PART B TEXAS INSTRUMENTS TI-85/TI-86 GRAPHING C ALCULATORS Setup When using this Guide, you should always, unless instructed otherwise, use the calculator setup specified below for both the TI-85 and TI-86. Before you begin, check the basic setup with 2nd MORE (MODE) . Use the following instructions to choose the settings shown in Figure 1. Check the window format with GRAPH ¥ MORE F3 (FORMT) . Choose the settings shown in Figure 2.
B-2 TI-85/TI-86 Graphing Calculators 7 (L) 2 ALPHA , ALPHA 7 (L) 3 ENTER . (See Figure 4.) Press EXIT or 2nd TI-85 Statistical Setup FIGURE 3 , ALPHA 7 (L) 4 , ALPHA EXIT (QUIT) to return to the home screen. TI- 86 Statistical Setup FIGURE 4 TI-86 Note: Because appropriate TI-86 computer linking software had not yet been perfected at the time this material was written, TI-86 screens in this Guide appear smaller than the TI85 screens.
Chapter 1 æ ö 1 Calculate ç ÷ ç 4*15 + 895 ÷ è 7 ø B-3 -1 using this nice shortcut. (If you wish to clear the home screen, press CLEAR .) 3. ANSWER DISPLAY When the denominator of a fraction has no more than three digits, your calculator can provide the answer in fraction form. When an answer is very large or very small, the calculator displays the result in scientific notation. The Òto a fractionÓ key is obtained by pressing 2nd ✕ (MATH) F5 (MISC) MORE F1 (‹Frac) .
B-4 TI-85/TI-86 Graphing Calculators Choose F1 (Goto) to position the cursor to the place the error occurred so that you can correct the mistake or choose F5 (Quit) to begin a new line on the home screen. Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 1 Ingredients of Change: Functions and Linear Models & 1.1 Fundamentals of Modeling There are many uses for a function that is entered in the graphing list. Graphing the function in an appropriate viewing window is one of these. Because you must enter all functions on one line (that is, you cannot write fractions and exponents the same way you do on paper) it is very important to have a good understanding of the calculatorÕs order of operations and to use parentheses whenever they are needed. 1.1.
B-6 TI-85/TI-86 Graphing Calculators Remove the lower menu with EXIT , and press F3 (ZOOM) MORE F4 (ZDECM) . Notice that the graphics screen is blank. TI-85 Press EXIT F2 (RANGE) to see the view set by ZDECM. TI-86 Press EXIT F2 (WIND) to see the view set by ZDECM. ¥ Both xMin and xMax are the settings of the left and right edges of the viewing screen, and yMin and yMax are the settings for the lower and upper edges of the viewing screen.
Chapter 1 Both Press F5 (GRAPH) to draw the graph of y1 = 1000(1 + 0.05^x) in the new view. CLEAR removes the menu from the bottom of the screen if you wish to see more of the graph. Copyright © Houghton Mifflin Company. All rights reserved.
B-8 TI-85/TI-86 Graphing Calculators 1.1.4 AUTOMATICALLY CHANGING THE VIEW OF THE GRAPH If your view of the graph is not good or if you do not see the graph, change the view using the built-in autoscaling feature of your calculator. This option will automatically find a view to see a l l the functions that you have turned on in the graphing list. Be sure the function you are graphing, y = 1000(1 + 0.05)x, is entered in the y1 location of t h e y(x)= list.
Chapter 1 B-9 Estimate the value of A where A = 1000(1 + 0.05)x when x = 5, x = 7, and x = 10. Press EXIT F2 (RANGE) (or WINDOW) . If you do not have the settings shown to the right, reset the values. Press F3 (ZOOM) F3 (ZOUT) . After the graph finishes drawing, press ENTER to once more enlarge your view of the graph. (Press EXIT F2 (RANGE) and observe the values now defining the graphics screen.) Note that (5, 1450) is a point on the screen, not a point on the graph of the function.
B-10 TI-85/TI-86 Graphing Calculators Press ENTER to display the output and see the point marked on the graph of the function. Repeat this process to find the outputs when x = 7 and x = 10. To evaluate an output from the home screen: First, press EXIT until you are on the home screen. Store the input value, say 5, in x by with 5 STO‹ x-VAR ENTER . Next, type the location in the y(x)= list in which the function is stored with 2nd ALPHA 0 (y) 1.
Chapter 1 B-11 Evaluate y1 at x = 7 by recalling the previous entry with 2nd ENTER (ENTRY) , edit the 5 to 7 by pressing and typing over the 5, and press ENTER . Repeat the process to evaluate y1 at x = 10. & 1.2 Functions and Graphs When you are asked to estimate or approximate an output or an input value, you can use your calculator in the following ways: ¥ ¥ tracing a graph close values obtained from a table of function values (Sections 1.1.5, 1.1.6) (End of Section 1.2.
B-12 TI-85/TI-86 Graphing Calculators The message that appears tells you to press ENTER to see the output. The output is in the form [ t, v(t) ]. We see that v(85) » $6945. If you had wanted to generate a list of values, starting at a value called TblMin and differing by DTbl, use the AUTO option in the program. Press ENTER to re-run the program, choose 1 for AUTO, and enter the values like those shown to the right.
Chapter 1 B-13 Press F1 (TABLE) , type in the x-value(s) at which the function is to be evaluated, and press ENTER . You can scroll through the table with , , , and/or . Unwanted input entries can be cleared with DEL . v(85) » $6945 1.2.2 SOLVING FOR INPUT VALUES Your calculator will solve for input values of an equation that you enter in the SOLVER. You can use any letter you wish for the input variable when using the SOLVER.
B-14 TI-85/TI-86 Graphing Calculators Notice that you could have typed the v(t), with input variable t, directly in the solver. The solution is, of course, the same. If there is more than one solution to an equation, you need to give the solver an approximate location for each answer. Suppose you are given q(x) = 8x2 + 54.65x Ð 163 and asked to find what input(s) correspond to an output of q(x) = 154. (The procedure outlined below also applies to finding where two functions are equal.) Enter 8x2 + 54.
Chapter 1 Press F4 (TRACE) and use and/or along the graph of the equation in y1. B-15 to move Press to jump from the graph of y1 to the graph of y2. Tracing reveals that guesses for the input values where these two graph intersect are x » –10.6, x »3.8. Return to the home screen with 2nd EXIT (QUIT) . Access the SOLVER and enter the equation y1 = y2 as shown on the right. Enter the guess for the leftmost intersection point and solve for x.
B-16 TI-85/TI-86 Graphing Calculators To graphically find an x-intercept, i.e., a value of x at which the graph crosses the horizontal axis, first press EXIT to return the menu to the bottom of the screen. TI-85 Press F1 (MATH) F3 (ROOT ) TI-86 Press F1 (MATH) F1 (ROOT ) Both Use to move the trace cursor near the first x-intercept, and press ENTER . To find the second x-intercept, press EXIT and the F-key under ROOT, use to move the trace cursor near the other x-intercept, and press ENTER . & 1.
Chapter 1 B-17 On the TI-85, ¥ The TI-85 interprets y1(y2) as the product of y1 and y2. To have the calculator graph t h e composite function f(g(x)), you should substitute g(x) everywhere an x appears in f(x). For instance, if f(x) = x2 - 1 is in y1 and g(x) = 0.3x + 5 is in y2, enter y7 = y22 Ð 1 for t h e composite function (fog)(x) . On the TI-86, ¥ Enter y1(y2 ) in y7 to obtain the composite function (fog)(x) = f(g(x)). Both Your calculator will evaluate and graph these constructed functions.
B-18 TI-85/TI-86 Graphing Calculators Because the two pieces are close together at x = 90, you may need to take a closer look to see the break. However, because the calculator draws graphs by connecting function outputs wherever the function is defined, it will connect the two pieces unless you tell it not to do so by pressing MORE F5 (FORMT) , use and to choose DrawDot. Press ENTER . Now, take a closer look with F5 (GRAPH) and F3 ZOOM F2 (ZIN ) .
Chapter 1 B-19 After setting the proper horizontal view, you can now use ZFIT to draw a graph of the piecewise function. You can find function values by evaluating outputs on the home screen or using the table. & 1.4 Linear Functions and Models Actual real-world data is used throughout Calculus Concepts. It is necessary that you use your calculator to find a curve that models the data.
B-20 TI-85/TI-86 Graphing Calculators To enter data using the list editor, Press 2nd – (LIST) F4 (EDIT) to access the list editor. Enter the name L2 at the name prompt and press ENTER . Enter the first y-value as e1, press ENTER or enter the second y-value as e2, etc. , If there are data values in this list, see 1.4.3 of this Guide and first delete the ÒoldÓ data.
Chapter 1 B-21 move down one location and a 0 is filled in at the insertion point. Type the data value to be inserted over the 0 and press ENTER . The 0 is replaced with the new value. ¥ To delete a single data value, move the cursor over the value you wish to delete, and press DEL . The values in the list below the deleted value move up one location. 1.4.3 DELETING OLD DATA Whenever you enter new data in your calculator, you should first delete any previously-entered data.
B-22 TI-85/TI-86 Graphing Calculators TI-86 Position the cursor over the L1 at the top of the first column. Replace the L1 values with L1 - 1992 values by pressing F3 (NAMES) , the menu key under L1, and then press – 1992 ENTER . Instead of an actual year, the input now represents the number of years after 1992. 1.4.5 PLOTTING DATA Any functions you have in the y(x)= list will graph when you plot data. Therefore, you should delete them or turn them off before drawing a scatter plot.
Chapter 1 B-23 To graph data on the TI-86: The TI-86 has a built-in command to graph data and autoscale the data window. Press 2nd + (STAT) F3 (PLOT) to display the STAT PLOTS screen. (Your screen may not look exactly like this one.) Note: When drawing a graph from the y(x)= list, you may get an error message or see a scatter plot of ÒoldÓ data as well as the function graph. If so, turn off the STAT PLOTS with F5 (PlOff) ENTER .
B-24 TI-85/TI-86 Graphing Calculators output values are constant. If the first differences are ÒcloseÓ to constant, this is an indication that a linear model may be appropriate. Program DIFF is given in the TI-85/TI-86 Appendix. To run the program, press PRGM F1 (NAMES) and then the F-key under the name of the program. Press ENTER . If you have not entered the data, stop the program by pressing F2 (Quit) . To continue, press F1 (Yes) .
Chapter 1 B-25 TI-85 It is possible to enter data while in the statistics mode of your calculator. However, the examples in this Guide assume that input data is already entered in list L1 and output data is in list L2. Press STAT F1 (CALC) and press the F-keys on the menu to enter L1 as the xlist Name and L2 as the ylist Name. Press ENTER after each entry. Choose the linear model with F2 (LINR) . The y-intercept, a, and the slope, b, of the model is displayed along with the number, n, of data points.
B-26 TI-85/TI-86 Graphing Calculators 1.4.8 PASTING A TI-85 MODEL INTO THE FUNCTION LIST The coefficients of the model found by the calculator should not be rounded. This is not a problem because t h e calculator will paste the entire model into the function list! TI-85 Press MORE F4 (STREG) . At the Name= prompt, type in y1 and press ENTER . Remember that you must use a lower-case y to refer to functions in the y(x)= graphing list. (Any function currently in y1 will be replaced with the linear model.
Chapter 1 B-27 Predict the tax owed in 1998 where the tax is found using the linear model computed from the data given in Section 1.4.1 of this Guide: Tax = 540.37143t + 2538.90476 dollars and t is the number of years since 1992. Note that 1998 is six years since 1992, so x = 6. Enter 6 in the x location, press to move to the y location, and press F5 (SOLVE) . The 1998 tax is predicted to be about $5781. Predict the year in which the tax will be $6000.
B-28 TI-85/TI-86 Graphing Calculators process with this program, it is better to use data that is not perfectly linear and data for which you do not know the best-fitting line. Before using program LSLINE, clear the y(x)= list and enter your data in lists L1 and L 2 . Next, draw a scatter plot. Reset xScl and yScl so that you can use the tick marks to help identify points on the graphics screen. Press GRAPH To activate program LSLINE, press PRGM to view the scatter plot.
Chapter 1 B-29 Pressing allows you to scroll through the list to see the portion that is not displayed. Type L1, press STO‹ , type the letters D A T E and press ENTER to store this list with the name DATE. This list should now appear in your list menu. If you later want to access this list, press 2nd – (LIST). Under NAMES, find DATE. Press the number corresponding to the location of the list, press STO‹ , and type the location you wish to move the list to (say, L1). Press ENTER .
