Basic Operations Correlation Coefficient
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Appendix GC GC-1 Graphing Calculator Appendix This appendix contains some keystroke suggestions for many graphing calcu- lator operations that are featured in this text. The keystrokes are for the TI-83/ TI-83 Plus calculators. The descriptions in the margin are the same as those used in the text and are arranged alphabetically. Please see your manual for additional information about your calculator. Basic Operations Numerical calculations are performed on the home screen. You can always re- turn to the home screen by pressing 2nd QUIT. Pressing CLEAR erases the home screen. To evaluate the expression –2(3 + 5) – 8 ÷ 4, use the following keystrokes. (–) 2 ( 3 + 5 ) – 8 ÷ 4 ENTER Note: There is a difference between the key to en- ter a negative number, (–) , and the key for subtraction, – . You cannot use these keys interchangeably. The 2nd key is used to access the commands written in gold above a key. For instance, to evaluate 49 , press 2nd 49 ) ENTER . The ALPHA key is used to place a letter on the screen. One reason to do this is to store a value of a variable. The following keystrokes give A the value of 5. 5 STO ALPHA A ENTER This value is now available in calculations. For in- stance, we can find the value of 3a2 by using the fol- lowing keystrokes: 3 ALPHA A x2 . To display the value of the variable on the screen, press 2nd RCL ALPHA A. Note: When using the ALPHA key, only capital letters are available on TI- 83 calculator. Correlation The value of the correlation coefficient for a regression equation calculation is Coefficient not shown unless the DiagnosticOn feature is enabled. To enable this fea- ture, press 2nd [catalog] D (scroll to DiagnosticOn) ENTER ENTER The correlation coefficient will appear on the screen along with a regression equation. GC-2 Graphing Calculator Appendix Evaluating Functions There are various methods of evaluating a function but all methods require that the expression be en- tered as one of the functions Y1 to Y7. To evaluate 2 x f ( x ) = ----------- when x = –3, enter the expression into, - x–1 for instance, Y1 and then press VARS 11 ( (–) 3 ) ENTER . Note: If you try to evaluate a function at a number that is not in the domain of the function, you will get an error message. For instance, 1 is not in the 2 x domain of f ( x ) = ----------- . If we try to evaluate the - x–1 function at 1, the error screen at the right appears. Evaluating logarithms Logarithms base 10 can be found using LOG . For instance, to find the value of 19 log ( 2 ) + 1 , press 19 LOG 2 ) + 1 ENTER . Natural logarithms can be found using LN . For instance, to find 3 ln ( 4 ) – 2 , press 3 LN 4 ) – 2 ENTER . Evaluating Variable To evaluate a variable expression, first store the val- Expressions ues of each variable. Then enter the variable expres- 2 sion on the home screen. To evaluate s + 2sl when s = 4 and l = 5, use the following keystrokes. 4 STO ALPHA S ENTER 5 STO ALPHA L ENTER ALPHA S x2 + 2 ALPHA S ALPHA L ENTER Financial Financial calculations on a TI-83 Plus are accessed by pressing APPS and se- Calculations lecting Finance. On a TI-83, press 2nd FINANCE. The process for all financial calculations are the same: Select the finance functions, enter the val- ues of the known variables, enter 0 for all other values, use the arrow keys to select the unknown variable, press ALPHA SOLVE. Here are some examples. Compound Interest Future Value Suppose an investment of $10,000 is made at an interest rate of 6.25% compounded daily. To find the value of the invest- ment in 5 years, access the finance functions (see above). Select TMV Solver (TMV is an abbreviation for time value of money), press ENTER . Use the up and down arrow keys to select the variables. For this problem, N = 5, I% = 6.25, PV = -10000, PMT = 0, FV = 0, P/Y = 1, and C/Y = 365. Now move the cursor to FV and press ALPHA SOLVE. The little square next to FV indicates that value was calculated. When using the ﬁnance functions of a TI-83 calculator, payments are entered as negative numbers, as we did for PV in the calculator screen at the left. The value of the investment in 5 years will be $13,668.01. Appendix GC GC-3 Compound Interest Present Value An investor wants to have $10,000 in 4 years. To find how much the investor must place in an account today that earns 8% interest compounded quarterly, select TMV Solver and press ENTER . Use the up and down arrow keys to enter values for the variables. For this problem, N = 4, I% = 8, PV = 0, PMT = 0, FV = 10000, P/Y = 1, and C/Y = 4 . Now move the cursor to PV and press ALPHA SOLVE. The little square next to PV indicates that value was calculated • Note that the result is a negative number. This Take Note is the amount that must The meaning of the variables in be deposited (paid) into a ﬁnancial calculation are: the account. N: For an annuity, the number of payments; for The investor must place $7261.74 in the account. compound interest, the number of years Monthly Car or Mortgage Payment A cabinet maker finances $12,500 for a car I%: Annual interest rate at an annual interest rate of 8.3% compounded monthly for 5 years. To find the PV: Present value FV: Future value monthly payment, select TMV Solver and press ENTER . Use the up and P/Y: Number of payments down arrow keys to enter values for the variables. For this problem, N = 60 per year. For compound ( 5 ⋅ 12 ), I% = 8.3, PV = -12500, PMT = 0, FV = 0, P/Y = 12, and C/Y = 12. Now interest, P/Y = 1 C/Y: Number of move the cursor to PMT and press ALPHA SOLVE. The little square next to compounding periods PMT indicates that value was calculated. per year PMT: END BEGIN - Select END if payments are made at the end of a period; select BEGIN if payments are made at the beginning of a period The monthly payment is $255.25. To calculate a monthly mortgage payment, follow the same steps as above. The present value is the amount of the mortgage. Calculate APR Suppose a management intern purchases an MP3 player for $250 and finances the purchase at an 8% simple interest rate for 12 months. The monthly payment is $22.50. To find the APR, select TMV Solver and press ENTER . Use the up and down arrow keys to enter values for the variables. For this problem, N = 12, I% = 0 (this is the APR, not the simple interest rate), PV = 250, PMT = – 22.5 , FV = 0, P/Y = 12, and C/Y = 12. Now move the cur- sor to I% and press ALPHA SOLVE. The little square next to I% indicates that value was calculated. The APR is 14.45%. GC-4 Graphing Calculator Appendix Calculating Mortgage or Loan Payoff An electrician has a 30-year mortgage at an annual interest rate of 6.5% and makes monthly payments of $1580.17. To find the mortgage payoff after making payments for 5 years (60 months), select TMV Solver and press ENTER . Use the up and down arrow keys to enter values for the variables. For this problem, N = 300 (360 – 60), I% = 6.5%, PV = 0 , PMT = – 1580.17 , FV = 0, P/Y = 12, and C/Y = 12. Now move the cursor to PV and press ALPHA SOLVE. The little square next to PV indicates that value was calculated. The mortgage payoff is $234,027.43. Graph To graph a function, use the Y= key to enter the expression for the func- tion, select a suitable viewing window, and then press GRAPH . For instance, 3 to graph f ( x ) = 0.1x – 2x – 1 in the standard viewing window, use the fol- lowing keystrokes. Y= . Θ 1 X,T,Θ,n ^ 3 – 2 Θ X,T,Θ,n – 1 ZOOM (scroll to 6) ENTER 10 –10 10 –10 Note: For the keystrokes above, you do not have to scroll to 6. Alternatively, use ZOOM 6. This will select the standard viewing window and automat- ically start the graph. Use the WINDOW key to create a custom window for a graph. x Some special functions such as e , ln ( x ) , and log ( x ) can be graphed by using – 2x the keys for these functions. For instance, to graph f ( x ) = 0.25e – 4 , press Y= . 25 2nd ex (–) 2 Θ X,T,Θ,n ) ZOOM (scroll to 6) ENTER 10 –10 10 –10 Appendix GC GC-5 Intersect The INTERSECT feature is used to solve a system of equations. To illustrate 2x – 3y = 13 this feature, we will use the system of equations . 3x + 4y = – 6 Note: Some equations can be solved by this method. See Solve an Equation below. Also, this method is used to ﬁnd a number in the domain of a func- tion for a given number in the range. See Find a domain element below. Solve each of the equations in the system of equations for y. In this case, we 2 13 3 3 have y = -- x – ----- and y = – -- x – -- . - - - - 3 3 4 2 2 13 Use the Y-editor to enter -- x – ----- into Y1 - - 3 3 3 3 and – -- x – -- into Y2. Graph the two func- - - 4 2 tions in the standard viewing window. (If the window does not show the point of intersection of the two graphs, adjust the window until you can see the point of in- tersection.) Press 2nd CALC (scroll to 5, intersect) ENTER . Alternatively, you can just press 2nd CALC 5. First curve? is shown at the bottom of the screen and identiﬁes one of the two graphs on the screen. Press ENTER . Second curve? is shown at the bot- tom of the screen and identiﬁes the sec- ond of the two graphs on the screen. Press ENTER . Guess?, shown at the bottom of the screen, asks you to use the left or right ar- row key to move the cursor to the approx- imate location of the point of intersection. (If there are two or more points of inter- section, it does not matter which one you choose ﬁrst.) Press ENTER . The solution of the system of equations is ( 2, – 3 ) . GC-6 Graphing Calculator Appendix Solve an Equation To illustrate the steps, we will solve the equation 2x + 4 = – 3x – 1 . The idea is to write the equation as the system of equations y = 2x + 4 and then use the steps for solving a system of equations. y = – 3x – 1 Use the Y-editor to enter the left and right sides of the equation into Y1 and Y2. Graph the two functions and then follow the steps for Intersect. The solution is –1, the x-coordinate of the point of intersection. Find a domain element For this example, we will find a number in the domain 2 of f ( x ) = – -- x + 2 that corresponds to 4 in the range of the function. This is - 3 2 like solving the system of equations y = – -- x + 2 and y = 4 . - 3 Use the Y-editor to enter the expression for the function in Y1 and the desired output, 4, in Y2. Graph the two functions and then follow the steps for Inter- sect. The point of intersection is (–3, 4). The number –3 in the domain of f produces an output of 4 in the range of f. Math Pressing MATH gives you access to many built-in functions. Frac function The Frac function converts a decimal to a fraction. The fol- 1 lowing keystrokes will convert 0.125 to the fraction -- . - 8 .125 MATH 1 ENTER . Factorial To find 8!, press 8 MATH . Use the right arrow key to highlight PRB and then use the down arrow key to select 4:. Press ENTER ENTER . Appendix GC GC-7 Permutations To find the permutations of 8 objects chosen 3 at a time, P ( 8, 3 ) , press 8 MATH . Use the right arrow key to highlight PRB (this is the menu that contains counting and probability functions) and then use the down ar- row key to select 2:. Press ENTER 3 ENTER Combinations To find the combinations of 10 objects chosen 4 at a time, C ( 10, 4 ) , press 10 MATH . Use the right arrow key to highlight PRB and then use the down arrow key to select 3:. Press ENTER 4 ENTER Additional built-in functions under MATH can be found by pressing MATH . For instance, to evaluate – – 25 , press (–) MATH 1 (–) 25 ) . See your owner’s manual for assistance with other functions under the MATH key. Radical expressions To evaluate a square root expression, press 2nd 2 . For instance, to evaluate .15 p + 4p + 10 when p = 100,000, first store 100,000 in P. Then press 0.15 2nd ALPHA P x2 + 4 ALPHA P + 10 ) ENTER . To evaluate a radical expression other than a square root, access x by pressing MATH . For instance, to evaluate 4 67 , press 4 (the index of the radical) MATH (scroll to 5) ENTER 67 ENTER . Statistics Mean, Median, Quartiles and Standard Deviation The values of the mean, median, quartiles, and standard deviation are calculated for a data set by se- lecting the 1-Var Stats function, which is one of the options that can be accessed by pressing the STAT key. GC-8 Graphing Calculator Appendix For instance, the results of an exam given to 20 students are given below. 96, 72, 76, 47, 92, 58, 66, 79, 44, 62, 76, 78, 50, 52, 90, 84, 63, 65, 75, 86 To calculate the mean, median, quartiles, or standard deviation for this data, press STAT to access the statistics menu. Press 1 to Edit or enter data. To de- lete data already in a list, press the up arrow to highlight the list name. For in- stance, to delete data in L1, highlight L1. Then press CLEAR and ENTER . Now enter each test score under L1, pressing ENTER after each entry. Use the up and down arrow keys to change a value. To calculate the mean, median, quartiles, or standard deviation, press STAT to access the CALC menu. The press 1 ENTER . The results are shown below. Notice the down arrow by n = 20. This indicates that more values follow. Use the down arrow key to scroll through all the values. For the calculation, we have the mean is 70.55, the sample standard deviation is approximately 15.35, the population standard deviation is approximately 14.96, the first quartile (Q1) is 60, the median is 73.5, and the third quartile (Q3) is 81.5. The lowest test score is 44 and the greatest test score is 96. Using these numbers, the range is 96 – 44 = 52 . Linear Regression A study is done to determine the number of grams of sugar that will dissolve in a liquid at various temperatures. The data is shown below. Temperature, x (in oC) 20 35 50 60 75 90 100 Grams of sugar, y 50 80 120 145 175 205 230 All calculations and graphs involving statistical data begin by entering the data using the Edit option which is accessed by pressing STAT . For the data above, press STAT to access the statistics menu. Press 1 to Edit or enter data. To delete data already in a list, press the up arrow to highlight the list name. For instance, to delete data in L1, highlight L1. Then press CLEAR and ENTER . Now enter each value of the independent variable in L1, pressing ENTER after each entry. Use the up and down arrow keys to change a value. When all values of the independent variable are entered, press . This will put you in the next column to enter the values of the depen- dent variable in L2. Appendix GC GC-9 Create a scatter diagram Press 2nd STATPLOT (use the down arrow key to select Plot1, Plot2, or Plot3) ENTER . Use arrow keys to move the cursor to On and then press ENTER . The first graph type is for a scatter dia- gram. Move the cursor over that symbol and press ENTER . Be sure that Xlist and Ylist are the names of the lists into which you stored data. You can change these by pressing 2nd and then selecting the appropriate list, L1 through L6. Prepare to graph the data by adjusting the viewing window by pressing WINDOW and entering appropriate values. Now press GRAPH . 250 0 110 0 Note: You can tell that STAT PLOTS is active by pressing Y= . For one screen at the right, ob- serve that PLOT1 is highlighted indicating it is active. To turn STAT PLOTSoff, use the up arrow key to highlight it and then press ENTER . Now use the arrow key to move the cursor to the right of the equal sign for Y1. Find a linear regression equation Press STAT (scroll to 4) ENTER 2nd L1 , 2nd L2 , VARS 1 1 ENTER .The values of the slope and y-intercept of the linear regression equation will be dis- played on the screen. If DiagnosticOn is enabled (See Correlation coeffi- 2 cient.), then the coefficient of determination r and the correlation coefficient r are also shown. Note: If data is stored in L1 and L2, the keystrokes 2nd L1 , 2nd L2 are not necessary. The keystrokes VARS 1 1 ENTER place the regression equation in Y1. These keystrokes are not necessary but are helpful if you need to graph the regression equation or evaluate the equation at a given value of the independent variable. See below for more details. Other regression equations can be calculated. For instance, to find a regression x equation of the form y = ab , called an exponential regression equation, enter the data, select ExpReg from the CALC menu under the STAT menu and then press ENTER ENTER . GC-10 Graphing Calculator Appendix Note: Because the data was entered into L1 and L2, it was not necessary to include them in ExpReg. We did include the optional Y1. This is good prac- tice because it makes evaluating and graphing a regression equation much easier. Graph a regression equation Press STAT 250 (scroll to 4) ENTER 2nd L1 , 2nd L2 , VARS 1 1 ENTER . This will store the regression equation in Y1. Now press GRAPH . It may be necessary to 0 110 0 adjust the viewing window. Evaluate a regression equation Complete the steps to graph a regression equation but do not graph the equation. To evaluate the equation when x = 50, press VARS 11 ( 50 ) ENTER . Table There are three steps in creating an input/output table for a function. First use the Y= editor to input the function. The second step is setting up the ta- ble, and the third step is displaying the table. To set up the table, press 2nd TBLSET. TblStart is the first value of the independent variable in the input/output table. ∆Tbl is the difference between successive values. Setting this to 1 means that, for this table, the input values are –2, – 1 , 0, 1, 2 ... . If ∆Tbl= 0.5, then the input values of would be –2, –1.5, –1, –0.5, 0, 0.5, ... . Indpnt is the independent variable. When this is set to Auto, values of the independent variable are automatically entered into the table. Depend is the dependent variable. When this is set to Auto, values of the dependent vari- able are automatically entered into the table. To display the table, press 2nd TABLE. An in- 2 put/output table for f ( x ) = x – 1 is shown at the right. Once the table is on the screen, the up and down ar- row keys can be used to display more values in the table. For the table at the right, we used the up arrow key to move to x = –7. Appendix GC GC-11 An input/output table for any given input can be created by selecting Ask for the independent vari- able. The cursor will be at X=. Enter values for x and press ENTER . The values will be displayed in the table. The table at the right shows an input/output 4x table for f ( x ) = ----------- for selected values of x. Note - x–2 the word ERROR when 2 was entered. This oc- curred because f is not defined when x = 2. Note: Using the table feature in Ask mode is the same as evaluating a function for given values of the independent variable. For instance, from the table at the right, we have f ( 4 ) = 8 . Trace Once a graph is drawn, pressing TRACE will place a cursor on the screen and the coordinates of the point below the cursor are shown at the bottom of the screen. Use the left and right arrow keys to move the cursor along the graph. For the graph of 3 f ( x ) = 0.1x – 2x + 2 , shown at the right, we have f ( 4.8 ) = 3.4592 . In TRACE mode, you can evaluate a function at any value of the independent variable that is within Xmin and Xmax. To do this, first graph the function. Now press TRACE (the value of x) ENTER . For the graph at the left below, we used x = –3.5. If a value of x is chosen outside the window, an error message is displayed. In the example above where we entered –3.5 for x, the value of the function was calculated as 4.7125. This means that f ( – 3.5 ) = 4.7125 . The keystrokes 2nd QUIT VARS 1 1 MATH 1 ENTER will convert the decimal value to a fraction. When the TRACE feature is used with two or more graphs, the up and down arrow keys are used to move between the graphs. The graphs below are for the 3 functions f ( x ) = 0.1x – 2x + 2 and g ( x ) = 2x – 3 . By using the up and down arrows, we can place the cursor on either graph. The right and left arrows are used to move along the graph. GC-12 Graphing Calculator Appendix Window The viewing window for a graph is controlled Ymax by pressing WINDOW . Xmin and Xmax are the Xscl Yscl minimum value and maximum value, respec- Xmin Xmax tively, of the independent variable shown on the graph. Xscl is the distance between tic marks on the x-axis. Ymin and Ymax are the Ymin minimum value and maximum value, respec- tively, of the dependent variable shown on the graph. Yscl is the distance be- tween tic marks on the y-axis. Leave Xres as 1. Note: In the standard viewing window, the distance between tic marks on the x-axis is different from the distance between tic marks on the y-axis. This will distort a graph. A more accurate picture of a graph can be created by us- ing a SQUARE viewing window. See ZOOM. Y= The Y= editor is used to enter the expression for a function. There are ten possible functions, la- beled Y1, Y2, Y3, . . ., Y0, that can be active at any one Take Note 2 Although there are ten possible time. For instance, to enter f ( x ) = x + 3x – 2 as Y1, functions, only seven can be use the following keystrokes. displayed at one time, as shown on the calculator screen at the Y= Θ X,T,Θ,n x2 + Θ 3 X,T,Θ,n – 2 right. Note: If an expression is already entered for Y1, place the cursor anywhere on that expression and press CLEAR . 2v – 1 To enter s = -------------- into Y2, place the cursor to the 3 - v –3 right of the equal sign for Y2. Then press ( 2 Θ X,T,Θ,n – 1 ) ÷ ( Θ X,T,Θ,n ^ 3 – 3 ) Note: When entering an equation, the independent variable, v in the expres- Θ sion above, is entered using X,T,Θ,n . The dependent variable, s in the ex- pression above, is one of Y1 to Y0. Also note the use of parentheses to ensure the correct order of operations. Observe the black rectangle that covers the equal sign for the two examples we have shown. This rect- angle means that the function is ‘active.’ If we were to press GRAPH , then the graphs of both functions would appear. You can make a function inactive by using the arrow keys to move the cursor over the equal sign of that function and then pressing ENTER . This will remove the black rectangle. We have done that for Y2, as shown above. Now if GRAPH is pressed, only Y1 will be graphed. It is also possible to control the appearance of the Default graph by moving the cursor on the Y= screen Bold graph to the left of any Y. With the cursor in this position, Shade above graph Shade below graph pressing ENTER will change the appearance of the Animate graph Animate graph graph. The options are shown at the right. Dashed graph Appendix GC GC-13 Zero The ZERO feature of a graphing calculator is used for various calculations: to find the x-intercepts of a function, to solve some equations, and to find the zero of a function. x-intercepts To illustrate the procedure for finding x-intercepts, we will use f(x) = x2 + x – 2. First, use the Y-editor to enter the expression for the function and then graph the function in the standard viewing window. (For some functions, it may be necessary to adjust this window so that the intercepts are visible). Once the graph is displayed, use the keystrokes below to find the x-intercepts of the graph of the function. Press 2nd CALC (scroll to 2 for zero of the function) ENTER . Alternatively, you can just press 2nd CALC 2. Left Bound?, shown at the bottom of the screen, asks you to use the left or right ar- row keys to move the cursor to the left of the desired x-intercept. Press ENTER . Right Bound?, shown at the bottom of the screen, asks you to use the left or right arrow keys to move the cursor to the right of the desired x-intercept. Press ENTER . Guess?, shown at the bottom of the screen, asks you to use the left or right ar- row keys to move the cursor to the ap- proximate location of the desired x- intercept. Press ENTER . The x-coordinate of an x-intercept is –2. Therefore, an x-intercept is (–2, 0). To ﬁnd the other x-intercept, follow the same steps as above. The screens for this calculation are shown below. A second x-intercept is (1, 0). Solve an equation To use the ZERO feature to solve an equation, first rewrite the equation with all terms on one side. For instance, one way to solve 3 3 x – x + 1 = – 2 x + 3 is to first rewrite the equation as x + x – 2 = 0 . Enter 3 x + x – 2 into Y1 and then follow the steps for finding x-intercepts. GC-14 Graphing Calculator Appendix Find the real zeros of a function To find the real zeros of a function, follow the steps for finding x-intercepts. Zoom Pressing ZOOM allows you to create some preset viewing windows. This key also gives you access to ZBox, Zoom In, and Zoom Out. These functions allow you to redraw a selected portion of a graph in a new window. Some win- dows used frequently in this text are shown below.