This workshop will Arapahoe Community College MAT 111 - Graphing Calculator Techniques for Survey of Algebra TI-83 Graphing Calculator Workshop #8 Solving Equations - Graphical Approach 1) demonstrate how to solve f(x) = 0 by graphing y = f(x) and using Zoom Box to estimate the x-intercepts, 2) show how to use the CALC zero menu selection to find the x-intercepts of y = f(x) and consequently estimate the solutions for f(x) = 0, 3) approximate solutions to f(x) = g(x) utilizing ZOOM Box on the points of intersection of y = f(x) and y = g(x) leading to solutions of f(x) = g(x). 4) practice using the CALC intersect menu selection to find the intersection points of y=f(x) and y=g(x) leading to solutions of f(x) = g(x). X-intercepts and Zoom Box The x-intercepts of the graph of y = f(x) correspond to the real solutions of f(x) = 0. To solve x 3 + 4x 2-11x - 20 = 0, graph the cubic function f(x) = x 3 + 4x 2-11x - 20 and estimate the x-intercepts using zoom box and trace. 1) Check that your TI-83 is in function graphing mode and clear all functions from your Y= menu. 2) Enter x^3 + 4x 2-11x - 20 intoy1. 3) Set your window parameters as follows:. 4) Graph Y1. You should see three x-intercepts. 5) To find the positive real solution to x 3 + 4x 2-11x - 20 = 0, press and form a box around the positive x-intercept of your graph. Box zoom on the x intercept again.
2 6) Report your current window parameters: Window Xmin = Xmax = Xscl = Ymin = Ymax = Yscl = Xres = 7) Use TRACE to estimate the positive solution to x 3 + 4x 2-11x - 20 = 0 which is about x = 2.73. 8) Return to the Window parameters used in step 3) and repeat the process in steps 4) through 7) to estimate the two negative solutions to x 3 + 4x 2-11x - 20 = 0. Solutions: x = and x =. 9) Solve the quadratic equation x 2-2x - 7 = 0. Put your function in Y2 and turn off Y1. Solution: x = and x =. X-Intercepts and the CALC zero Menu Selection The x-intercepts of the graph of y = f(x) correspond to the real solutions of f(x) = 0. To solve x 3 + 4x 2-11x - 20 = 0, graph the cubic function f(x) = x 3 + 4x 2-11x - 20 = 0 and estimate the x-intercepts using the CALC zero menu option. 1) Turn on Y1 and turn off Y2. 2) Set your window parameters as follows: 3) Graph Y1. You should see three x-intercepts.
3 4) Press which is the CALC menu selection and then or arrow down to zero and press. 5) The zero option needs a left boundary of where to look for an x-intercept. Use the arrow key to define a left bound which should be a value of x to the left of the positive root we seek. Press to anchor it. Notice the triangle that appears on the left side of the graph, marking your boundary. 6) The zero option also needs a right boundary. Use your arrow keys again and press to anchor it. 7) The zero option needs a point (guess) to start the numerical algorithm to find the x-intercept. The guess must be between the left and right boundary. 8) Press to approximate the solution as x = 2.726. You have not finished the process until the "Zero" appears in the lower left corner. 9) Repeat the process in steps 4) through 9) to estimate the two negative solutions to x 3 + 4x 2-11x - 20 = 0. Solutions: x=, x = 10) Turn on Y2 and turn off Y1. 11) Solve the quadratic equation x 2-2 x - 7 = 0. Solutions : x =, x = Intersections and Box Zoom The real solutions of f(x) = g(x) correspond to the x coordinates of the points of intersection of the graphs of y = f(x) and y = g(x). Suppose you want to solve x-3 = x 2-2x-7. You could graph the two functions and estimate the points of intersection. The x coordinates of the point of intersection are the real solutions to x-3 = x 2-2x-7. 1) Clear all functions from your y= menu. 2) Enter y1=x-3.
3) Enter y2=x 2-2x-7. 4) Graph y1 and y2 on the standard viewing window. 4 5) Box zoom a couple of times on the left point of intersection. 6) Use trace to estimate the point of intersection. The x coordinate of the point of intersection is a solution to x-3 =x 2-2x-7, so the solution is approximately x = -0.996. You will need to learn to suspect this solution may be x = -1. 7) Return to the standard viewing window and estimate the other solution to x-3=x 2-2x-7. solution x= 8) Solve x 3 +4x 2-11x-20 = x 2-2x-7. Use y2=x 2-2x-7 y3= x 3 +4x 2-11x-20 and turn off y1. There are three solutions. You may have to adjust your viewing window to see all three solutions. solutions x=, x=, x= Intersections and CALC intersect Menu Selection Let s return to the problem of solving x-3=x 2-2x-7. You could graph the two functions and estimate the points of intersection using the CALC Intersect menu selection. 1) Turn y1 and y2 on and turn y3 off. 2) Graph y1 and y2 on the standard viewing window.
5 3) Press to select the CALC menu. 4) Press to select the intersect menu selection. 5) You will find the left intersection point of y1 and y2. ISECT needs you to identify the two curves that have the point of intersection you are interested in and an initial guess of the point of intersection. Use the arrow keys to get the cursor somewhere on the first function (notice the y1=x-3 in the upper left corner). 6) Press. 7) Use the arrow keys to get the cursor somewhere on the second function (notice the y2=x^2-2x-7 in the upper left hand corner). 8) Press. 9) Use the arrow keys to enter a guess of the point of intersect to start the numerical algorithm 10) Press. 11) The x coordinate of the left point of intersection is a solution to x-3=x 2-2x-7, so the solution is approximately x=-1.
6 12) Estimate the other solution to x-3=x 2-2x-7 by repeating steps 2) through 10) for the right hand point of intersection. solution x= 13) Solve x 3 +4x 2-11x-20 = x 2-2x-7. Turn on y2 and y3 and turn off y1. There are three solutions. You may have to adjust your viewing window to see all three solutions. solutions x=, x=, x= Here are some more problems to practice what you have learned in this workshop. 5x + 27 = 3 - x answer x=-4 3x 2 + 2x - 1 = -x 2 + 4 answers x = 1+ 21 0. 8956 4 x = 1 21 1.3956 4 x 4 + 7x 3 + 9x 2-17x - 20 = 0 answers x=-4 x=-1 x = 1+ 6 1.45 x = 1 6 3.45