THE MATHEMATICS DIVISION OF LEHIGH CARBON COMMUNITY COLLEGE PRESENTS WORKSHOP II Graphing Functions on the TI-83 and TI-84 Graphing Calculators
Graphing Functions on the TI-83 or 84 Graphing Calculators INTRODUCTION Graphs provide us with a wonderful and easy way to see trends and relationships. Let's use a graph to analyze the following situation. Suppose you took two tests and your grades were 82% and 91%. At this point you have an 82 91 86.5% average 86.5 2. Let's use the graphing capabilities of your calculator to determine what you must get on your third test to maintain this B-average. If you let x represent the third test grade and let y represent the average of your three tests, the following equation will show the relationship between these two unknowns. 8291x y 3 1. Clear or turn off all functions in the Y= menu. 2. Enter Y 1 = (82+91+x)3. 3. Adjust the viewing rectangle. Press WINDOW and enter the following: Xmin = 0 Xmax = 94 (This is a good choice since there are 94 pixels from left to right.) Xscl = 10 (This will give you a tick-mark every 10 units.) Ymin = 0 Ymax = 100 Yscl = 10 (This will give you a tick-mark every 10 units.) 4. Press GRAPH to see your graph. Note the upward trend of your graph. This is reasonable since an increase in x (the third test grade) causes an increase in y (the average of the three test grades). 5. To see some specific x and y-values, press TRACE. Note that the cursor is now on the graph and the coordinates of the point are displayed at the bottom of the screen. Since you are interested in scores that will maintain your B-average (i.e., x-values that will make 79.5 y < 89.5), press the right-arrow until you see the first point where the y-value is at least 79.5. Note the x-value is 66. Now press the right-arrow until you see the last point where the y-value is still less than 89.5. (Note that the viewing rectangle selfadjusts as you move past the right-end of the screen!) Note the x-value is 95. Conclusion: A third test grade that is greater than or equal to 66 but less than or equal to 95 will maintain your B-average. 1
FIND RELATIVE MINIMUMS AND MAXIMUMS PROBLEM: To the nearest 0.001, find the relative minimum point of the function f (x) = 1 3 x3-4x + 3. Steps: 1. Clear or turn off all functions in Y=. 2. Enter Y 1 = (1/3)x^3-4x+3 3. Look at the graph in the standard viewing window ( Zoom 6 ) 4. Use the CALC menu to find the minimum point: a) Access CALC menu: 2nd [CALC]. b) Choose 3 to calculate the minimum value. c) Note the bottom of the screen! To answer the question, "Left Bound?", ("Lower Bound?" on the TI-82) move cursor to the left of the minimum point, and press ENTER. (Note the arrow that has been inserted at the top of the screen. d) Note the bottom of the screen! To answer the question, "Right Bound?", move cursor to the right of the minimum point, and press ENTER. (Note the second arrow that has been inserted at the top of the screen.) e) Note the bottom of the screen! To answer the question: "Guess?", move cursor close to the minimum point, and press ENTER. f) The coordinates of the minimum point are now displayed at the bottom of the screen : (2, -2.333) PRACTICE: Find the relative maximum point of the function f (x) = 1 3 x3-4x+3. Answer: (-2, 8.333) 2
FIND X-INTERCEPTS PROBLEM: To the nearest hundredth, find the x-intercepts of the function f (x) = 1 3 x3-4x + 3. Steps: 1. Enter and graph the function. (See steps 1 through 3 on page 2.) Note that this function has three x-intercepts. Let us find the left-most x-intercept. 2. Use the CALC menu to find an x-intercept, which is also called a zero of the function f(x). a) Access CALC menu: 2ND [CALC]. b) Choose 2 to calculate the x-intercept. c) Note the bottom of the screen! To answer the question, "Lower Bound?", move cursor to the left of the x-intercept, and press ENTER. (Note the arrow that has been inserted at the top of the screen.) d) Note the bottom of the screen! To answer the question, "Upper Bound?", move cursor to the right of the x-intercept, and press ENTER. (Note the second arrow that has been inserted at the top of the screen.) e) Note the bottom of the screen! To answer the question: "Guess?", move cursor close to the x-intercept, and press ENTER. f) The coordinates of the x-intercept are now displayed at the bottom of the screen: (-3.79, 0). PRACTICE: To the nearest hundredth, find the other two x-intercepts of the function. Answer: (0.79, 0) and (3.00, 0) 3
FIND POINTS OF INTERSECTION PROBLEM: To the nearest hundredth, find the intersection point of f (x) = 1 3 x3-4x + 3 and g(x) = -4. Steps: 1. Enter and graph the two functions. (See steps 1 through 3 on page 2.) 2. Use the CALC menu to find the intersection point. a) Access CALC menu: 2nd [CALC]. b) Choose 5 to calculate the intersection point. c) Note the bottom of the screen! To answer the question: "First Curve?", place cursor anywhere on the first graph with which you are working. [Note that the equation is in the top left corner], and press ENTER Note: to choose a different graph, press either the up-arrow or down-arrow until you have the graph you want. Note that the calculator marks the curve you have just selected by placing a "+" on the curve. d) Note the bottom of the screen! To answer the question: "Second Curve?", place cursor anywhere on the second graph with which you are working, and press ENTER. Note that the calculator marks the curve you have just selected by placing a "+" on the curve. e) Note the bottom of the screen! To answer the question: "Guess?", move cursor close to the intersection point, and press ENTER f) The coordinates of the intersection point are now displayed at the bottom of the screen: (-4.13, -4). PRACTICE: To the nearest hundredth, find the intersection points of the functions f(x) = x 4-5 and g(x) = 2x. Answer: (-1.26, -2.51) and (1.70, 3.41) 4
SOLVE EQUATIONS BY GRAPHING PROBLEM A: Solve x 3 5x = 0. Give all answers accurate to the nearest hundredth. Steps: Solving f(x) = 0 is the same as finding the zero(s); i.e., the x-intercept(s) (see page 2 for the steps). Answer: x = -2.24, 0.00 or 2.24 PRACTICE A: Solve x 2 = 8. Give all answers accurate to the nearest hundredth. (HINT: First rewrite the problem as x 2 8 = 0.) Answer: x = -2.83 or 2.83 PROBLEM B: Solve x 3 5x = x 2 8. Give all answers accurate to the nearest hundredth. Method 1: Find the x coordinate(s) of the intersection point(s) of the two functions y = x 3 5x and y = x 2 8 (see page 3 for the steps). Method 2: First rewrite the problem as x 3 - x 2-5x + 8 = 0, and find the x-intercept(s) of the function y = x 3 - x 2-5x + 8 (see page 2 for the steps). Answer: x = -2.42 PRACTICE B: Solve 1 6 x2 - x - 5 = 2 3 x3. Answer: x = -1.64 DISCUSSION PROBLEM: Use your calculator to solve the following two problems that are algebraically equivalent. x 3 2 0 2 x 6x9 0 5
SOLVE SYSTEMS BY GRAPHING PROBLEM: Solve the system y = (x 3) 2 and x + y = 5. Method 1 (easy): Steps: 1. Solve each equation for y, and enter each equation into Y= menu: Y 1 = (x 3)^2 and Y 2 = 5 x 2. Find the intersection points of these two functions (see page 3). Method 2 (more difficult): Steps: 1. First solve each equation for y and rewrite the problem as (x 3) 2 = 5 x, which is equivalent to (x 3) 2 + x 5 = 0. 2. Find the x-intercept(s) of y = (x 3) 2 + x 5 (see page 2). This will give the x-coordinates of the solutions, i.e., x = l and x = 4. 3. Find the corresponding y-coordinates of the solutions by evaluating one of the original functions at the x-values you found in step 2, i.e., at x = 1 find that y = 4, and at x = 4 find that y = 1. Answer: (1, 4) and (4, 1) PRACTICE: Solve the system x 2 + y 2 = 25 and x y = 1 Answer: (-3, -4) and (4, 3) Discussion Problem: (See what calculator does when there is no solution to a system.) Solve the system: x 2 y = 0 and x y = 1. 6
FIND SEVERAL VALUES OF A FUNCTION PROBLEM: Evaluate f(x) = 0.6x 2 + 8.2 for x = 0, 1, 2, 3, 4, and 5. Method 1 (review): Steps: 1. Enter Y 1 = 0.6x 2 + 8.2 in the Y= menu. 2. Go to the home screen, store 0 in the x-variable, and enter Y 1. Repeat this step for each x-value. Method 2 (review): Steps: 1. Enter Y 1, = 0.6x 2 + 8.2 in the Y= menu. 2. Go to the home screen, enter Y 1 (0). Repeat this step for each x-value. Method 3 Using the Table feature: Steps: 1. Enter Y 1 = 0.6x 2 + 8.2 in the Y= menu, and turn off all other functions. 2. Use the Table feature to find the corresponding y-values: a) Set up the table of values: 2nd [TblSet]. 1) Set the initial value to be used for the x-variable: For our problem we want TblMin = 0. [Note: on the TI-83, it is TblStart = 0] 2) Set increments to be used for the x-variable: For our problem we want Tbl =1. 3) We want everything to be generated automatically, so set Indpnt: Auto and Depend: Auto. b) See the table of values: 2nd [TABLE]. X Y 1 0 8.2 1 8.8 2 10.6 3 13.6 4 17.8 5 23.2 6 29.8 Note: You can use your up and down arrows to see more values for this table. Answer: f(0) = 8.2, f(1) = 8.8, f(2) = 10.6, f(3) = 13.6, f(4) = 17.8, f(5) = 23.2 PRACTICE: Evaluate f(x) = x 4-2x for x = -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6, and 08. Round all answers to the nearest tenth. Answer: f(-0.6) = 1.3, f(-0.4) = 0.8, f(-0.2) = 0.4, f(0) = 0, f(0.2) = 0.4, f(0.4) = 0.8, f(0.6) = 1.1, f(0.8) = 1.2 7
GET A CLOSER LOOK AT A PORTION OF A GRAPH WITH "ZBOX" PROBLEM: Determine whether or not the graphs of f(x) = x 2 4x + 1 and g(x) = -2.1x intersect. Steps: 1. Enter and graph the functions. (see steps 1 through 3 on page 1). 2. Use the ZOOM menu to investigate the area of the graph where the curves seem to touch. a) Access the Zoom menu: Zoom b) Choose 1 to select Zbox. c) Use your arrows to move the cursor to a corner of a box that you would like to form. Press ENTER. d) Use your arrows to sweep out a box, and when you have the box you want, press ENTER. Note that the box you drew has now filled the screen. Repeat step 2 until you can see that the two curves do not intersect! 8
DETERMINE AND SET UP AN APPROPRIATE WINDOW FOR YOUR GRAPH PROBLEM: An open box is to be formed from a 26 inch square piece of metal by folding up sides of length x and cutting out the corners (see the figures below). The volume of the newly formed box is a function of x and can be modeled by the function: V(x) = x(26 2x) 2 where 0 < x < 13. Graph this function. Method 1 Steps: 1. Enter Y 1 = x(26 2x) 2 in the Y= menu and clear or turn off all other functions. 2. To estimate an appropriate window for your graph, use the TABLE feature to find the corresponding y-values for the following sample of x-values: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13. (see Method 3 on page 6): X Y 1 0 0 1 576 2 968 3 1200 4 1296 5 1280 6 1176 7 1008 8 800 9 576 10 360 11 176 12 48 13 0 smallest y-value in this sample largest y-value in this sample From this sample, you can see that you need a scale on the y-axis that goes from zero to at least 1296. But 1296 may not be the largest y-value in the graph, so give yourself a little extra space and make the scale on the y-axis go up to 1600, perhaps. 9
3. Use the WINDOW menu to adjust the viewing rectangle. a) Press WINDOW. b) Enter the following: Xmin = 0 Xmax = 13 Xscl = 1 (This will give you a tick mark on every unit on the x-axis) Ymin = 0 Ymax = 1600 Yscl = 500 (This will give you a tick mark every 500 units on the y-axis.) 4. Press GRAPH to see your graph. Note - you are looking at the portion of the graph of V(x) = x(26 2x) 2 that is meaningful to your application. To quickly see more of the graph, press ZOOM, 3, ENTER for Zoom Out. (Now to "zoom-out" again, just press ENTER to quickly replay this sequence of steps.) To quickly zoom-in on the graph, use the arrow keys to place the cursor in the center of the area about which you want to zoom-in, and press ZOOM, 2, ENTER for Zoom In. (Now to "zoom in" again, just press ENTER to quickly replay this sequence of steps.) Method 2 Using ZoomFit Steps: 1 2. Same as above 3a) 3b) Enter the following: Xmin = 0 Xmax = 13 4. Press ZOOM [ZoomFit] to see your graph. (Note that this feature determines the values for Ymin and Ymax for you!) PERM14J-k Kradel revised 4/22/05 10