# EL-9650/9600c/9450/9400 Handbook Vol. 1

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1 Graphing Calculator EL-9650/9600c/9450/9400 Handbook Vol. Algebra EL-9650 EL-9450

2 Contents. Linear Equations - Slope and Intercept of Linear Equations -2 Parallel and Perpendicular Lines 2. Quadratic Equations 2- Slope and Intercept of Quadratic Equations 2-2 Shifting a Graph of Quadratic Equations 3. Literal Equations 3- Solving a Literal Equation Using the Equation Method (Amortization) 3-2 Solving a Literal Equation Using the Graphic Method (Volume of a Cylinder) 3-3 Solving a Literal Equation Using Newton s Method (Area of a Trapezoid) 4. Polynomials 4- Graphing Polynomials and Tracing to Find the Roots 4-2 Graphing Polynomials and Jumping to Find the Roots 5. A System of Equations 5- Solving a System of Equations by Graphing or Tool Feature 6. Matrix Solutions 6- Entering and Multiplying Matrices 6-2 Solving a System of Linear Equations Using Matrices 7. Inequalities 7- Solving Inequalities 7-2 Solving Double Inequalities 7-3 System of Two-Variable Inequalities 7-4 Graphing Solution Region of Inequalities 8. Absolute Value Functions, Equations, Inequalities 8- Slope and Intercept of Absolute Value Functions 8-2 Shifting a graph of Absolute Value Functions 8-3 Solving Absolute Value Equations 8-4 Solving Absolute Value Inequalities 8-5 Evaluating Absolute Value Functions 9. Rational Functions 9- Graphing Rational Functions 9-2 Solving Rational Function Inequalities 0. Conic Sections 0- Graphing Parabolas 0-2 Graphing Circles 0-3 Graphing Ellipses 0-4 Graphing Hyperbolas

3 Read this first. 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. OPTION E 2 CL Note: Since all memories will be deleted, it is advised to use the CE-LKP 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 OPTION C and select data to be deleted from the menu. Other keys to delete data: CL : to erase equations and remove error displays QUIT : to cancel previous function 3. Initial settings Initial settings are as follows: Set up ( SET UP ): Rad, FloatPt, 9, Rect, Decimal(Real), Equation Format ( FORMAT ): RectCoord, OFF, OFF, Connect, Sequen Stat Plot ( STAT PLOT E ): 2. PlotOFF Shade ( DRAW G ): 2. INITIAL Zoom ( ZOOM A ): 5. Default Period ( FINANCE C ):. PmtEnd Note: returns to the default setting in the following operation. ( OPTION ) E 4. Using the keys Press to use secondary functions (in yellow). To select sin - : ed as follows: Press ALPHA to use the alphabet keys (in blue). To select A: ALPHA sin ed as follows: sin sin - 5. 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. ALPHA A

5 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 (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.. Graph the equations y = x and y = 2x. 2. Graph the equations y = x and y = x Graph the equations y = x and y = - x. 4. Graph the equations y = x and y = x + 2. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Enter the equation y = x for Y and y = 2x for Y2. Y= X/ /T/n 2 X/ /T/n -2 View both graphs. The equation Y = 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>) to make the line steeper. 2- Enter the equation y = 2 x for Y Y= CL a/b 2 View both graphs. X/ /T/n Notice how Y2 becomes less steep or climbs slower. Decrease the size of the slope (0<m<) to make the line less steep. -

6 EL-9650/9600c/9450/9400 Graphing Calculator Use either pen touch or cursor to operate. 3- Enter the equation y = - x for Y2. Y= CL (-) X/ /T/n 3-2 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. 4- Enter the equation y = x + 2 for Y2. Y= CL X/ /T/n View both graphs. 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. -

7 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 slopes that are negative reciprocals of each other (m = - m ). These characteristics can be verified by graphing these lines. Example Graph parallel lines and perpendicular lines.. Graph the equations y = 3x + and y = 3x Graph the equations y = 3x - and y = - x +. 3 Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM C ( ALPHA ) 7 Use either pen touch or cursor to operate Enter the equations y = 3x + for Y and y = 3x + 2 for Y2. Y= 3 X/ /T/n + 3 X/ /T/n + 2 View the graphs. These lines have an equal slope but different y-intercepts. They are called parallel, and will not intersect. 2- Enter the equations y = 3x - for Y and y = - x + for Y2. 3 Y= CL 3 X/ /T/n CL (-) a/b 3 X/ /T/n + -2

8 EL-9650/9600c/9450/9400 Graphing Calculator 2-2 Use either pen touch or cursor to operate. View the graphs. These lines have slopes that are negative reciprocals of each other (m = - m ). They are 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. -2

9 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.. Graph y = x 2 and y = (x - 2) Graph y = x 2 and y = x Graph y = x 2 and y = 2x Graph y = x 2 and y = - 2x 2. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Enter the equation y = x 2 for Y. Y= X/ /T/n x 2-2 Enter the equation y = (x - 2) 2 for Y2 using Sub feature. EZ ALPHA C SUB 2 ( ) 0-3 View both graphs. 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-

10 EL-9650/9600c Graphing Calculator 2- Use either pen touch or cursor to operate. Change the equation in Y2 to y = x Y= SUB 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. 3- Change the equation in Y2 to y = 2x 2. Y= SUB View both graphs. Notice that the multiplication of 2 pinches or closes the basic y = x 2 graph. This demonstrates the fact that multiplying an a (> ) in the standard form y = a (x - h) 2 + k will pinch or close the basic graph. 4- Change the equation in Y2 to y = - 2x 2. Y= SUB (-) View both graphs. 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 (<-) 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-

11 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.. Shift the graph y = x 2 upward by Shift the graph y = x 2 to the right by Pinch the slope of the graph y = x 2. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Access Shift feature and select the equation y = x 2. SHIFT/CHANGE A -2 Move the graph y = x 2 upward by Save the new graph and observe the Notice that upward movement changes in the graph and the of the basic y = x 2 graph by 2 equation. units in the direction of the y- axis means addition of 2 to the ALPHA 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

12 EL-9650/9600c/9450/9400 Graphing Calculator Use either pen touch or cursor to operate. 2- Move the graph y = x 2 to the right by 3. CL (three times) 2-2 Save the new graph and observe the changes in the graph and the equation ALPHA Notice that movement of the basic y = x 2 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- Access Change feature and select the equation y = x 2. SHIFT/CHANGE B 3-2 Pinch the slope of the graph. 3-3 Save the new graph and observe the changes in the graph and the equation. ALPHA Notice that pinching or closing the basic y = x 2 graph is equivalent to increasing an a (>) within the standard form y = a (x - h) 2 + k and broadening the graph is equivalent to decreasing an a (<). 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

13 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 - -(+ 2 ) I / 2 P= monthly payment L= loan amount I= interest rate N=number of months. Find the monthly payment on a \$5,000 car loan, made at 9% interest over four years (48 months) using the Equation method. 2. Save the formula as AMORT. 3. Find amount of loan possible at 7% interest over 60 months with a \$300 payment, using the saved formula. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Access the Solver feature. SOLVER This screen will appear a few seconds after SOLVER is displayed. -2 Select the Equation method for solving. SOLVER A -3 Enter the amortization formula. SOLVER P = L ALPHA ( a/b ( + ALPHA a/b 2 a b ALPHA (-) ALPHA N a b (-) I 2 ) ) 3-

14 EL-9650/9600c Graphing Calculator Use either pen touch or cursor to operate. -4 Enter the values L=5,000, I=0.09, N= Solve for the payment(p). EXE ( CL ) The monthly payment (P) is \$ Save this formula. SOLVER C 2-2 Give the formula the name AMORT. A M O R T Recall the amortization formula. 0 SOLVER B Enter the values: P = 300, I = 0.0, N = Solve for the loan (L). The amount of loan (L) is \$ 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-

15 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 2 h ( V = volume r = radius h = height). Find the radius of a cylinder with a volume of 30in 3 and a height of 0in, using the Graphic method. 2. Save the formula as V CYL. 3. Find the radius of a cylinder with a volume of 200in 3 and a height of 5in, using the saved formula. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Access the Solver feature. SOLVER This screen will appear a few seconds after SOLVER is displayed. -2 Select the Graphic method for solving. SOLVER A 3-3 Enter the formula V = πr 2 h. ALPHA V ALPHA ALPHA R x 2 ALPHA H = π -4 Enter the values: V = 30, H = 0. Solve for the radius (R) EXE 3-2

16 EL-9650/9600c Graphing Calculator -5 Use either pen touch or cursor to operate. Set the variable range from 0 to The graphic solver will prompt with a variable range for solving r 2 = = <3 0π π r = r 2 = 2 = <3 r =2 r 2 = 2 2 = 4 >3 Use the larger of the values to be safe. -6 Solve. ( ) EXE CL The solver feature will graph the left side of the equation (volume, y = 30), then the right side of the equation (y = 0r 2 ), and finally will calculate the intersection of the two graphs to find the solution. The radius is 0.98 in. 2 Save this formula. Give the formula the name V CYL. SOLVER C V SPACE C Y L 3- Recall the formula. Enter the values: V = 200, H = 5. SOLVER B Solve the radius setting the variable range from 0 to 4. EXE EXE r 2 = = < 4 5π π r = 3 r 2 = 3 2 = 9 < 4 r = 4 r 2 = 4 2 = 6 > 4 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

17 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. The formula : A= h(b+c) (A = area h = height b = top face c = bottom face) 2. Find the height of a trapezoid with an area of 25in 2 and bases of length 5in and 7in using Newton's method. (Set the starting point to.) 2. Save the formula as A TRAP. 3. Find the height of a trapezoid with an area of 50in 2 with bases of 8in and 0in using the saved formula. (Set the starting point to.) Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Access the Solver feature. SOLVER This screen will appear a few seconds after SOLVER is displayed. -2 Select Newton's method for solving. SOLVER A 2-3 Enter the formula A = h(b+c). 2 ALPHA A = ALPHA a/b 2 ALPHA H ( ALPHA B + ALPHA C ) -4 Enter the values: A = 25, B = 5, C =

18 EL-9650/9600c Graphing Calculator Use either pen touch or cursor to operate. -5 Solve for the height and enter a starting point of. -6 Solve. EXE Newton's method will prompt with a guess or a starting point. The answer is : h = 4.7 EXE ( CL ) 2 Save this formula. Give the formula the name A TRAP. SOLVER C A SPACE T R A P 3- Recall the formula for calculating the area of a trapezoid. 0 SOLVER B 3-2 Enter the values: A = 50, B = 8, C = Solve. The answer is : h = 5.56 EXE 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

19 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.. Graph the polynomial y = x 3-3x 2 + x Approximate the left-hand root. 3. Approximate the middle root. 4. Approximate the right-hand root. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM A ( ALPHA ) 7 Setting the zoom factors to 5 : ZOOM B 5 5 QUIT Use either pen touch or cursor to operate. - Enter the polynomial y = x 3-3x 2 + x +. Y= EZ Enter the coefficients. SUB (-) 3 Return to the equation display screen. It may take few seconds for the graph to be drawn. Enter each coefficients when the cursor is displayed. EXE -4 View the graph. 4-

20 EL-9650/9600c Graphing Calculator Use either pen touch or cursor to operate Move the tracer near the left-hand root. TRACE (repeatedly) Zoom in on the left-hand root. Tracer Note that the tracer is flashing on the curve and the x and y coordinates are shown at the bottom of the screen. ZOOM A Move the tracer to approximate the root. Tracer The root is : x TRACE or (repeatedly) 3- Return to the previous decimal viewing window. ZOOM H Move the tracer to approximate the middle root. TRACE (repeatedly) Tracer The root is exactly x =. (Zooming is not needed to find a better approximate.) Tracer 4 The root is : x 2.42 Move the tracer near the righthand root. Zoom in and move the tracer to find a better approximate. (repeatedly) ZOOM A 3 TRACE 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-

21 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.. Graph the polynomial y = x 4 + x 3-5x 2-3x Find the four roots one by one. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Setting the zoom factors to 5 : ZOOM A A A QUIT Use either pen touch or cursor to operate. - Enter the polynomial y = x 4 + x 3-5x 2-3x + Y= X/ /T/n a b 4 + X/ /T/n 3 5 X/ /T/n x 2 3 X/ /T/n + a b -2 View the graph. 2- Find the first root. CALC 5 x Y is almost but not exactly zero. Notice that the root found here is an approximate value. 2-2 Find the next root. CALC 5 x

22 EL-9650/9600c/9450/9400 Graphing Calculator Use either pen touch or cursor to operate. 2-3 Find the next root. x 0.24 CALC Find the next root. x 2.05 CALC 5 The calculator allows jumping to find the roots by graphing a polynomial and using the Calculate feature, without tracing the graph. 4-2

23 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.. Solve the system using the Calculate feature. y = x 2 - { y = 2x 2. Solve the system using the Tool feature. { 5x + y = -3x + y = -5 Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default and delete all data. Choose the viewing window -5 < X < 5, -0 < Y < 0 using Rapid window feature WINDOW EZ 5 ( ALPHA ) 7 4 As the Tool feature is only available on the EL-9650/9600c, the example 2 does not apply to the EL-9450/9400. Use either pen touch or cursor to operate Enter the system of equations y = x 2 - for Y and y = 2x for Y2. Y= X/ /T/n x 2 2 X/ /T/n View the graphs. -3 The find the left-hand intersection using the Calculate feature. CALC 2 Note that the x and y coordinates are shown at the bottom of the screen. The answer is : x = y = Find the right-hand intersection by The answer is : x = 2.4 accessing the Calculate feature again. y = 4.83 CALC 2 5-

24 EL-9650/9600c/9450/9400 Graphing Calculator 2- Use either pen touch or cursor to operate. Access the Tool menu. Select the number of variables. TOOL B 2 Using the system function, it is possible to solve simultaneous linear equations. Systems up to six variables and six equations can be solved. 2-2 Enter the system of equations. 5 (-) 3 (-) Solve the system. EXE x = 0.75 y = A system of equations can be solved easily by using the Calculate feature or Tool feature. 5-

25 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 2. Enter a 3 x 3 matrix B 3. Multiply the matrices A and B B Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Access the matrix menu. MATRIX B -2 Set the dimension of the matrix at three rows by three columns Enter the elements of the first row, the elements of the second row, and the elements of the third row. 2 2 (-) (-) 2 6-

26 EL-9650/9600c Graphing Calculator Use either pen touch or cursor to operate. 2 Enter a 3 x 3 matrix B. MATRIX B Multiply the matrices A and B together at the home screen. MATRIX A 2 A X MATRIX 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 ( ) is placed in the, (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. 3-2 Delete the input matrices for future use. 2 OPTION QUIT C Matrix multiplication can be performed easily by the calculator. 6-

27 Example Use matrix multiplication to solve a system of linear equations.. Enter the 3 x 3 identity matrix in matrix A. 2. Find the inverse matrix of the matrix B. 3. Solve the equation system. x + 2y + z = 8 2x + y - z = x + y - 2z = -3 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 - B. Note that the multiplication is order sensitive and the correct answer will be obtained by multiplying BA -. An inverse matrix A - is a matrix that when multiplied by A results in the identity matrix I (A - x A=I). The identity matrix I is defined to be a square matrix (n x n) where each position on the diagonal is and all others are 0. { B Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Set up 3 x 3 identity matrix at the home screen. MATRIX C Save the identity matrix in matrix A. STO MATRIX A -3 Confirm that the identity matrix is stored in matrix A. MATRIX B 6-2

28 EL-9650/9600c Graphing Calculator Use either pen touch or cursor to operate. 2- Enter a 3 x 3 matrix B. MATRIX B (-) (-) Exit the matrix editor and find the Some square matrices have no inverse of the square matrix B. inverse and will generate error statements when calculating the inverse. QUIT CL MATRIX A 2 x - (repeatedly) B - = Enter the constants on the right side of the equal sign into matrix C (3 x ). MATRIX B (-) 3 The system of equations can be expressed as 2 x y = -2 z -3 Let each matrix B, X, C : BX = C B - BX = B - C (multiply both sides by B - ) I = B - (B - B = I, identity matrix) X = B - C 3-2 Calculate B - C. CL x - MATRIX X A 2 MATRIX A 3 The is the x coordinate, the 2 the y coordinate, and the 3 the z coordinate of the solution point. (x, y, z)=(, 2, 3) 3-3 Delete the input matrices for future use. 2 OPTION QUIT C 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

29 Solving Inequalities EL-9650/9600c Graphing Calculator 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.. Solve 3(4-2x) 5 - x, by rewriting the right-hand side of the inequality as Solve 3(4-2x) 5 - x, by shading the solution region that makes the inequality true. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Rewrite the equation 3(4-2x) 5 - x so that the right-hand side becomes 0, and enter y = 3(4-2x) x for Y. Y= 3 ( 4 2 X/ /T/n ) 5 + X/ /T/n 3(4-2x) 5 - x 3(4-2x) x 0-2 View the graph. -3 Find the location of the x-intercept The x-intercept is located at and solve the inequality. the point (.4, 0). CALC Since the graph is above the x-axis to the left of the x-intercept, the solution to the inequality 3(4-2x) x 0 is 5 all values of x such that x.4. 7-

30 EL-9650/9600c Graphing Calculator Use either pen touch or cursor to operate. 2- Enter y = 3(4-2x) for Y and y = 5 - x for Y2. Y= (7 times) DEL (4 times) 5 X/ /T/n 2-2 View the graph. 2-3 Access the Set Shade screen. DRAW G 2-4 Set up the shading. Since the inequality being solved is Y Y2, the solution is where the graph of Y is on the top and Y2 is on the bottom. 2-5 View the shaded region. 2-6 Find where the graphs intersect and solve the inequality. CALC 2 The point of intersection is (.4, 3.6). Since the shaded region is to the left of x =.4, the solution to the inequality 3(4-2x) 5 - x is all values of x such that x.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-

31 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 - 5-2x -5 7 Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. Enter y = - for Y, y = 2x - 5 for Y2, and y = 7 for Y3. Y= (-) 2 X/ /T/n 5 7 The double inequality given can also be written to - 2x View the lines. 3 Find the point of intersection. y = 2x - 5 and y = - intersect at (2, -). CALC 2 7-2

32 EL-9650/9600c/9450/9400 Graphing Calculator Use either pen touch or cursor to operate. 4 Move the tracer and find another intersection. y = 2x - 5 and y = 7 intersect at (6,7). CALC 2 5 Solve the inequalities. The solution to the double inequality - 2x 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

33 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 x 2 + y Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM A ( ) 7 Use either pen touch or cursor to operate. Rewrite each inequality in the system so that the left-hand side is y : 2 Enter y = - 2x for Y and y = - x 2 for Y2. Y= 2 X/ /T/n x 2 X/ /T/n 3 Access the set shade screen 2x + y y - 2x x 2 + y y - x 2 4 DRAW G Shade the points of y -value so that Y y Y2. 5 Graph the system and find the intersections. The intersections are (0, ) and (2, -3) CALC 2 CALC 2 6 Solve the system. 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

34 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.. Graph the solution region of a system of inequalities: x + 2y x 2 + y 4 2. Which of the following points are within the solution region? (-.6,.8), (-2, -5), (2.8, -.4), (-8,4) Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Rewrite the inequalities so that the left-hand side is y. x + 2y y -x 2 x 2 +y 4 y 4 - x 2-2 Enter y = -x for Y and 2 y = 4 - x 2 for Y2. Y= a/b X/ /T/n 2 4 X/ /T/n x 2-3 Set the shade and view the solution region. Y2 y Y DRAW G 2- Set the display area (window) to : -9 < x < 3, -6 < y < 5. WINDOW (-) 9 3 (-)

35 EL-9650/9600c/9450/9400 Graphing Calculator Use either pen touch or cursor to operate Use the cursor to check the position of each point. (Zoom in as necessary). or or or Substitute points and confirm whether they are in the solution region. (-) X 8... (Continuing key operations omitted.) Points in the solution region are (2.8, -.4) and (-8, 4). Points outside the solution region are (-.6,.8) and (-2, -5).. (-.6,.8): = 2 This does not materialize.. (-2, -5): (-5) = -2 (-2) 2 + (-5) = - This does not materialize.. (2.8, -.4): (-.4) = 0 (2.8) 2 + (-.4) = 6.44 This materializes.. (-8, 4): = 0 (-8) = 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

36 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.. Graph y = x 2. Graph y = x - and y = x - using the Rapid Graph feature. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Set the zoom to the decimal window: ZOOM A ( ) 7 Use either pen touch or cursor to operate. - Enter the function y = x for Y. Y= MATH B X/ /T/n -2 View the graph. 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 in the range of X > 0 and - in the range of X Enter the standard form of an absolute value function for Y2 using the Rapid Graph feature. Y= EZ Substitute the coefficients to graph y = x -. SUB 0 8-

37 EL-9650/9600c Graphing Calculator 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 x- axis. 2-4 Change the coefficients to graph y = x -. Y= SUB (-) 2-5 View the graph. 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-

38 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.. 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. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. - Access the Shift feature. Select y = x. SHIFT/CHANGE ( ALPHA A ) 8-2 Move the graph downward by 2. y = x changes to y = x -2-3 Save the new graph and look at the relationship of the function and the graph. ALPHA The graph of the equation that is highlighted is shown by a solid line. Notice that the y- intercept k in the standard form y = a x - h + k takes charge of vertical movement of the graph. 8-2

39 EL-9650/9600c/9450/9400 Graphing Calculator Use either pen touch or cursor to operate. 2- Move the original graph to the right by 2. y = x changes to y = x-2 ALPHA 2-2 Save the new graph and look at the relationship of the function and the graph. ALPHA Notice that the function h in the standard form y = a x - h + k takes charge of horizontal movement of the graph. 3- Access the Change feature. SHIFT/CHANGE B 3-2 Select y = x Make the slope of the graph steeper. Save the new graph. y = x y = 2 x 3-4 Make the slope of the graph minus. Save the new graph. y = x y = - x 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

40 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 = 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. Example Solve an absolute value equation 5-4x = 6 Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Use either pen touch or cursor to operate. Enter y = 5-4x for Y. Enter y = 6 for Y2. Y= MATH B 5 4 X/ /T/n 6 2 View the graph. There are two points of intersection of the absolute value graph and the horizontal line y = 6. 3 Find the points of intersection of the two graphs and solve. CALC 2 CALC 2 The solution to the equation 5-4x = 6 consists of the two values and 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 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

41 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.. 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 > 0 by shading the solution region. Before There may be differences in the results of calculations and graph plotting depending on the setting. Starting Return all settings to the default value and delete all data. Set viewing window to -5< x <50, and -0< y <0 using Rapid Window feature to solve Q. WINDOW EZ Use either pen touch or cursor to operate. - Rewrite the equation. -2 Enter y = 20-6x - 8 for Y. 5 Y= MATH B X/ /T/n 5 8 a/b 6x 20 - < x - 8 < View the graph, and find the x-intercepts. CALC CALC 5 5 x = 0, y = 0 x = y = ( Note) The intersections with the x- axis are (0, 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.) -4 Solve the inequality. Since the graph is below the x-axis for x in between the two x-intercepts, the solution is 0 < x <

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