TI-83 Graphing Calculator Manual to Accompany. Precalculus Graphing and Data Analysis

Transcription

1 TI-83 Graphing Calculator Manual to Accompany Precalculus Graphing and Data Analysis How to Use This Manual This manual is meant to provide users of the text with the keystrokes necessary to obtain the output provided by the TI-83 graphing calculator as that obtained in the text. This manual by no means covers all of the features of this calculator. In addition, once a particular feature is introduced, it will not be discussed again. For example, graphing is introduced in Section 1.2, therefore, when graphing is again required in Section 6.4, the keystrokes required are not reviewed and are assumed to be known. Introduction to the TI-83 Before learning how to use the TI83 as a graphing calculator, an introduction to the TI83 as a scientific calculator may be worthwhile. In this introduction, basic calculator skills will be discussed. Turning the Calculator On and Off The ON button is located in the lower left hand corner of the keyboard. To turn the calculator off, press the 2 nd button and then the ON button. Notice that all 2 nd functions are in yellow on the keyboard. 2 nd Function ON button An Introduction to the TI83 Keyboard If you look at the bottom portion of the TI83, you should recognize that it looks similar to a standard scientific calculator. This portion of the calculator is enclosed in the box shown on the calculator above.

2 When treating the graphing calculator as a scientific calculator, we typically work from the HOME screen. This simply means there are no menus on the screen. Below I have the HOME screen labeled. Notice the blinking cursor. In order to get to the HOME screen from anywhere within the calculator simply press 2 nd QUIT (QUIT is a 2 nd function located directly to the right of the yellow 2 nd button). HOME screen Let s review the keys associated with a scientific calculator starting with the 4 basic functions of addition, subtraction, multiplication and division. The four binary operations are located on the right side of the keyboard (they are blue). The traditional equal sign is replaced with ENTER. So, to perform the operation 4 + 5, simply type and hit ENTER. The TI-83 is aware of the order of operations, so to evaluate: ( 4 5*6) 3 ( 4-5 * 6 ) 3 simply type: and hit ENTER. Note the * is the symbol for multiplication. The parenthesis are the buttons located above the 8 and 9.

3 EXPONENTS 2 Exponents can be handled using either the x button or the caret key (^). For example, to evaluate 5 simply type 5 and then push the x button and hit ENTER. To evaluate 5, 2 type 5 ^ 4 and hit ENTER. NOTE: The x button may only be used to raise a quantity to the 2 nd power. 1 x 2 x Caret (^) Negative button 1 1 To compute the reciprocal of a number use the x button. For example to compute 5, 1 type 5 and then push the x button. Alternatively, you could use the caret key with the negative button. Notice the TI-83 automatically puts the solution in decimal form. Should you want your answer to be in a fraction, use the FRAC option under the MATH menu. For example, to compute the reciprocal of 5 with the solution in fraction 1 form, type: 5 and then push the x button, but instead of hitting ENTER, press the MATH button and you see the following menu: MATH button

4 We wish to access option 1:Frac. Since this is the option currently highlighted, we select it by pressing ENTER or pressing the number 1 on the keyboard. You should obtain the following: Press ENTER to see the result: Suppose we computed the reciprocal of 5 without using the FRAC option as follows: We then realize that we wanted our answer as a fraction. Rather than retyping the previous command, we could simply access the MATH menu and select FRAC and obtain the following:

5 The calculator automatically stores the most recent output in a memory locator call Ans. So, the calculator will take the output in Ans and convert it to a fraction! NOTE: Ans can also be retrieved using 2 nd and the negative button. RADICALS You should see the square root ( ) as a 2 nd function above the button. So to compute 36 simply press 2 nd then press the the parenthesis and press ENTER. To evaluate radicals whose index is different than 2 we again go to the MATH button. Notice options 4 and 5 are radicals: 2 x 2 x button, then 3 6 close So, to compute 4 81, type 4 then select option 5 from the MATH menu, then type 81, close the parenthesis and hit ENTER. EXPONENTIALS AND LOGARITHMS The LOG and LN buttons on the keyboard on located in the x x bottom left portion of the keypad. Notice that 10 and e are 2 nd functions above these keys. It s fairly straightforward to use these features. LOG LN

6 For example to compute log 3, simply press the LOG button followed by a 3 and a close 4 parenthesis and hit ENTER. To compute e, press 2 nd LN followed by 4 and hit ENTER. TRIGONOMETRIC FUNCTIONS Notice the SIN, COS, and TAN functions. To use these features, we need to know whether our angle is in Degrees or Radians. Setting the angle is done by pressing MODE. SIN, COS, TAN MODE After pressing the MODE button you should see the following: Currently, the calculator is in radian mode. If we wish to change the calculator to degree mode we use the arrows on the keyboard to move the blinking cursor down to highlight Degree and then press ENTER. You should have the following:

7 To get back to the HOME screen press 2 nd o QUIT. Suppose we wish to evaluate: sin 30, we press the SIN button then 3 0 ) and hit ENTER. The inverse trigonometric functions are second functions above their corresponding 1 function. So, sin is a 2 nd function of sin, etc. OTHER FEATURES If your HOME screen ever becomes too cluttered, we can clear it by using the CLEAR button. Another neat feature is the 2 nd ENTER feature. Press 2 nd ENTER a couple of times. You should see that previous entries appear on the command line. The TI83 can remember about the last 7 10 commands entered!!! These commands can then be edited. The remainder of the manual discusses features as they arise in the text. 1.2 Graphs of Equations Example 3 Graphing an Equation on a Graphing Utility 2 Graph the equation 6x + 3y = 36 SOLUTION Step 1: Done on page 15. Step 2: To enter the expression y 2 = 2x + 12 into a TI-83. Press Y =

8 Y = You should see the following on your screen: The highlighted Plot1 means StatPlot is on. To turn it off use the up arrow to put the cursor on Plot1 and hit enter. Move the cursor back down to Y 1 = and your screen should look like this: In Y 1, type in the following order (consult the figure for the location of these buttons) 1. ( - ) X,T,θ,n 4. x

9 x 2 This squares the quantity preceding it. X,T,θ,n ( - ) NOTE: This is not the subtraction sign. This button represents the negative of a number. Your screen should have the following: STEPS 3 and 4: To obtain a graph with the standard viewing window press ZOOM and then select option 6. Zstandard. Immediately a graph should start to appear on the standard window. STEP 5: To adjust the viewing window, press WINDOW. Use the arrows to move the cursor to the values you wish to change. We wish to change Ymin to -12, so move the cursor to Ymin and make its value -12. Press GRAPH and you should obtain the following:

10 USING THE TABLE FEATURE The table feature can be activated by pressing 2 nd GRAPH. In order for the feature to work, you must have at least one equation entered in the Y = menu. Before proceeding to the table, I first want to introduce the TABLESET feature. This is activated by pressing 2 nd WINDOW on your TI-83. TABLE TaBLeSET After pressing 2 nd WINDOW, the following should appear. Independent Variable I first want to direct your attention to the Independent variable (x-variable) setting. Currently AUTO is highlighted. This means the calculator will automatically fill in a table starting at x = 0 and increasing by values of 1. To see this concept we will reproduce Table 2 on page 18. Let Y 1 = x 2-4. Press 2 nd WINDOW to access table setup and change TblStart to -3 to obtain the following.

11 Now press 2 nd GRAPH to obtain the following TABLE: You can use the arrows to scroll around the TABLE. Notice that whatever quantity the cursor lies on is displayed in the bottom row of the TABLE. Let's go back to TableSetUp to see the affect of changing the independent variable to ASK. Bring up TableSetUp and use the cursor to highlight Ask under the independent variable. Now press ENTER. Your screen should look as follows: Now press 2 nd GRAPH to display a TABLE. The TABLE should be empty. If it is not, press delete with the cursor in the X-column until it is empty. Your screen should look as follows: Enter values of x into the X-column and the corresponding y values get automatically filled in. For example we could enter x = -2, 0, and 2 and obtain the following:

12 These are the intercepts of the graph of the equation!! EXAMPLE 8 Using a Graphing Utility to Find Intercepts Use a graphing utility to find the intercepts of the equation y = x 3-8. SOLUTION Graph Y 1 = x 3-8 using Xmin = -5, Xmax = 5, Xscl = 1, Ymin = -20, Ymax = 10, Yscl = 5. The VALUE feature on a TI-83 evaluates an equation for any value of x and locates the corresponding ordered pair on the graph of the equation. Letting x = 0 allows us to find the y-intercept of the graph of an equation. After graphing the equation, press 2 nd TRACE (this activates the CALC menu) CALC menu You should obtain the following on your screen:

13 We want to select option 1: value. After selecting this option, we obtain the following: After the X= type 0 and press enter to obtain the following: Thus, when x = 0, y = -8. We can use the ZERO feature on a TI-83 to determine the x-intercepts of the graph of the equation. Again, go to the CALC menu, but this time select option 2: ZERO. The calculator asks for a LEFT BOUND - this is a value of x to the left of the x-intercept. To establish the left bound use the left and right arrows to move the cursor until it is left of the x-intercept you wish to approximate, then press ENTER. x-intercept

14 Notice the arrow. This is telling you the calculator will look for an x-intercept to the right of this value. The calculator now wants a RIGHT BOUND - this is a value of x to the right of the x-intercept. Use the left and right arrows to move the cursor somewhere to the right of the x-intercept you wish to approximate and press ENTER. You obtain the following: You now need to give the calculator an initial guess as to the value of the x-intercept. This can be any x-coordinate you wish provided it is between the left and right bounds. Move the cursor near the x-intercept and press ENTER. The calculator then thinks for a while and provides a solution: The x-intercept is Solving Equations EXAMPLE 1 Using ZERO (or ROOT) to Approximate Solutions of an Equation Simply follow the same procedures as were used in Example 8 from Section 1.2

15 EXAMPLE 2 Using INTERSECT to Approximate Solutions of an Equation Find the solution(s) to the equation 3(x - 2) = 5(x - 1). Round answers to two decimal places. SOLUTION Graph each side of the equation: Y 1 = 3(x - 2); Y 2 = 5(x - 1) using the viewing window Xmin = -4, Xmax = 4, Xscl = 1, Ymin = -15, Ymax = 5, Yscl = 5. To determine the solution, find the x-coordinate of the point of intersection. Select the CALC menu (2 nd TRACE). Select option 5: INTERSECT by either highlighting INTERSECT or Pressing 5. The calculator displays the following: If you press the left or right arrow, you should notice the cursor TRACEing along Y 1. Move the cursor near the point of intersection of the two graphs and press ENTER. The screen displays the following:

16 Again move the cursor near the point of intersection and press ENTER. The screen now displays the following: Simply press ENTER a third time and the screen displays the coordinates of the point of intersection: The solution to the equation is x = Check: The calculator now has x = -0.5 automatically stored. To check the answer type 3(x - 2) on the home screen and press enter. Do the same for 5(x - 1). You obtain the following: The solution checks. 1.5 Solving Inequalities Example 8 Solving Combined Inequalities Solve the inequality -5 < 3x - 2 < 1 and draw a graph to illustrate the solution.

17 SOLUTION Graph Y 1 = -5, Y 2 = 3x - 2, and Y 3 = 1 using the viewing window Xmin = -8, Xmax = 8, Xscl = 1, Ymin = -8, Ymax = 4, Yscl = 1. We wish to find the point of intersection between Y 1 and Y 2 and the point of intersection between Y 2 and Y 3. We will first find the point of intersection between Y 1 and Y 2. After graphing all three equations, bring up the CALC menu and select 5: INTERSECT. When First Curve? appears, the cursor should be on Y 1. Move the cursor near the point of intersection and press ENTER. We know we are TRACEing on Y 1 because of this!! After pressing ENTER, the cursor automatically jumps to Y 2. Again, move the cursor near the point of intersection. Press ENTER a third time and you obtain the point of intersection as (-1, -5). We now wish to find the point of intersection between Y 2 and Y 3. Bring up the CALC menu and select 5: INTERSECT. The first curve is on Y 1, but we want it on Y 2. This is accomplished by pressing the down arrow. Move the cursor near the point of intersection and press ENTER. The cursor jumps to Y 3. Again, move the cursor near the point of intersection and press ENTER. Press ENTER a third time to obtain your solution, (1, 1).

18 The graph of Y 2 is between the graphs of Y 1 and Y 3 for -1 < x < 1. EXAMPLE 9 Solving an Inequality Involving Absolute Value Solve the inequality x < 4 SOLUTION Graph Y 1 = x and Y 2 = 4. To graph Y 1 = x, enter the Y = menu. With the cursor in Y 1, press MATH. MATH You should see the following on your screen: Press the right arrow to highlight NUM:

19 Select 1: abs( and the absolute value function will appear in Y 1. NOTE: If abs( appears on your home screen rather than in Y 1, you forgot to press Y =! You should have the following on your screen: Press X, T, θ, n to put the variable x after the abs( and finally press ). Hit ENTER. In Y 2, we put 4 to graph Y 2 = 4. Set the viewing window so Xmin = -6, Xmax = 6, Xscl = 1, Ymin = 0, Ymax = 6, and Yscl = 1. Press graph and use INTERSECT twice to obtain the solution -4 < x < 4 since the graph of Y 1 is below the graph of Y 2 in this interval. 1.6 Lines Square Screens To get a square screen on the TI-83 press ZOOM and select option 5: ZSquare. ZOOM

20 The calculator will automatically graph any equations entered in the Y = menu. A note about ZSquare - it will square the viewing window that is established. You may need to adjust the initial window setting before using ZSquare. 1.7 Scatter Diagrams; Linear Curve Fitting Example 1 Drawing a Scatter Diagram The data listed in Table 9 (see page77) represent the apparent temperature versus relative humidity in a room whose actual temperature is 72 o Fahrenheit. Draw a scatter diagram using a graphing calculator. Solution First press the STAT button. STAT On the screen you should see the following: Select 1: Edit by either pressing ENTER or pressing 1. You should now be in the data editor.

21 In L1 we will enter the value of the x-variable (relative humidity). Enter 0, 10, 20, etc. into L1. Use the arrows to move the cursor to L2 and enter the values of the y-variable, apparent temperature. After the data has been entered, press 2 nd QUIT. You should now be at the HOME screen. To draw the scatter diagram, we have to turn STAT PLOTS on and select the scatter diagram option. Press 2 nd Y =. QUIT Y = 2nd ARROWS Your screen should look like this: Select 1: Plot 1 by pressing ENTER or Pressing 1. Now you should have the following screen:

22 Move the blinking cursor to On and press ENTER. Leave the Type where it is above (if you have a different type selected, use the arrows to highlight the same icon as above and press ENTER). Your Xlist should be L1 and the Ylist should be L2. (If they are not, use the arrows to move the cursor and change them. L1 is a 2 nd function above 1, so to get L1 type 2 nd 1.) The mark can be any of the three, but I find the box to be easiest to read. We are now ready to graph the scatter diagram. Press ZOOM. ZOOM You should see the following on your screen: 2 nd 1 is L1 Now scroll down (or up) until you reach option 9: ZoomStat Press ENTER and, if everything goes according to plan, you will obtain your scatter diagram. NOTE: We could have also obtained ZoomStat by pressing 9 in the ZOOM menu.

23 If you are wondering what the viewing window is press WINDOW: WINDOW GRAPH You should see the following on your screen: So Xmin = -2.5, etc. You could change this viewing window by typing in any values you wish. To obtain the window used in Figure 17(b), we would let Xmin = -10, Xmax = 110, Xscl = 10, Ymin = 62, Ymax = 78, Yscl = 2. Make these changes, then press GRAPH to obtain the scatter diagram. EXAMPLE 4 FINDING THE LINE OF BEST FIT With the data from Example 1, find the line of best fit using a graphing utility. Solution First, we enter the data into the graphing utility in the STAT menu. This is done in Example 1. We then execute the Linear Regression option on a TI-

24 83, by pressing STAT to obtain the following window: Press the right arrow to highlight CALC and select option 4: LinReg(ax + b). Press ENTER or 4: With the data in L1 as the independent variable and the data in L2 as the dependent variable, press ENTER to obtain the line of best fit. The TI-83 does not automatically provide the user with the correlation coefficient. If you desire this value, you must perform the following steps. Press the CATALOG option (this is a 2 nd function above 0 at the bottom of the keyboard)

25 Scroll down the screen under you reach DiagnosticOn Press ENTER to insert the phrase "DiagnosticOn" on the HOME SCREEN. Press ENTER a second time to enact the command. A neat feature on TI graphing calculators is that they have memory of recent keystrokes. These keystrokes can be "brought back" by pressing 2 nd ENTER. Press the 2 nd Function and then ENTER and the most recent command "DiagnosticOn" comes up on the HOME SCREEN. Press 2 nd ENTER a second time and the next prior command enacted (LinReg) comes up on the HOME SCREEN. Once you have the command LinReg (ax + b) on your HOME SCREEN, press ENTER to obtain the following: Page Using a Graphing Calculator to Evaluate any Function 2 Use a TI-83 to evaluate the function G( x) = 2x 3x at x = 3. That is, find G(3).

26 Solution Enter the function into the Y = menu to obtain: Press 2 nd Quit to exit the Y = menu. You should now be at the HOME screen. To evaluate the function in Y 1, press VARS and obtain the following screen: Press the right arrow to highlight Y-VARS and obtain the following screen: Select 1: Function by pressing ENTER or 1 to obtain: Select 1: Y 1 and obtain:

27 Now press ( 3 ) to obtain: Hit ENTER to obtain the solution: 2.2 Characteristics of Functions EXAMPLE 5 Using a Graphing Utility to Locate Local Maxima and Minima and to Determine Where a Function is Increasing and Decreasing 3 (a) Use a graphing utility to graph f ( x) = 6x 12x + 5 for -2 < x < 2. Determine where f has a local maximum and where f has a local minimum. (b) Determine where f is increasing and where it is decreasing. Solution (a) Graphing utilities have a feature that finds the maximum or minimum point of a graph within a given interval. Graph the function f with Xmin = -2 and Xmax = 2, Xscl = 1, Ymin = -10, Ymax = 30, and Yscl = 5. You should obtain the following:

28 To find the maximum, press 2 nd CALC as shown below: 2nd CALC is above TRACE You should obtain the following on your screen: We want to select option 4: maximum by either pressing 4 or scrolling down using the arrows until the 4 is highlighted and pressing ENTER. Upon selecting 4: maximum, you obtain the following: We can see a local maxima occurs near x = -1. The query Left Bound? from the calculator means the TI-83 wants you to select a point somewhere left of the local maxima. Use the arrows to move the cursor anywhere left of the local maxima.

29 Once the cursor is to the left of the local maxima, press ENTER. Your screen displays the following: Left Bound Notice two things (1) The arrow indicates the calculator will look for a maximum value to the right of the left bound. (2) The calculator now wants a right bound. So, move the cursor right of the local maxima using the arrows, then press ENTER. Your screen should be similar to the one below. Left Bound Right Bound You now have the left and right bounds established. The calculator prompts the user for a Guess?. Move the cursor near the local maxima and press ENTER. The calculator then provides the local maxima. These results agree with Figure 18(a) on page 121. The procedure for finding the local minima is similar to that of finding the local maxima, except we select option 3: minimum in the CALC menu. Try this yourself to see if you obtain the results shown in Figure 18(b).

30 2.3 Library of Functions; Piecewise-defined Functions EXAMPLE 1 Analyzing a Piecewise-defined Function For the following function f, x + 1 f ( x) = 2 2 x (c) Graph f using a graphing utility. if if if 1 x < 1 x = 1 x > 1 SOLUTION Press Y =. After Y 1 = Type the following: ( x +1)(x We now need to insert a > symbol. To get the inequalities, you must access the TEST menu. This is a 2 nd function above MATH on the keyboard. The TEST menu is obtained by pressing 2 nd MATH After pressing 2 nd TEST, you should see the following on your screen:

31 Select item 4 as shown below: Continue typing the function to be graphed so that you have the following in the Y= menu. The calculator must be in DOT MODE. Press the MODE button on the keyboard and use the arrows to put the cursor on DOT and press ENTER: Alternatively, we could change the graphing style from the Y = menu by 3 dots indicate

32 using the arrows to put the cursor to the left of Y 1 = and pressing ENTER until the line style changes to 3 dots as shown: Now set the viewing window as shown: Press GRAPH and you should obtain the result shown in Figure 29(a) on page Operations on Functions; Composite Functions EXAMPLE 2 Evaluating a Composite Function 2 Suppose that f ( x) = 2x 3 and g( x) = 4x. Find (a) ( f o g)( 1) 2 Solution Let Y = f ( x) = 2x 3 and Y = g( x) = 4x 1 2 as shown. f o on the TI-83, we first need to insert Y 1 on the home screen. Press the VARS button To compute ( g)( 1)

33 VARS You should see the following: Hit the right arrow to highlight Y-VARS. Select option 1: Function by pressing ENTER.

34 Select Y 1 by pressing ENTER. Y 1 appears on your HOME screen. Press the open parenthesis located directly above the 8 on the keyboard. Repeat the process described above to obtain Y 1 in order to place Y 2 on the HOME screen. Press the open parenthesis key again, the number 1 and two close parenthesis and press ENTER. You should obtain the results shown in Figure 45 on page Quadratic Functions; Curve Fitting EXAMPLE 11 Fitting a Quadratic Function to Data The data in Table 2 on page 184 represent the enrollment (in millions) in public high schools for the years (a) Using a graphing utility, draw a scatter diagram of the data. (b) Find the quadratic function of best fit. (c) Using a graphing utility, draw the quadratic function of best fit on the scatter diagram. Solution (a) To draw a scatter diagram of the data, refer back to the steps provided in Section 1.7. (b) In Section 1.7, you found the Line of Best Fit. In this section, we find the Quadratic Function of Best Fit. The keystrokes are essentially the same, but instead of selecting option 4: LinReg (ax + b), we select option 5: QuadReg.

35 Press ENTER to get QuadReg on the HOME SCREEN. In L1 we have the independent variable and in L2, we have the dependent variable. In anticipation of graphing the quadratic function of best fit on our scatter diagram, we can perform the following steps to have the calculator automatically store the quadratic function of best fit into Y 1. With QuadReg on the HOME SCREEN, press VARS, scroll right to highlight Y-VARS, select 1: Function and select 1: Y 1. After doing this, you should have the following on your HOME SCREEN: This means the calculator will find the quadratic function of best fit and store the function into Y 1. Press ENTER to enact the program. The following output appears on your screen. If you press Y =, you see the equation in Y 1. (c) To graph the quadratic function of best fit on the scatter diagram is as simple as pressing GRAPH with the function already stored in Y 1.

36 3.2 Power Functions; Curve Fitting EXAMPLE 3 Fitting Data to a Power Function In order to find the power function of best fit repeat the steps given for finding the quadratic function of best fit. However, instead of selecting 5:QuadReg, select A:PwrReg. 3.3 Polynomial Functions; Curve Fitting EXAMPLE 7 A Cubic Function of Best Fit In order to find the power function of best fit repeat the steps given for finding the quadratic function of best fit. However, instead of selecting 5:QuadReg, select 6:CubicReg. 4.2 Complex Numbers; Quadratic Equations with a Negative Discriminant EXAMPLE 1 Adding and Subtracting Complex Numbers (a) (3 + 5i) + (-2 + 3i) SOLUTION From the HOME screen, press ( To get the i, press 2 nd and then press the decimal point.

37 2 nd Function Decimal Point You should now see the following on your HOME screen: Now press ) + ( i) and hit ENTER You should see the following on your HOME screen: EXAMPLE 4 Writing the Reciprocal of a Complex Number in Standard Form 1 Write 3 + 4i 3 + 4i. in standard form a + bi; that is, find the reciprocal of

38 SOLUTION Enter the expression 1 / ( i ) on the HOME screen: To obtain a fraction, press MATH, then select option 1: Frac under the MATH menu. MATH Hit ENTER to select 1: Frac. You should see the following on your HOME screen: Press ENTER to obtain the following result:

39 5.2 EXPONENTIAL FUNCTIONS EXAMPLE 1 Using a Calculator to Evaluate Powers of 2 Using a calculator, evaluate: (a) SOLUTION On the HOME screen type 2. Now press the caret (^) key. Caret key Finally, type 1.4 as shown: Press ENTER to obtain the result shown in Figure Exponential, Logarithmic, and Logistic Curve Fitting EXAMPLE 1 Fitting a Curve to an Exponential Model (a) Using a graphing utility, draw a scatter diagram (b) Using a graphing utility, fit an exponential model to the data.

40 SOLUTION (a) Enter the independent variable (year) in L1 and the dependent variable (price) in L2. The scatter diagram is shown below. Scatter diagrams are discussed in Section 1.7 if you need a review. (b) Press STAT, scroll right to highlight CALC and select 0;ExpReg as shown below: You should obtain the following on your HOME screen Hit ENTER to obtain the following output:

41 EXAMPLE 2 Fitting a Curve to a Logarithmic Model To perform logarithmic curve fitting, repeat the steps described above for Exponential curve fitting, but select 9: LnReg as shown below: EXAMPLE 3 Fitting a Curve to a Logistic Growth Model To perform logistic curve fitting, repeat the steps described above for Exponential curve fitting, but select B: Logistic as shown below: NOTE: In order for the output to include the correlation coefficient, r, you must turn DIAGNOSTICS ON in the catalog. See Section 1.7 for a discussion of this. 6.1 Angles and Their Measure EXAMPLE 2 Converting Between Degrees, Minutes, Seconds and Decimal Forms o (a) Convert t a decimal in degrees. o o (b) Convert to the D M S form.

42 SOLUTION (a) On the HOME screen type: 50. You should see the following: Now, access the angle menu. On the TI83 Plus, it is above the APPS button as shown: ANGLE Now select 1: O You should have the following on your HOME screen:

43 Now type 6 to obtain the following: Go back to the angle menu and select item 2: You now have the following on your HOME screen: Type 21. To obtain the double quotes representing the seconds, press ALPHA then the plus sign as shown. ALPHA The double quotes are above the + sign.

44 After pressing ENTER you obtain the following result: (c) Type on your HOME screen. Then select the angle menu and choose option 4: DMS You should now have the following on your HOME screen: Press ENTER to obtain the solution.

45 6.2 TRIGONOMETRIC FUNCTIONS: UNIT CIRCLE APPROACH EXAMPLE 9 Using a Calculator to Approximate the Value of Trigonometric Functions Use a calculator to find the approximate value of: (a) o cos 48 (c) tan π 12 SOLUTION (a) Set the MODE to degrees. This is done by pressing the MODE button on the calculator. Now use the arrows to put the cursor on Degree and press ENTER. You should have the output shown in Figure 30(a). The trigonometric functions sine, cosine and tangent are located in the middle of the keyboard: Sine Tangent Cosine Press the COS button, then press 4 8 ). Hit ENTER to obtain the results shown in Figure 30(b). (b) Set the MODE to receive radians. Press the TAN button. Now press π. π is obtained by pressing 2 nd ^

46 Now type / 1 2 ) Hit ENTER to obtain the result shown in Figure Sinusoidal Graphs; Sinusoidal Curve Fitting EXAMPLE 10 Finding the Sine Function of Best Fit Use a graphing utility to find the sine function of best fit for the data in Table 11 on page 429. SOLUTION To draw the scatter diagram follow the same steps as those presented in Section 1.7. To find the sinusoidal function of best fit, follow the same procedure as that presented in Section 5.8 (this section shows how to store the function of best fit in Y 1) except you should select C: SinReg under the STAT, CALC menu: 7.5 The Inverse Trigonometric Functions (I) EXAMPLE 3 Finding an Approximate Value of an Inverse Sine Function Find the approximate value of: (a) 1 sin 1 3 SOLUTION Set the MODE to radians. From the HOME screen, press the 1 sin button. This is a 2 1 sin is accessed by pressing 2 nd sin

47 You should have the following on your HOME screen: Now type 1 / 3 ) and press ENTER to obtain the following: 9.1 Polar Coordinates EXAMPLE 5 Converting from Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates of the points with the following polar coordinates: (a) ( 6, π / 6) SOLUTION From the HOME screen, access the ANGLE menu (2 nd APPS on the TI-83 Plus) and select 7: P Rx(. You should have the following on your HOME screen:

48 Now type 6, π following: / 6 ) and hit ENTER to obtain the Repeat the process to obtain the y-coordinate, but select 8:P Ry( instead. EXAMPLE 6 Converting from Rectangular Coordinates to Polar Coordinates Find polar coordinates of a point whose rectangular coordinates are (0, 3). SOLUTION From the HOME screen access the ANGLE menu. Select item 5:R Pr(. You should have the following on your HOME screen: Now type 0, 3 ) and hit ENTER to obtain the following:

49 To obtain the angle, repeat the above procedure, but select 6:R Pθ(. 9.2 Polar Equations and Graphs EXAMPLE 4 Graphing a Polar Equation Using a Graphing Utility Use a graphing utility to graph the polar equation r sin θ = 2. SOLUTION We must change the MODE of the calculator to POLAR mode. This is done by pressing MODE and using the arrows to put the cursor on POL and pressing ENTER as shown: Now press the Y = button and enter the expression in r 1 : NOTE: Because the calculator is in POLAR mode, the θ shown in r 1 is obtained by pressing the X,T, θ, n button. Set the viewing window as shown on page 571 and press GRAPH. You should obtain the following:

50 10.7 Plane Curves and Parametric Equations EXAMPLE 2 Graphing a Curve Defined by Parametric Equations Using a Graphing Utility Graph the curve defined by the parametric equations x = 3t 2 y = 2t 2 t 2 SOLUTION We must first put the calculator into parametric mode. Press the MODE button and use the arrows to highlight PAR and press ENTER: Now press the Y = button. Enter the equations as shown: NOTE: When in parametric mode, pressing the X,T,θ,n button results in a T on the screen. Press the WINDOW button and enter the values shown on page 692. Now press GRAPH to obtain the following:

51 EXAMPLE 5 Projectile Motion Suppose that Jim hit a golf ball with an initial velocity of 150 feet o per second at an angle of 30 to the horizontal. (d) Using a graphing utility, simulate the motion of the golf ball by simultaneously graphing the equations found in part (a). SOLUTION Press the MODE button. You should already have PAR highlighted. Scroll down and highlight SIMUL and press ENTER. Enter the equations x = 75 3t and y = 16t t into the calculator by pressing the Y = button. Set the viewing window as follows as stated on page 696 and press GRAPH. One interesting feature of the calculator is that you can pause the actual drawing of the graph by pressing ENTER. Press ENTER again to resume drawing. Below you can see what the screen looks like if you do this. These dots mean the graph is paused.

52 11.3 Systems of Linear Equations: Matrices EXAMPLE 6 Solving a System of Linear Equations Using Matrices Solve: ( 1) ( 2 ) ( ) x y+ z= 8 2x+ 3y z= 2 3x 2y 9z = 9 3 Graphing Solution The augmented matrix of the system is To enter this matrix into the TI-83, press the MATRX button. On the TI83+, the MATRX button is a 2 nd Function.

53 MATRX on the TI83 You should see the following on the screen: To enter the augmented matrix into the calculator, highlight EDIT: After pressing ENTER you obtain the following:

54 We wish to enter a 3 by 4 matrix, so press 3 and hit ENTER, then press 4 and hit ENTER. You should see the following on your screen: Now enter the entries of the augmented matrix. You can use the arrows to move around. Notice the bottom of the screen indicates your location within the matrix. For example 1,1 means the cursor is in row 1, column 1. So, 2, 3 would mean the cursor is in row 2, column 3. After entering all the elements of the augmented matrix, press 2 nd QUIT to return to the HOME screen. To see matrix [A] on the HOME screen, press MATRX to obtain the following: You can now see matrix [A] is a 3 by 4 matrix. Press ENTER and obtain the following: Press ENTER again to see the entries of matrix [A]:

55 This is the same as FIGURE 7(a) on page 732. To obtain the ref( command, press MATRX and scroll right to highlight MATH. Scroll down to item A:ref(. Select it by pressing ENTER. You should see the following on your home screen: Now select matrix A by pressing MATRX and highlighting matrix A under the name menu. Then press ENTER:

56 Now press the close parenthesis ) - this is a 2 nd function above the number 9. We want our matrix in fractional form, so press MATH and select 1: Frac. Now press ENTER to obtain the result shown in Figure 7(b). You can scroll right using the arrow key to see the rest of the matrix. If you want your matrix in reduced row echelon form, press MATRX, then highlight the MATH menu and scroll down until you select B:rref(. Hit ENTER and now press MATRX, and select matrix A. Close the parenthesis and select the FRAC option. Hit ENTER to obtain the following: 11.4 Systems of Linear Equations: Determinants EXAMPLE 2 Using a Graphing Utility to Evaluate a 2 by 2 Determinant Use a graphing utility to evaluate SOLUTION

57 We enter the matrix A. If you don't remember how to enter a matrix, see Section 11.3 of this TI-83 manual. You should have the following in A: Select MATRX, scroll right to highlight MATH. Now select 1:det(. You should have the following on your home screen: Select matrix A by press MATRX and selecting 1:[A] from the NAMES menu. Close your parenthesis and hit ENTER to obtain the result. EXAMPLE 3 Solving a System of Linear Equations Using Determinants Use Cramer's Rule, if applicable, to solve the system 3x 2y = 4 6x+ y = 13 Graphing Solution Enter all three matrices as defined in the text. For example,

58 3 2 A = 6 1 and 4 2 B = 6 1 To find the value of x, we evaluate det([b])/det([a]) as shown below: 11.5 Matrix Algebra EXAMPLE 3 Adding and Subtracting Matrices Suppose that A = and B = Find (a) A + B (b) A - B SOLUTION Enter the matrices into the graphing utility. For a review of this, see Section 6.3 tutorial. Name each matrix A and B. To add matrix A and B 1) Press MATRX. Under the NAMES menu, select 1.[A] by pressing 1 or hitting ENTER. You should have the following on your home screen: 2) Press +.

59 3) Press MATRX. Under the NAMES menu, select 2. [B] by pressing 2 or highlighting 2 and hitting ENTER. You should have the following on your home screen: 4) Now press ENTER to obtain the result. To find A - B, repeat the above steps, but select the subtraction sign instead of the addition sign. EXAMPLE 5 Operations Using Matrices Suppose that A = B = C 9 0 = 3 6 Find (a) 4A (b) 1 C (c) 3A - 2B 3 SOLUTION Type everything as seen in Figure 12 on page 755. EXAMPLE 13 Finding the Inverse of a Matrix The matrix

60 1 1 0 A = is nonsingular. Find its inverse. SOLUTION Enter the matrix A into your TI-83 as done in Section Press MATRX and select matrix [A] so it appears on the home screen as follows: Press the inverse button on your calculator: Inverse x -1 You should have the following on your home screen: Press ENTER to obtain the result:

61 EXAMPLE 15 Using the Inverse Matrix to Solve a System of Linear Equations Solve the system of equations: x+ y = 3 x + 3y + 4z = 3 4y + 3z = 2 SOLUTION Enter the following matrices into your TI-83: A = B = 3 2 Obtain the following on your home screeen: Now hit ENTER to obtain the result:

62 Section 11.8 Systems of Linear Inequalities; Linear Programming EXAMPLE 2 Graphing a Linear Inequality Using a Graphing Utility Use a graphing utility to graph 3x+ y 6 0. SOLUTION Graph Y1 = 3x + 6 (this is obtained by solving 3x + y - 6 = 0 for y). Use the same viewing window as that used in the text. Now test the point (-1, 2) by typing the following: The asterisk is the multiplication symbol. We now need the inequality. It can be found in the TEST menu (this is a second function located above the MATH button): TEST menu Once you select the TEST menu you should obtain the following on your screen:

63 Select 6: < by pressing 6 or scrolling down and highlighting 6 and then pressing ENTER. You should now have the following on your home screen: Finally, press 0 and hit ENTER: The 1 as output means the statement is true so that (-1, 2) is a solution to the inequality. Therefore, we shade the side containing (-1, 2). The side containing (-1, 2) is below the line. To shade below the line go back to the Y= menu: Place cursor here Notice to the left of Y 1, there is a solid line. Use the arrows to place the cursor on the solid line as shown above. While the cursor is in this location press ENTER. You should notice the line style changes to a thick line. Pressing ENTER a second time causes the

64 calculator to shade above the line. Press ENTER one more time and we see the calculator will now shade below the line: Press GRAPH with the line style as shown above to obtain the following graph: 12.1 Sequences EXAMPLE 2 Using a Graphing Utility to Write the First Several Terms of a Sequence Use a graphing utility to write the first six terms of the following sequence and graph it. SOLUTION On your TI83, press 2 nd LIST: { an} n 1 = n

65 2 nd LIST You will see the following on the HOME screen: Scroll right to highlight OPS and select 5: seq( so you see the following on your HOME screen: The sequence command requires the following input: Seq(function,variable,starting point, ending point, step) Enter the following in your calculator:

66 After pressing ENTER, we obtain the following: If you want the terms of the sequence as fractions, press MATH and select 1.4Frac to obtain the following: Press ENTER to obtain the following: If you wish to see the remaining terms in the sequence, use the arrow keys to scroll right.

67 To graph the sequence, first put the TI83 in sequence mode. This is done by pressing the MODE button: MODE Use the arrows to highlight Seq and press ENTER to select sequence mode. Press Y= to obtain the following: Enter the function in u(n)= as follows:

68 NOTE: The variable n is obtained by pressing the X,T,θ,n button. For u(nmin), type the value of the function at n = nmin. Typically, the value of n will be 1 since we require the sequence be defined for positive integers (1,2,3, ). Now set the viewing window by pressing WINDOW. Fill in the screen as follows: Now press GRAPH: You can TRACE along the graph to see the terms in the sequence. The Factorial Symbol Suppose we wish to evaluate 8!. This is done by typing 8 on the HOME screen as follows: We now wish to enter the factorial symbol. The factorial symbol is located in the MATH menu. Press MATH, then use the arrows to highlight PRB. You should see the following on your screen:

69 Select 4:!. You now see the following on your HOME screen: To obtain the result press ENTER. EXAMPLE 7 Using a Graphing Utility to Write the Terms of a Recursively Defined Sequence Use a graphing utility to write down the first five terms of the following recursively defined sequence. SOLUTION s = 1 s = 4s 1 n n 1 We put the TI83 in sequence mode. Press Y= so we can enter the recursive formula. The only difference between this function and the one from Example 2, is that this function is recursive. We enter u(n) by putting the cursor on u(n)=. Press 4 u ( n - 1 ) The u seen in the formula is a 2 nd function above 7 on the keyboard. See Figure 8(a).

70 u (2 nd 7) Again, the n is obtained by pressing the X,T,θ,n button. Finally, u(nmin) is set equal to s 1. Next, set the viewing window. Since we want the first 5 terms, nmin = 1 and nmax = 5. Now press GRAPH to obtain the figure. To obtain a TABLE of values in the sequence, set up the table as shown below (remember, TBLSET is 2 nd WINDOW).

71 We start the table at 1 since the recursive sequence begins at n = 1 and Tbl = 1 because the domain of the sequence is the positive integers. Now press TABLE to obtain: EXAMPLE 12 Using a Graphing Utility to Find the Sum of a Sequence Using a graphing utility, find the sum of the sequence SOLUTION Press 2 nd STAT to obtain the LIST menu. Then highlight MATH as shown and select 5:sum(. You should have sum( on your HOME screen as shown. 5 k= 1 3k Again press 2 nd STAT to obtain the LIST menu, but now highlight the OPS menu and select 5:seq( so you have sum(seq( on the HOME screen.

72 Now type the rest of the command line as shown below and press ENTER to obtain the result. NOTE: The sequence command has the following syntax: Seq(function, variable, start, end, step) 12.5 The Binomial Theorem EXAMPLE 1 Evaluating n j Find: (d) SOLUTION From your HOME screen type 65:

73 Press MATH and highlight PRB and select 3: ncr: You should have the following on your HOME screen: Now type 15 and press ENTER to obtain the result Permutations and Combinations EXAMPLE 5 Computing Permutations Evaluate P(52, 5) SOLUTION From the HOME screen, type 5 2 to obtain the following:

74 Now press the MATH button and scroll right or left highlighting PRB and select 2:nPr: You should have the following on your HOME screen: Now type 5 and press ENTER obtaining: EXAMPLE 8 Using Formula (2) Evaluate C(52, 5) SOLUTION Use the same procedures as were followed to evaluate P(52, 5) except choose 3: ncr under the MATH, PRB menu Obtaining Probabilities from Data Exploration on page 890

75 In order to perform a simulation, we must first set the seed. Typically, I suggest that students use the last four digits of their social security number as the seed. From the HOME screen type the last four digits of your social security number (or any number you wish). Now press the STO button. Finally, press the MATH button, highlight PRB and select 1:rand. Press ENTER and you ve set the seed! STO Now press the MATH button again, highlight PRB and select 5:randInt(. After randint( appears on the HOME screen, type 1, 2, 100 ) Do not hit ENTER yet, we want to store the results in L1. Now press STO following by L1. Remember, L1 is obtained by pressing 2 nd 1. Now press ENTER.

76 We now want to draw a histogram of the data in L1 so we can determine the number of 1s and 2s. Press 2 nd Y = to access the STAT PLOT menu. Select 1:Plot1. After selecting Plot1 you obtain the following: Highlight On Histogram Icon Turn Plot1 on by using the arrows to place the cursor on On. Press the down arrow to place the cursor in the Type: field. Highlight the histogram icon and press ENTER. Notice the screen now changes a little. Your screen should match what I have below: Now we need to set the viewing window. Press the WINDOW button. Make sure your calculator is in Function MODE and the Y = menu is cleared of any functions. Let Xmin = 0, Xmax = 4, Xscl = 1, Ymin = 0, Ymax = 75, Yscl = 25. Press GRAPH.

77 Press TRACE and use the right arrow to move the cursor to the first box. Hit the right arrow again and you ll be on the second box. You should obtain a result similar (although not the same unless you used the same seed) as shown below: We had 52 1s and 48 2s The Tangent Problem; The Derivative EXAMPLE 3 Finding the Derivative of a Function Use a graphing utility to find the derivative of 2. That is, find f (2). f ( x) = 2x 2 5x at SOLUTION From the HOME screen, press the MATH button. Scroll down the MATH menu until you reach 8:Nderiv( and press ENTER. Now type everything you see shown below and press ENTER to obtain the result.

78 The syntax for the numerical derivative of f at x = c is : NDeriv(function, variable, c) 14.5 The Area Problem; the Integral EXAMPLE 3 Using a Graphing Utility to Approximate an Integral Use a graphing utility to approximate the area under the graph of 2 f ( x) = x from 1 to 5. That is, evaluate the integral 5 1 x 2 dx SOLUTION From the HOME screen, press the MATH button. Scroll down the MATH menu until you reach 9:fnInt( and press ENTER. You should have the following on your HOME screen:

79 Now type the expression shown below and press ENTER: To get the answer in fraction form, press MATH and select 1:Frac and press ENTER to obtain the result shown in Figure 29. The syntax for the fnint( command is: FnInt(function, variable, a, b) Appendix Section 1 EXAMPLE 9 Evaluating an Algebraic Expression Evaluate each expression if x = 3 and y = -1. (a) x + 3y (d) -4x + y SOLUTION (a) First, we must store the values of x and y in the calculator. From the HOME screen, type 3 then press the STO button. Finally, press the X,T,θ,n button and press ENTER to store the value of 3 in the variable x. Next, type (-) 1, then press the STO button. Finally, press the ALPHA button and 2 nd 1 to access the variable Y (note the Y in green above the 1) and press ENTER.

80 X,T,θ,n ALPHA Notice the green Y above the 1. Store button This is the negative button. Here is what you should see on your screen: Now press the X,T,θ,n button, followed by + 3 ALPHA 1 After hitting ENTER, you should have the following on your screen: (d)with the values of x and y still stored in memory, from the HOME screen press the MATH button and scroll to the right to highlight NUM:

81 Select 1:abs( by pressing ENTER or pressing the number 1 on the calculator keyboard. You should have the following on your HOME screen: Now type - 4 X + Y ) and hit ENTER to obtain the following: Appendix Section 2 EXAMPLE 4 Exponents on a Graphing Calculator Evaluate: ( 2.3) 5 SOLUTION From the HOME screen type: 2. 3 ^ 5 and hit ENTER to obtain the following:

82 2 The x button raises a number to the second power The caret (^) is used for exponents Decimal Point