The answer is : h = 4.17 Solve. 2nd F Newton's method will prompt with a guess or a starting point. EXE ( ) CL ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2 Save this formula. Give the formula the name “A TRAP”. C 2nd F SOLVER A SPACE * ENTER A R T P ENTER ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Recall the formula for calculating the area of a trapezoid. B 2nd F SOLVER 0 3-2 3-3 * 1 Enter the values: A = 50, B = 8, C = 10. ENTER 5 0 ENTER ENTER 1 0 ENTER * * Solve. The answer is : h = 5.56 * ENTER 8 2nd F 2nd F EXE 1 EXE ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ One very useful feature of the calculator is its ability to store and recall equations. The solution from various values for known variables can be easily obtained by recalling an equation which has been stored and giving values to the known variables in the Solver mode. If a starting point is known, Newton's method is useful for quick solution of a complicated equation. 3-3 EL-9650/9600c Graphing Calculator Graphing Polynomials and Tracing to Find the Roots A polynomial y = f (x) is an expression of the sums of several terms that contain different powers of the same originals. The roots are found at the intersection of the x-axis and the graph, i. e. when y = 0. Example Draw a graph of a polynomial and approximate the roots by using the Zoom-in and Trace features. 1. Graph the polynomial y = x - 3x 2. Approximate the left-hand root. 3. Approximate the middle root. 4. Approximate the right-hand root. 3 2 + x + 1. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM A ( ENTER ALPHA * Setting the zoom factors to 5 : ZOOM B Step & Key Operation * ENTER 5 ENTER Display * 5 ) 7 ENTER * 2nd F QUIT Notes *Use either pen touch or cursor to operate. 1-1 Enter the polynomial y = x 3 - 3x 2 + x + 1. Y= 5 1-2 EZ ENTER * * ENTER * It may take few seconds for the graph to be drawn. Enter each coefficients when the cursor is displayed. SUB 1 ENTER 1 ENTER * * (-) 1 3 ENTER * ENTER Return to the equation display screen. 2nd F 1-4 ENTER Enter the coefficients. 2nd F 1-3 * EXE View the graph. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 4-1 EL-9650/9600c Graphing Calculator Step & Key Operation Notes Display *Use either pen touch or cursor to operate. 2-1 Tracer TRACE 2-2 * (repeatedly) Zoom in on the left-hand root. A ZOOM 2-3 Note that the tracer is flashing on the curve and the x and y coordinates are shown at the bottom of the screen. Move the tracer near the left-hand root. * 3 * Tracer Move the tracer to approximate the root. TRACE * or * The root is : x -0.42 (repeatedly) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Return to the previous decimal viewing window. H ZOOM 2 3-2 * Tracer * Move the tracer to approximate the middle root. TRACE * The root is exactly x = 1. (Zooming is not needed to find a better approximate.) (repeatedly) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 4 Tracer Move the tracer near the righthand root. Zoom in and move the tracer to find a better approximate. * ZOOM TRACE The root is : x 2.42 (repeatedly) A * 3 * * or * (repeatedly) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The calculator allows the roots to be found (or approximated) visually by graphing a polynomial and using the Zoom-in and Trace features. 4-1 EL-9650/9600c/9450/9400 Graphing Calculator Graphing Polynomials and Jumping to Find the Roots A polynomial y = f (x) is an expression of the sums of several terms that contain different powers of the same originals. The roots are found at the intersection of the x- axis and the graph, i. e. when y = 0. Example Draw a graph of a polynomial and find the roots by using the Calculate feature. 1. Graph the polynomial y = x + x 2. Find the four roots one by one. 4 3 - 5x 2 - 3x + 1. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Setting the zoom factors to 5 : ZOOM 1-1 * ENTER A ENTER Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) A ENTER 2nd F QUIT Notes Enter the polynomial y = x 4 + x 3 - 5x 2 - 3x + 1 Y= X/ /T/n ab ab 3 — 1-2 A * 4 — + 3 X/ /T/n * + 5 X/ /T/n X/ /T/n x2 1 View the graph. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-1 Find the first root. 2nd F CALC 5 * x -2.47 Y is almost but not exactly zero. Notice that the root found here is an approximate value. 2-2 Find the next root. 2nd F CALC 5 x -0.82 * 4-2 EL-9650/9600c/9450/9400 Graphing Calculator 2-3 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Find the next root. 2nd F CALC 2-4 5 5 x 0.24 x 2.05 * Find the next root. 2nd F CALC Notes * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The calculator allows jumping to find the roots by graphing a polynomial and using the Calculate feature, without tracing the graph. 4-2 EL-9650/9600c/9450/9400 Graphing Calculator Solving a System of Equations by Graphing or Tool Feature A system of equations is made up of two or more equations. The calculator provides the Calculate feature and Tool feature to solve a system of equations. The Calculate feature finds the solution by calculating the intersections of the graphs of equations and is useful for solving a system when there are two variables, while the Tool feature can solve a linear system with up to six variables and six equations. Example Solve a system of equations using the Calculate or Tool feature. First, use the Calculate feature. Enter the equations, draw the graph, and find the intersections. Then, use the Tool feature to solve a system of equations. 1. Solve the system using the Calculate feature. { y = x2- 1 y = 2x 2. Solve the system using the Tool feature. { 5x + y = 1 -3x + y = -5 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default and delete all data. Choose the viewing window “-5 < X < 5”, “-10 < Y < 10” using Rapid window feature ) 7 WINDOW EZ 5 ENTER ( ALPHA ENTER 4 ENTER * * * * * * As the Tool feature is only available on the EL-9650/9600c, the example 2 does not apply to the EL-9450/9400. 1-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Notes Enter the system of equations y = x 2 - 1 for Y1 and y = 2x for Y2. Y= X/ /T/n x2 — 1 ENTER * 2 X/ /T/n 1-2 View the graphs. GRAPH 1-3 The find the left-hand intersection using the Calculate feature. 2nd F CALC 1-4 2 * Find the right-hand intersection by accessing the Calculate feature again. 2nd F CALC 2 Note that the x and y coordinates are shown at the bottom of the screen. The answer is : x = - 0.41 y = - 0.83 The answer is : x = 2.41 y = 4.83 * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 5-1 EL-9650/9600c/9450/9400 Graphing Calculator 2-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Access the Tool menu. Select the number of variables. 2nd F 2-2 TOOL B * 2 Notes Using the system function, it is possible to solve simultaneous linear equations. Systems up to six variables and six equations can be solved. * Enter the system of equations. 5 ENTER 1 ENTER 1 3 ENTER 1 ENTER (-) ENTER (-) 5 ENTER 2-3 Solve the system. 2nd F x = 0.75 y = - 2.75 EXE ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ A system of equations can be solved easily by using the Calculate feature or Tool feature. 5-1 EL-9650/9600c Graphing Calculator Entering and Multiplying Matrices A matrix is a rectangular array of elements in rows and columns that is treated as a single element. A matrix is often used for expressing multiple linear equations with multiple variables. Example Enter two matrices and execute multiplication of the two. A Enter a 3 x 3 matrix A 1 2 1 1 Enter a 3 x 3 matrix B 2 1 -1 4 1 1 -2 7 Multiply the matrices A and B 1. 2. 3. B 2 3 5 6 8 9 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-1 Access the matrix menu. MATRIX 1 1-2 * * Set the dimension of the matrix at three rows by three columns. 3 1-3 B ENTER 3 ENTER Enter the elements of the first row, the elements of the second row, and the elements of the third row. 1 ENTER 2 ENTER 1 2 ENTER 1 ENTER (-) 1 ENTER 1 ENTER 1 ENTER (-) 2 ENTER ENTER ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 6-1 EL-9650/9600c Graphing Calculator Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 2 Enter a 3 x 3 matrix B. MATRIX B * 2 3 * ENTER 3 ENTER 1 ENTER 2 ENTER 3 ENTER 4 ENTER 5 ENTER 6 ENTER 7 ENTER 8 ENTER 9 ENTER ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Multiply the matrices A and B together at the home screen. MATRIX A 3-2 * 2 * A * 1 * X MATRIX ENTER Matrix multiplication can be performed if the number of columns of the first matrix is equal to the number of rows of the second matrix. The sum of these multiplications (1 1 + 2 4 + 1 7) is placed in the 1,1 (first row, first column) position of the resulting matrix. This process is repeated until each row of A has been multiplied by each column of B. . . . Delete the input matrices for future use. 2nd F OPTION 2 * 2nd F ENTER C * ENTER QUIT ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Matrix multiplication can be performed easily by the calculator. 6-1 EL-9650/9600c Graphing Calculator Solving a System of Linear Equations Using Matrices Each system of three linear equations consists of three variables. Equations in more than three variables cannot be graphed on the graphing calculator. The solution of the system of equations can be found numerically using the Matrix feature or the System solver in the Tool feature. A system of linear equations can be expressed as AX = B (A, X and B are matrices). The solution matrix X is found by multiplying A-1 B. Note that the multiplication is “order sensitive” and the correct answer will be obtained by multiplying BA-1. An inverse matrix A-1 is a matrix that when multiplied by A results in the identity matrix I (A-1 x A=I). The identity matrix I is defined to be a square matrix (n x n) where each position on the diagonal is 1 and all others are 0. Example Use matrix multiplication to solve a system of linear equations. B Enter the 3 x 3 identity matrix in matrix A. 1 2 1 Find the inverse matrix of the matrix B. 2 1 -1 Solve the equation system. 1 1 -2 x + 2y + z = 8 2x + y - z = 1 x + y - 2z = -3 1. 2. 3. { There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-1 Set up 3 x 3 identity matrix at the home screen. MATRIX 1-2 * 0 5 * 3 ENTER Save the identity matrix in matrix A. STO 1-3 C MATRIX A * 1 * ENTER Confirm that the identity matrix is stored in matrix A. MATRIX B * 1 * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 6-2 EL-9650/9600c Graphing Calculator Step & Key Operation Notes Display *Use either pen touch or cursor to operate. 2-1 Enter a 3 x 3 matrix B. MATRIX 2-2 B * 2 3 * 3 ENTER ENTER 1 ENTER 2 ENTER 1 2 ENTER 1 ENTER (-) 1 ENTER 1 ENTER 1 ENTER (-) 2 ENTER ENTER Exit the matrix editor and find the inverse of the square matrix B. 2nd F QUIT MATRIX A * Some square matrices have no inverse and will generate error statements when calculating the inverse. CL 2 x-1 2nd F * - 0.17 0.83 - 0.5 0.5 B = 0.5 - 0.5 0.17 0.17 - 0.5 ENTER -1 (repeatedly) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Enter the constants on the right side of the equal sign into matrix C (3 x 1). MATRIX 8 B * ENTER 3 3 * 1 1 ENTER ENTER (-) ENTER ENTER 3 The system of equations can be expressed as 1 2 1 2 1 -1 1 1 -2 x y z = 8 1 -3 Let each matrix B, X, C : BX = C B-1BX = B-1C (multiply both sides by B-1) I = B-1 (B-1B = I, identity matrix) X = B-1 C 3-2 Calculate B-1C. CL 2nd F 3-3 x -1 MATRIX X A * MATRIX 2 A * * 3 * ENTER The 1 is the x coordinate, the 2 the y coordinate, and the 3 the z coordinate of the solution point. (x, y, z)=(1, 2, 3) Delete the input matrices for future use. 2nd F OPTION 2 * C * ENTER 2nd F QUIT ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The calculator can execute calculation of inverse matrix and matrix multiplication. A system of linear equations can be solved easily using the Matrix feature. 6-2 EL-9650/9600c Graphing Calculator Solving Inequalities To solve an inequality, expressed by the form of f (x) ≤ 0, f (x) ≥ 0, or form of f (x) ≤ g (x), f (x) ≥ g (x), means to find all values that make the inequality true. There are two methods of finding these values for one-variable inequalities, using graphical techniques. The first method involves rewriting the inequality so that the right-hand side of the inequality is 0 and the left-hand side is a function of x. For example, to find the solution to f (x) < 0, determine where the graph of f (x) is below the x-axis. The second method involves graphing each side of the inequality as an individual function. For example, to find the solution to f (x) < g(x), determine where the graph of f (x) is below the graph of g (x). Example Solve an inequality in two methods. 1. Solve 3(4 - 2x) ≥ 5 - x, by rewriting the right-hand side of the inequality as 0. 2. Solve 3(4 - 2x) ≥ 5 - x, by shading the solution region that makes the inequality true. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 1-1 1-2 Rewrite the equation 3(4 - 2x) ≥ 5 - x so that the right-hand side becomes 0, and enter y = 3(4 - 2x) - 5 + x for Y1. Y= 3 ( — 5 + 4 — 2 X/ /T/n 3(4 - 2x) ≥ 5 - x ➞ 3(4 - 2x) - 5 + x ≥ 0 ) X/ /T/n View the graph. GRAPH 1-3 Find the location of the x-intercept and solve the inequality. 2nd F CALC 5 * The x-intercept is located at the point (1.4, 0). Since the graph is above the x-axis to the left of the x-intercept, the solution to the inequality 3(4 - 2x) - 5 + x ≥ 0 is all values of x such that x ≤ 1.4. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 7-1 EL-9650/9600c Graphing Calculator Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 2-1 Enter y = 3(4 - 2x) for Y1 and y = 5 - x for Y2. Y= ENTER 2-2 * * (7 times) DEL — 5 (4 times) X/ /T/n View the graph. GRAPH 2-3 Access the Set Shade screen. G 2nd F DRAW 1 2-4 * Set up the shading. — 2-5 * — * * — * Since the inequality being solved is Y1 ≥ Y2, the solution is where the graph of Y1 is “on the top” and Y2 is “on the bottom.” View the shaded region. GRAPH 2-6 Find where the graphs intersect and solve the inequality. 2nd F CALC 2 * The point of intersection is (1.4, 3.6). Since the shaded region is to the left of x = 1.4, the solution to the inequality 3(4 - 2x) ≥ 5 - x is all values of x such that x ≤ 1.4. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The EL-9650/9600c allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily. 7-1 EL-9650/9600c/9450/9400 Graphing Calculator Solving Double Inequalities The solution to a system of two inequalities in one variable consists of all values of the variable that make each inequality in the system true. A system f (x) ≥ a, f (x) ≤ b, where the same expression appears on both inequalities, is commonly referred to as a “double” inequality and is often written in the form a ≤ f (x) ≤ b. Be certain that both inequality signs are pointing in the same direction and that the double inequality is only used to indicate an expression in x “trapped” in between two values. Also a must be less than or equal to b in the inequality a ≤ f (x) ≤ b or b ≥ f (x) ≥ a. Example Solve a double inequality, using graphical techniques. 2x - 5 ≥ -1 2x -5 ≤ 7 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. 1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Enter y = -1 for Y1, y = 2x - 5 for Y2, and y = 7 for Y3. (-) Y= 2 2 X/ /T/n ENTER 1 — 5 Notes The “double” inequality given can also be written to -1 ≤ 2x - 5 ≤ 7. * ENTER * 7 View the lines. GRAPH 3 Find the point of intersection. 2nd F CALC 2 y = 2x - 5 and y = -1 intersect at (2, -1). * 7-2 EL-9650/9600c/9450/9400 Graphing Calculator 4 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Move the tracer and find another intersection. 2nd F CALC 5 2 Notes y = 2x - 5 and y = 7 intersect at (6,7). * Solve the inequalities. The solution to the “double” inequality -1 ≤ 2x - 5 ≤ 7 consists of all values of x in between, and including, 2 and 6 (i.e., x ≥ 2 and x ≤ 6). The solution is 2 ≤ x ≤ 6. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The EL-9650/9600c/9450/9400 allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily. 7-2 EL-9650/9600c/9450/9400 Graphing Calculator System of Two-Variable Inequalities The solution region of a system of two-variable inequalities consists of all points (a, b) such that when x = a and y = b, all inequalities in the system are true. To solve two-variable inequalities, the inequalities must be manipulated to isolate the y variable and enter the other side of the inequality as a function. The calculator will only accept functions of the form y = . (where y is defined explicitly in terms of x). Example Solve a system of two-variable inequalities by shading the solution region. 2x + y ≥ 1 x2 + y ≤ 1 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM Display (When using EL-9650/9600c) 2 Enter y = 1 - 2x for Y1 and y = 1 - x 2 for Y2. Y= 1 — 1 — 2 X/ /T/n X/ /T/n ENTER * ) 7 Notes 2x + y ≥ 1 ➞ y ≥ 1 - 2x x2 + y ≤ 1 ➞ y ≤ 1 - x2 * x2 Access the set shade screen G 2nd F DRAW 1 * * Shade the points of y -value so that Y1 ≤ y ≤ Y2. — 5 ( ENTER 2nd F (When using EL-9650/9600c) *Use either pen touch or cursor to operate. Rewrite each inequality in the system so that the left-hand side is y : 4 * Step & Key Operation 1 3 A * * — — * Graph the system and find the intersections. The intersections are (0, 1) and (2, -3) GRAPH 2nd F CALC 6 2 * 2nd F CALC Solve the system. 2 * The solution is 0 ≤ x ≤ 2. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The EL-9650/9600c/9450/9400 allows the solution region to be indicated visually using the Shade feature. Also, the points of intersection can be obtained easily. 7-3 EL-9650/9600c/9450/9400 Graphing Calculator Graphing Solution Region of Inequalities The solution region of an inequality consists of all points (a, b) such that when x = a, and y = b, all inequalities are true. Example Check to see if given points are in the solution region of a system of inequalities. 1. Graph the solution region of a system of inequalities: 2. x + 2y ≤ 1 x2 + y ≥ 4 Which of the following points are within the solution region? (-1.6, 1.8), (-2, -5), (2.8, -1.4), (-8,4) There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. 1-1 1-2 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Rewrite the inequalities so that the left-hand side is y. a/b 2 X/ /T/n — 1 ENTER X/ /T/n — 4 * x2 Set the shade and view the solution region. G 2nd F DRAW — — * * 1 * — * GRAPH 2-1 Set the display area (window) to : -9 < x < 3, -6 < y < 5. WINDOW ENTER 7-4 x + 2y ≤ 1 ➞ y ≤ 1-x 2 x 2+y ≥ 4 ➞ y ≥ 4 - x 2 Enter y = 1-x for Y1 and 2 y = 4 - x 2 for Y2. Y= 1-3 Notes (-) (-) 9 ENTER 6 ENTER 3 ENTER 5 ENTER Y2 ≤ y ≤ Y1 EL-9650/9600c/9450/9400 Graphing Calculator 2-2 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Use the cursor to check the position of each point. (Zoom in as necessary). or GRAPH 2-3 or or Substitute points and confirm whether they are in the solution region. (-) 2 X 1 1 • • 6 8 ... + (Continuing key operations omitted.) Notes Points in the solution region are (2.8, -1.4) and (-8, 4). Points outside the solution region are (-1.6, 1.8) and (-2, -5). .(-1.6, 1.8): -1.6 + 2 1.8 = 2 ➞ This does not materialize. .(-2, -5): -2 + 2 (-5) = -12 (-2) + (-5) = -1 ➞ This does not materialize. .(2.8, -1.4): 2.8 + 2 (-1.4) = 0 (2.8) + (-1.4) = 6.44 ➞ This materializes. .(-8, 4): -8 + 2 4 = 0 ✕ ✕ 2 ✕ 2 ✕ (-8)2 + 4 = 68 ➞ This materializes. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often very difficult to solve algebraically. The EL-9650/9600c/9450/9400 allows the solution region to be indicated visually using the Shading feature. Also, the free-moving tracer or Zoom-in feature will allow the details to be checked visually. 7-4 EL-9650/9600c Graphing Calculator Slope and Intercept of Absolute Value Functions The absolute value of a real number x is defined by the following: |x| = x if x ≥ 0 -x if x ≤ 0 If n is a positive number, there are two solutions to the equation |f (x)| = n because there are exactly two numbers with the absolute value equal to n: n and -n. The existence of two distinct solutions is clear when the equation is solved graphically. An absolute value function can be presented as y = a|x - h| + k. The graph moves as the changes of slope a, x-intercept h, and y-intercept k. Example Consider various absolute value functions and check the relation between the graphs and the values of coefficients. 1. Graph y = |x| 2. Graph y = |x -1| and y = |x|-1 using the Rapid Graph feature. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM Step & Key Operation A * ( ENTER 2nd F Display * ) 7 Notes *Use either pen touch or cursor to operate. 1-1 Enter the function y =|x| for Y1. Y= 1-2 MATH B * 1 * X/ /T/n Notice that the domain of f(x) = |x| is the set of all real numbers and the range is the set of non-negative real numbers. Notice also that the slope of the graph is 1 in the range of X > 0 and -1 in the range of X ≤ 0. View the graph. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-1 Enter the standard form of an absolute value function for Y2 using the Rapid Graph feature. Y= EZ ENTER 2-2 * ENTER * ENTER * Substitute the coefficients to graph y = |x - 1|. 2nd F 0 8-1 8 SUB ENTER 1 ENTER 1 ENTER * EL-9650/9600c Graphing Calculator Step & Key Operation Display Notes *Use either pen touch or cursor to operate. 2-3 View the graph. Notice that placing an h (>0) within the standard form y = a|x - h|+ k will move the graph right h units on the xaxis. GRAPH 2-4 Change the coefficients to graph y =|x|-1. Y= ENTER 2-5 2nd F (-) 1 View the graph. GRAPH SUB ENTER 1 ENTER Notice that adding a k (>0) within the standard form y=a|x-h|+k will move the graph up k units on the y-axis. The EL-9650/9600c shows absolute values with | |, just as written on paper, by using the Equation editor. Use of the calculator allows various absolute value functions to be graphed quickly and shows their characteristics in an easy-to-understand manner. 8-1 EL-9650/9600c/9450/9400 Graphing Calculator Shifting a graph of Absolute Value Functions The absolute value of a real number x is defined by the following: |x| = x if x ≥ 0 -x if x ≤ 0 If n is a positive number, there are two solutions to the equation |f (x)| = n because there are exactly two numbers with the absolute value equal to n: n and -n. The existence of two distinct solutions is clear when the equation is solved graphically. An absolute value function can be presented as y = a|x - h|+ k. The graph moves as the changes of slope a, x-intercept h, and y-intercept k. Example Move and change graphs of absolute value function y =|x| to check the relation between the graphs and the values of coefficients. 1. Move the graph y = |x| downward by 2 using the Shift feature. 2. Move the graph y = |x| to the right by 2 using the Shift feature. 3. Pinch the slope of y = |x| to 2 or minus using the Change feature. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. 1-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Access the Shift feature. Select y = |x|. 2nd F SHIFT/CHANGE ( 1-2 ENTER A * ) 8 * Move the graph downward by 2. ALPHA y =|x|changes to y = |x|-2 * Save the new graph and look at the relationship of the function and the graph. ENTER 8-2 * ALPHA ENTER 1-3 Notes The graph of the equation that is highlighted is shown by a solid line. Notice that the yintercept k in the standard form y = a|x - h|+ k takes charge of vertical movement of the graph. EL-9650/9600c/9450/9400 Graphing Calculator 2-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Move the original graph to the right by 2. ALPHA 2-2 ENTER * ALPHA * y = |x| changes to y = |x-2| * Save the new graph and look at the relationship of the function and the graph. ENTER Notes * Notice that the function h in the standard form y = a|x - h|+ k takes charge of horizontal movement of the graph. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Access the Change feature. 2nd F SHIFT/CHANGE 3-2 * Select y = |x|. 3 3-3 B * Make the slope of the graph steeper. Save the new graph. y = |x|➞ y = 2|x| ENTER ENTER 3-4 Make the slope of the graph minus. Save the new graph. y = |x|➞ y = - |x| ENTER ENTER 3-5 Look at the relationship of the function and the graph. ALPHA * * Notice that the coefficient a in the standard form y = a |x - h| + k takes charge of changing the slope. * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ EL-9650/9600c/9450/9400 shows absolute values with | |, just as written on paper, by using the Equation editor. Use of the calculator allows various absolute value functions to be graphed quickly and shows their characteristics in an easy-to-understand manner. The Shift/Change feature of the EL-9650/9600c/9450/9400 allows visual understanding of how graph changes affect the form of absolute value functions. 8-2 EL-9650/9600c/9450/9400 Graphing Calculator Solving Absolute Value Equations The absolute value of a real number x is defined by the following: x if x ≥ 0 -x if x ≤ 0 |x| = If n is a positive number, there are two solutions to the equation |f (x)| = n because there are exactly two numbers with the absolute value equal to n: n and -n. The existence of two distinct solutions is clear when the equation is solved graphically. Example Solve an absolute value equation |5 - 4x| = 6 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. 1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Enter y = |5 - 4x| for Y1. Enter y = 6 for Y2. Y= X/ /T/n 2 MATH ENTER B * * 1 * 5 — 4 6 View the graph. GRAPH 3 Notes Find the points of intersection of the two graphs and solve. CALC 2 2nd F CALC 2 2nd F * * There are two points of intersection of the absolute value graph and the horizontal line y = 6. The solution to the equation |5 - 4x|= 6 consists of the two values -0.25 and 2.75. Note that although it is not as intuitively obvious, the solution could also be obtained by finding the x-intercepts of the function y = |5x - 4| - 6. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The EL-9650/9600c/9450/9400 shows absolute values with | |, just as written on paper, by using the Equation editor. The graphing feature of the calculator shows the solution of the absolute value function visually. 8-3 EL-9650/9600c/9450/9400 Graphing Calculator Solving Absolute Value Inequalities To solve an inequality means to find all values that make the inequality true. Absolute value inequalities are of the form |f (x)|< k, |f (x)|≤ k, |f (x)|> k, or |f (x)|≥ k. The graphical solution to an absolute value inequality is found using the same methods as for normal inequalities. The first method involves rewriting the inequality so that the right-hand side of the inequality is 0 and the left-hand side is a function of x. The second method involves graphing each side of the inequality as an individual function. Example Solve absolute value inequalities in two methods. 1. Solve 20 - 6x < 8 by rewriting the inequality so that the right-hand side of 5 the inequality is zero. 2. Solve 3.5x + 4 > 10 by shading the solution region. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set viewing window to “-5< x <50,” and “-10< y <10” using Rapid Window feature to solve Q1. EZ WINDOW 1-1 1-2 ENTER * 3 ENTER * 3 ENTER Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Notes 6x |< 8 5 6x ➞|20 | - 8 < 0. 5 Rewrite the equation. |20 - Enter y = |20 - 6x | - 8 for Y1. 5 6 MATH B X/ /T/n — * * 5 * 2 0 — 5 a/b * 8 View the graph, and find the x-intercepts. GRAPH CALC 5 2nd F CALC 5 2nd F * * ➞ x = 10, y = 0 ➞ x = 23.33333334 y = 0.00000006 ( Note) 1-4 * Step & Key Operation Y= 1-3 3 Solve the inequality. The intersections with the xaxis are (10, 0) and (23.3, 0) ( Note: The value of y in the x-intercepts may not appear exactly as 0 as shown in the example, due to an error caused by approximate calculation.) Since the graph is below the x-axis for x in between the two x-intercepts, the solution is 10 < x < 23.3. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 8-4 vol.1 8.4_10.4.layout.p65 Page 2 02.10.28, 1:15 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator 2-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Enter the function y =|3.5x + 4|for Y1. Enter y = 10 for Y2. Y= 2-2 3 • 1 0 CL MATH 5 X/ /T/n — * + 1 * 4 ENTER * Since the inequality you are solving is Y1 > Y2, the solution is where the graph of Y2 is “on the bottom” and Y1 in “on the top.” G — * * 1 * * — * Set viewing window to “-10 < x < 10” and “-5 < y < 50” using Rapid Window feature and view the graph. WINDOW 3 2-4 B Set up shading. 2nd F DRAW 2-3 Notes 2 EZ ENTER ENTER * 5 ENTER * * Find the points of intersection. Solve the inequality. CALC 2 2nd F CALC 2 2nd F * * ➞ x = -4, y = 10 ➞ x = 1.714285714 y = 9.999999999 ( Note) The intersections are (-4, 10) and (1.7, 10.0). The solution is all values of x such that x <- 4 or x >1.7. ( Note: The value of y in the intersection of the two graphs may not appear exactly as 10 as shown in the example, due to an error caused by approximate calculation.) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The EL-9650/9600c/9450/9400 shows absolute values with | |, just as written on paper, by using the Equation editor. Graphical solution methods not only offer instructive visualization of the solution process, but they can be applied to inequalities that are often difficult to solve algebraically. The Shade feature is useful to solve the inequality visually and the points of intersection can be obtained easily. 8-4 vol.1 8.4_10.4.layout.p65 Page 3 02.10.28, 1:15 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator Evaluating Absolute Value Functions The absolute value of a real number x is defined by the following: x if x ≥ 0 -x if x ≤ 0 Note that the effect of taking the absolute value of a number is to strip away the minus sign if the number is negative and to leave the number unchanged if it is nonnegative. Thus, |x|≥ 0 for all values of x. Example |x| = Evaluate various absolute value functions. 1. Evaluate |- 2(5-1)| 2. Is |-2+7| = |-2| + |7|? 3. Evaluate each side of the equation to check your answer. Is |x + y| =|x|+ |y| for all real numbers x and y ? If not, when will |x + y| = |x|+|y| ? Is |6-9 | = |6-9| ? 1+3 |1+3| Evaluate each side of the equation to check your answer. Investigate with more examples, and decide if you think |x / y|=|x|/|y| There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) 1-1 Access the home or computation screen. 1-2 Enter y = |-2(5-1)| and evaluate. B MATH 1 1 * ) * (-) 2 ( 5 Notes The solution is +8. — ENTER ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-1 Evaluate|-2 + 7|. Evaluate|-2|+|7|. ➞|-2 + 7| ≠ |-2| + |7|. CL MATH 1 MATH 1 1 * |-2 + 7| = 5, |-2| + |7| = 9 7 * * (-) 2 (-) 2 + 7 + ENTER MATH ENTER 8-5 vol.1 8.4_10.4.layout.p65 Page 4 02.10.28, 1:15 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator 2-2 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Notes Is |x + y| = |x| +|y|? Think about this problem according to the cases when x or y are positive or negative. If x ≥ 0 and y ≥ 0 [e.g.; (x, y) = (2,7)] |x +y| = |2 + 7| = 9 |x|+|y| = |2| + |7| = 9 If x ≤ 0 and y ≥ 0 [e.g.; (x, y) = (-2, 7)] |x +y| = |-2 + 7| = 5 |x|+|y| = |-2| + |7| = 9 If x ≥ 0 and y ≤ 0 [e.g.; (x, y) = (2, -7)] |x +y| = |2-7| = 5 |x|+|y| = |2| + |-7| = 9 If x ≤ 0 and y ≤ 0 [e.g.; (x, y) = (-2, -7)] |x +y| = |-2-7| = 9 |x|+|y| = |-2| + |-7| = 9 ➞|x + y| = |x| + |y|. ➞|x + y| ≠ |x| + |y|. ➞|x + y| ≠ |x| + |y|. ➞|x + y| = |x| + |y|. Therefore |x +y|=|x|+|y|when x ≥ 0 and y ≥ 0, and when x ≤ 0 and y ≤ 0. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3-1 Evaluate 6-9 . Evaluate 6-9 . 1+3 1+3 CL * 3-2 MATH 1 1 + MATH 1 MATH 1 * * * a/b 3 — 6 9 6-9 6-9 1+3 = 0.75 , 1+3 = 0 .75 ➞ 6-9 1+3 = 6-9 1+3 ENTER 6 — 9 1 + 3 * a/b ENTER Is |x /y| = |x|/|y|? Think about this problem according to the cases when x or y are positive or negative. If x ≥ 0 and y ≥ 0 [e.g.; (x, y) = (2,7)] |x /y| = |2/7| = 2/7 |x|/|y| = |2| /|7| = 2/7 If x ≤ 0 and y ≥ 0 [e.g.; (x, y) = (-2, 7)] |x /y| = |(-2)/7| = 2/7 |x|/|y| = |-2| /|7| = 2/7 If x ≥ 0 and y ≤ 0 [e.g.; (x, y) = (2, -7)] |x /y| = |2/(-7)| = 2/7 |x|/|y| = |2| /|-7| = 2/7 If x ≤ 0 and y ≤ 0 [e.g.; (x, y) = (-2, -7)] |x /y| = |(-2)/-7| = 2/7 |x|/|y| = |-2| /|-7| = 2/7 ➞|x /y| = |x| / |y| ➞|x /y| = |x| / |y| ➞|x /y| = |x| / |y| ➞|x /y| = |x| / |y| The statement is true for all y ≠ 0. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The EL-9650/9600c/9450/9400 shows absolute values with | |, just as written on paper, by using the Equation editor. The nature of arithmetic of the absolute value can be learned through arithmetical operations of absolute value functions. 8-5 vol.1 8.4_10.4.layout.p65 Page 5 02.10.28, 1:15 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator Graphing Rational Functions p (x) where p (x) and q (x) are two q (x) polynomial functions such that q (x) ≠ 0. The domain of any rational function consists of all A rational function f (x) is defined as the quotient values of x such that the denominator q (x) is not zero. A rational function consists of branches separated by vertical asymptotes, and the values of x that make the denominator q (x) = 0 but do not make the numerator p (x) = 0 are where the vertical asymptotes occur. It also has horizontal asymptotes, lines of the form y = k (k, a constant) such that the function gets arbitrarily close to, but does not cross, the horizontal asymptote when |x| is large. The x intercepts of a rational function f (x), if there are any, occur at the x-values that make the numerator p (x), but not the denominator q (x), zero. The y-intercept occurs at f (0). Example Graph the rational function and check several points as indicated below. x-1 Graph f (x) = x 2-1 . Find the domain of f (x), and the vertical asymptote of f (x). Find the x- and y-intercepts of f (x). Estimate the horizontal asymptote of f (x). 1. 2. 3. 4. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM 1-1 A * ( ENTER ALPHA Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) * ) 7 * Notes Enter y = x2 - 1 for Y1. x -1 Y= a/b — 1-2 X/ /T/n — 1 * X/ /T/n x2 1 View the graph. GRAPH The function consists of two branches separated by the vertical asymptote. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 9-1 vol.1 8.4_10.4.layout.p65 Page 6 02.10.28, 1:15 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator 2 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Find the domain and the vertical asymptote of f (x), tracing the graph to find the hole at x = 1. (repeatedly) TRACE Notes Since f (x) can be written as x-1 , the domain (x + 1)(x - 1) consists of all real numbers x such that x ≠ 1 and x ≠ -1. There is no vertical asymptote where x = 1 since this value of x also makes the numerator zero. Next to the coordinates x = 0.9, y = 0.52, see that the calculator does not display a value for y at x = 1 since 1 is not in the domain of this rational function. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 3 Find the x- and y-intercepts of f (x). 2nd F CALC 6 * The y-intercept is at (0 ,1). Notice that there are no x-intercepts for the graph of f (x). ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 4 Estimate the horizontal asymptote of f (x). The line y = 0 is very likely a horizontal asymptote of f (x). ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The graphing feature of the EL-9650/9600c/9450/9400 can create the branches of a rational function separated by vertical asymptote. The calculator allows the points of intersection to be obtained easily. 9-1 vol.1 8.4_10.4.layout.p65 Page 7 02.10.28, 1:15 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator Solving Rational Function Inequalities p (x) where p (x) and q (x) are two q (x) polynomial functions such that q (x) ≠ 0. The solutions to a rational function inequality can A rational function f (x) is defined as the quotient be obtained graphically using the same method as for normal inequalities. You can find the solutions by graphing each side of the inequalities as an individual function. Example Solve a rational inequality. x Solve 1 - x 2 ≤ 2 by graphing each side of the inequality as an individual function. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM 1 * ( ENTER ALPHA Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) * ) 7 * Notes x Enter y = for Y1. Enter y = 2 1- x 2 for Y2. Y= 1 2 A B MATH — 1 * X/ /T/n x * 2 a/b ENTER X/ /T/n * * 2 Set up the shading. G 2nd F DRAW — 1 * — * * — * Since Y1 is the value “on the bottom” (the smaller of the two) and Y2 is the function “on the top” (the larger of the two), Y1 < Y < Y2. View the graph. 3 GRAPH 4 Find the intersections, and solve the inequality. 2nd F CALC 2 * Do this four times The intersections are when x = -1.3, -0.8, 0.8, and 1.3. The solution is all values of x such that x ≤ -1.3 or -0.8 ≤ x ≤ 0.8 or x ≥ 1.3. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The EL-9650/9600c/9450/9400 allows the solution region of inequalities to be indicated visually using the Shade feature. Also, the points of intersections can be obtained easily. 9-2 vol.1 8.4_10.4.layout.p65 Page 8 02.10.28, 1:15 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator Graphing Parabolas The graphs of quadratic equations (y = ax 2 + bx + c) are called parabolas. Sometimes the quadratic equation takes on the form of x = ay 2 + by + c. There is a problem entering this equation in the calculator graphing list for two reasons: a) it is not a function, and only functions can be entered in the Y= list locations, b) the functions entered in the Y= list must be in terms of x, not y. There are, however, two methods you can use to draw the graph of a parabola. Method 1: Consider the "top" and "bottom" halves of the parabola as two different parts of the graph because each individually is a function. Solve the equation of the parabola for y and enter the two parts (that individually are functions) in two locations of the Y= list. Method 2: Choose the parametric graphing mode of the calculator and enter the parametric equations of the parabola. It is not necessary to algebraically solve the equation for y. Parametric representations are equation pairs x = F(t), y = F(t) that have x and y each expressed in terms of a third parameter, t. Example Graph a parabola using two methods. 1. Graph the parabola x = y 2. Graph the parabola x = y 2 2 -2 in rectangular mode. -2 in parametric mode. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM * ( ENTER ALPHA Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) 1-1 Solve the equation for y. 1-2 Enter y = √x+2 for Y1 and enter y = -Y1 for Y2. Y= ENTER 1-3 A 2nd F * (-) √ VARS View the graph. GRAPH + X/θ /T/n A * * ) 7 * Notes x = y 2 -2 x + 2 = y2 y=+ –√ x + 2 2 ENTER 1 * The graph of the equation y = √x+ 2 is the "top half" of the parabola and the graph of the equation y = - √x + 2 gives the "bottom half." ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 10-1 vol.1 8.4_10.4.layout.p65 Page 10 02.10.28, 1:15 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator 2-1 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Change to parametric mode. 2nd F SET UP 2 2-2 Notes E * * Rewrite x = y 2 -2 in parametric form. Enter X1T = T 2 -2 and Y1T = T. Y= X/ /T/n x2 — 2 ENTER Let y = T and substitute in x = y 2 - 2, to obtain x = T 2- 2. * X/ /T/n 2-3 View the graph. Consider why only half of the parabola is drawn. (To understand this, use Trace feature.) GRAPH 2-4 ) TRACE Set Tmin to -6. WINDOW 2-5 ( The graph starts at T =0 and increases. Since the window setting is T ≥ 0, the region T < 0 is not drawn in the graph. (-) 6 ENTER * View the complete parabola. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ The calculator provides two methods for graphing parabolas, both of which are easy to perform. 10-1 vol.1 8.4_10.4.layout.p65 Page 11 02.10.28, 1:16 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator Graphing Circles The standard equation of a circle of radius r that is centered at a point (h, k) is (x - h) 2 + (y - k) 2 = r 2. In order to put an equation in standard form so that you can graph in rectangular mode, it is necessary to solve the equation for y. You therefore need to use the process of completing the square. Example Graph the circles in rectangular mode. Solve the equation for y to put it in the standard form. 1. Graph x 2. Graph x 2 + y 2 = 4. 2 - 2x + y 2 + 4y = 2. There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM 1-1 * ( ENTER ALPHA Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) ENTER 2nd F * (-) √ — 4 VARS A * X/ /T/n ENTER * ) 7 * Notes Solve the equation for y. Enter y = √ 4 - x 2 for Y1 (the top half). Enter y = -√ 4 - x 2 for Y2. Y= 1-2 A y2 = 4 - x2 y= + – √4 - x 2 x2 1 * This is a circle of radius r , centered at the origin. View the graph. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 2-1 Solve the equation for y, completing the square. x 2 - 2x + y 2 + 4y = 2 Place all variable terms on the left and the constant term on the right-hand side of the equation. x 2 -2 x + y 2 +4 y +4 = 2 + 4 Complete the square on the y-term. x 2 - 2x + (y+2)2 = 6 Express the terms in y as a perfect square. (y+2)2 = 6 -x 2 + 2x Leave only the term involving y on the left hand side. y+2 = ±√6-x 2+2x y = ±√6-x 2+2x -2 Take the square root of both sides. Solve for y. 10-2 vol.1 8.4_10.4.layout.p65 Page 12 02.10.28, 1:16 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator 2-2 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Enter y = √6 - x 2 + 2x for Y1, y = Y1 - 2 for Y2, and y = -Y1 -2 for Y3. Y= CL x2 + A VARS 2 2-3 2 6 X/ /T/n ENTER 1 ENTER * ENTER (-) √ 2nd F Notice that if you enter y = √6 - x 2 + 2x - 2 for Y1 and y = - Y1 for Y2, you will not get the graph of a circle because the “±” does not go with the “-2”. CL * — * * VARS 1 ENTER * — 2 "Turn off" Y1 so that it will not graph. * 2-4 X/ /T/n — Notes ENTER Notice that “=” for Y1 is no longer darkened. You now have the top portion and the bottom portion of the circle in Y2 and Y3. * View the graph. GRAPH 2-5 Adjust the screen to see the bottom part of the circle using the Rapid feature. EZ ENTER 2-6 * * * ENTER ENTER * Wait until the graph is displayed after each operation. (It takes few seconds to graph) * ENTER * View the graph in the new window. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Graphing circles can be performed easily on the calculator display. Also, the Rapid Zoom feature of the EL-9650/9600c/9450/9400 allows shifting and adjusting display area (window) of a graph easily. 10-2 vol.1 8.4_10.4.layout.p65 Page 13 02.10.28, 1:16 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator Graphing Ellipses The standard equation for an ellipse whose center is at the point (h, k) with major and 2 2 minor axes of length a and b is (x -2 h) + (y -2 k) = 1. a b There is a problem entering this equation in the calculator graphing list for two reasons: a) it is not a function, and only functions can be entered in the Y = list locations. b) the functions entered in the Y = list locations must be in terms of x, not y. To draw a graph of an ellipse, consider the “top” and “bottom” halves of the ellipse as two different parts of the graph because each individual is a function. Solve the equation of the ellipse for y and enter the two parts in two locations of the Y = list. Example Graph an ellipse in rectangular mode. Solve the equation for y to put it in the standard form. Graph the ellipse 3(x -3) 2 + (y + 2) 2 = 3 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM 1 * ( ENTER ALPHA Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Solve the equation for y, completing the square. Enter Y1 = √3 - 3(x - 3)2 Y2 = Y1 - 2 Y3 = -Y1 -2 Y= 2nd F X/ /T/n — VARS 2 1 2 A A ) (-) 3 x2 1 ENTER * — 3 3 * ENTER — √ * * ) 7 * Notes 3(x - 3)2 + (y + 2)2 = 3 (y + 2)2 = 3 - 3(x - 3)2 y + 2 = + √3 - 3(x - 3)2 y = + √3 - 3(x - 3)2 - 2 ( ENTER * — VARS ENTER 2 Turn off Y1 so that it will not graph. * ENTER * 10-3 vol.1 8.4_10.4.layout.p65 Page 14 02.10.28, 1:16 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator 3 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Notes View the graph. GRAPH 4 Adjust the screen to see the bottom part of the ellipse using the Rapid Zoom feature. EZ 5 * ENTER Wait until the graph is displayed after each operation. (It takes few seconds to graph) * View the graph in the new window. GRAPH ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Graphing an ellipse can be performed easily on the calculator display. In addition to the Zoom-in/Zoom-out features, the EL-9650/9600c/9450/9400 have the Rapid Zoom feature to adjust the display easily. 10-3 vol.1 8.4_10.4.layout.p65 Page 15 02.10.28, 1:16 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator Graphing Hyperbolas The standard equation for a hyperbola can take one of two forms: ( x - h ) 2 - ( y - k ) 2 = 1 with vertices at ( h ± a, k ) or a2 b2 2 ( y - h) 2 (x-k) = 1 with vertices at ( h, k ± b ). a2 b2 There is a problem entering this equation in the calculator graphing list for two reasons: a) it is not a function, and only functions can be entered in the Y= list locations. b) the functions entered in the Y= list locations must be in terms of x, not y. To draw a graph of a hyperbola, consider the “top” and “bottom” halves of the hyperbola as two different parts of the graph because each individual is a function. Solve the equation of the hyperbola for y and enter the two parts in two locations of the Y= list. Example Graph a hyperbola in rectangular mode. Solve the equation for y to put it in the standard form. Graph the hyperbola x 2 + 2x - y 2 - 6y + 3 = 0 There may be differences in the results of calculations and graph plotting depending on the setting. Before Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM 1 * ( ENTER ALPHA Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Solve the equation for y completing the square. Enter 2nd F X/θ /T/n + (-) A * √ X/ /T/n 1 2 VARS A * ENTER 1 ENTER * + x2 1 ) 7 * Notes 2 * — ENTER * x 2 + 2x - y 2 -6y = -3 x 2 + 2x - (y 2 + 6y + 9) = -3 -9 x 2 + 2x - (y +3)2 = -12 (y + 3)2 = x 2 + 2x + 12 y + 3 = + √x 2 + 2x + 12 y = + √x 2 + 2x + 12 - 3 Y1 = √x 2 + 2x + 12 Y2 = Y1 -3 Y3 = -Y1 -3 Y= VARS 2 A 3 * — ENTER * 3 Turn off Y1 so that it will not graph. * ENTER * 10-4 vol.1 8.4_10.4.layout.p65 Page 16 02.10.28, 1:16 PM Adobe PageMaker 6.5J/PPC EL-9650/9600c/9450/9400 Graphing Calculator 3 Step & Key Operation Display (When using EL-9650/9600c) *Use either pen touch or cursor to operate. (When using EL-9650/9600c) Notes View the graph. GRAPH 4 Zoom out the screen. ZOOM A * 4 * ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Graphing hyperbolas can be performed easily on the calculator display. In addition to the Zoom-in/Zoom-out features, the EL-9650/9600c/9450/9400 have the Rapid Zoom feature to adjust the display easily. (See the section “Graphing Ellipses (No. 10-3)” about how to use the Rapid Zoom feature.) 10-4 vol.1 8.4_10.4.layout.p65 Page 17 02.10.28, 1:16 PM Adobe PageMaker 6.5J/PPC Key pad for the SHARP EL-9650/9600c Calculator Vol.1 Front.Back.layout.p65 Graphing keys Cursor movement keys Power supply ON/OFF key Clear/Quit key Alphabet specification key Variable enter key Secondary function specification key Calculation execute key Display screen Communication port for peripheral devices Page 5 02.10.28, 1:14 PM Adobe PageMaker 6.5J/PPC Key pad for the SHARP EL-9450/9400 Calculator Vol.1 Front.Back.layout.p65 Graphing keys Cursor movement keys Power supply ON/OFF key Clear/Quit key Alphabet specification key Variable enter key Secondary function specification key Calculation execute key Display screen Communication port for peripheral devices Page 6 02.10.28, 1:14 PM Adobe PageMaker 6.5J/PPC ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Dear Sir/Madam We would like to take this opportunity to invite you to create a mathematical problem which can be solved with the SHARP graphing calculator EL-9650/9600c/9450/9400. For this purpose, we would be grateful if you would complete the form below and return it to us by fax or mail, specifying which calculator you are writing problems for, the EL-9650/9600c or 9450/9400. If your contribution is chosen, your name will be included in the next edition of The EL-9650/9600c/9450/ 9400 Graphing Calculator Handbook. We regret that we are unable to return contributions. We thank you for your cooperation in this project. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Use this form to send us your contribution Ms. ) Mr. ○ ○ ○ Name: ( ○ ○ School/College/Univ.: ○ ○ ○ Address: Country: ○ ○ Post Code: Fax: ○ ○ ○ Phone: * You are making this sheet for the ( EL-9650/9600c, EL-9450/9400). SUBJECT : Write a title or the subject you are writing about. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ E-mail: ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ INTRODUCTION : Write an explanation about the subject. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ EXAMPLE : Write example problems. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ SHARP Graphing Calculator ○ ○ Vol.1 Insert.layout.p65 1 02.10.28, 1:14 PM ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ BEFORE STARTING : Write any conditions to be set up before solving the problems. ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ NOTES ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 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○ ○ ○ SHARP Graphing Calculator ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Fax: ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ STEP 2 02.10.28, 1:14 PM ○ Vol.1 Insert.layout.p65 SHARP CORPORATION OSAKA, JAPAN Vol.1 Front.Back.layout.p65 Page 7 02.10.28, 1:14 PM Adobe PageMaker 6.5J/PPC

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