Documents
Resources
Learning Center
Upload
Plans & pricing Sign in
Sign Out

Basic Operations Correlation Coefficient

VIEWS: 9 PAGES: 14

  • pg 1
									                                                                                      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.

								
To top