Graphics Calculator Workshop

TI-84Plus

Roseanne Hofmann. Ed.D.

Montgomery County Community College

GRAPHICS CALCULATOR (TI-84PLUS Silver) Introduction          

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DISPLAY CONTRAST

To increase the contrast, press and release the blue ` key (row 2, column A), press and hold : key (row 2, between column D and E).

To decrease the contrast, press and release the blue ` key, press and hold ; key (row 3, between columns D and E).

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Check the operating system on the calculator:

Press [MEM] ((` key, then + key) (row 9 column E), choose 1:  About.  The calculator below has 2.41 operating system. 

To upload a different operating system from another calculator, connect the link cable, then go to[LINK] (` key (row 2, column A) then x key (row 3, column B)), on the receiving calculator move the cursor right to Receive.  The receiving calculator is now passive.

On the sending calculator, go down to G:  Send OS

It will take awhile, but a gauge will indicate the progress of the upload.

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SET Defaults

When you reset defaults on the TI-84 Plus, all defaults in RAM are restored to the factory settings.  Stored data and programs are not changed. 

Press [MEM] (` key then + key (row 9, column E)); 

choose 7:Reset, then 2:Defaults, then 2:Reset.

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SETTINGS

For this worksheet, change the MODE settings of your calculator to look like the screen below.  Press M key (row 2, column B), move the arrow keys ; > (rows 2 and 3, columns D and E) to highlight the selection.  Press the e key (row 10, column E) to select what you have highlighted.  Press î (` M, row 2, column B)) to return to the home screen.

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Many keys on the calculator have three functions:

a)  what is written on the key in white lettering (just press the key)

b)  what is written above and to the left of the key in blue lettering (press ` (row 2, column A) and the desired “blue” function)

c)  what is written above and to the right of the key in green lettering (press a (row 3, column A) and the desired “green” letter or symbol).

OPERATIONS

2+3*4, e (row 10, column E)                                                  14.000

3², (square function x² is q key 6A), e 9.000

Ö9, (square root `, q), e 3.000

2⁵ (2^5, use ^ (5E) to raise a number to a power)                    32.000

 (32 ^ (1/5))                                                                              2.000

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EDITING

(2+3)*4, e                                                                                   20.000

To recall the last expression, press [ENTRY] (`, e (10E)).  The last expression will reappear.  To edit the last expression, place the cursor over the 3 and replace it with 6.

(2+6)*4, e                                                                                   32.000

Try it again!  Recall the last expression.  To delete +6, use the arrow keys to move the cursor so that the + sign is highlighted.  Press the d key (2C) to delete the + sign and the 6.  Press the [INS] key (`, d, (2C)), then insert .1+5, the decimal point (10C), the addition symbol (9E), and the numeral 5 (7C).

(2.1+5)*4, e                                                                               28.4000

Subtraction vs Opposite of (additive inverse)

-3² (white _ key, (10D), Square q (6A))                                               -9.000

2*-6 (grey - key, (8E))                                                                   ERR:SYNTAX

Go to error and type white _, e                                            -12.000

2 _ 6 (white key)                                                                            ERR:SYNTAX

Note:  The “subtract” operation is 5 pixels wide, “opposite of” is 3 pixels wide and 1 pixel higher than the subtract symbol.

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FRACTIONS

1/2 + 1/3 (Use m key (4A), 1: FRAC), e 5/6

1/2+1/3, e                                                                                  .833

m, 1: FRAC, e                                                                       5/6

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ASSIGNMENT STATEMENTS & EVALUATING AN EXPRESSION

To make x = 5

5, = (9A), X, = key (3B)), e                             5

2X²-2X+5, e                                                                              45.000

Remember, X was 5.  To evaluate the same expression with X = -3, input

-3, =, X, e                                                                            -3.000

recall the expression by pressing [ENTRY] twice (`, e), then `

e                                                                                      29.000

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CONCATENATION AND DEEP RECALL

The colon “:” (`, ., (10C)) allows two or more commands to be put together.

2, =, x : X²-3X+7, e                                                  5.000

[ENTRY] (`, e) allows you to recall many of your previous expressions in the stack. 

To bring back the last entry, press [ENTRY], edit the line to input -2.75 for X

-2.75, =, = : X²-3X+7, e 22.813 

Solve 3 + x = 10 - (x + 2). 

The equation 3+x=10-(x+2) may be solved using different methods on the calculator.  Let us look at Graphing, Tables, and the Solve Command.

GRAPHING

Let y1 be the left hand side of the equation, namely, y1 = 3+x and let y2 = 10-(x+2), the right hand side of the equation.  Look for the point where the two lines intersect.

To input functions, press ! key (1A), key in:

Y1=3+X

Y2=10-(X+2)

By pressing the arrow keys, move the cursor to the left of Y1.  Use the e key to cycle through the “style” choices for the graph.  Choose the “thick” for Y1 and the “animated with a path” for Y2.

If you were graphing using a pencil and paper, you would need to set up a graphing grid or viewing window.  To use a viewing window of [-10,10] by [-10,10], press the # key (1C), choose 6:ZStandard from the menu.  Press the $ key (1D) to see the (x, f(x)) values of the function along the bottom of the screen.  Notice the spider-like cursor!  The left () and right () arrow keys move the cursor along the function.  The up () and down () arrow keys switch between the functions.  Notice the equation of the function being traced is printed in the upper left side of the screen when you press the Trace option!  Press the [FORMAT] key (1C) to select ExprOn or ExprOff.  When you press the $ key, you can either use the arrow keys or you can enter a specific value for x, press e; the cursor will move to the specified location and the x and y coordinates will be displayed.

Press @ (1B), notice that the viewing window is [-10, 10] by [-10,10].  To change to a more “friendly” window, press @ and change Xmin to -18.8, Xmax to 18.8, Xscl to 4, Ymin to -12.4, Ymax to 12.4, Yscl to 4.  Press the $ key and look at the “nice” x values

The intersection point seems “close” to x = 2.4 and y = 5.4, but what is the exact intersection point?  While you are on the graph screen, press [CALC] (`, $ (1D)).  Choose 5:intersect from the menu.  In response to “First curve?”, move cursor to curve Y1, press e, in response to “Second curve?”, move to curve Y2, press e, “Guess?” move close to point of intersection, press e.  Notice the intersection point printed along the bottom of the screen.

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TABLES

The solution to the equation 3+x=10-(x+2) may be found using tables.  To input functions, press ! key (1A), key in:

Y1=3+X

Y2=10-(X+2)

Press [TBLSET] (`,@ (1B)), set TblStart = 2 and DTbl=.1.  Press [TABLE] (`,% (1E)), use the arrow keys to look for where Y1 values and Y2 values are the same.  Press the M key (2B) to select G-T to display the graph and the table simultaneously.  Press the [TABLE] key and then the arrow keys to move the cursor in the table, press the % key or the $ key to move the cursor along the graph.

SOLVE command

Another method to find the solution of the equation 3+x=10-(x+2) is to use the solve command.  From the home screen, press the m key (4A), select 0:Solver... from the menu.  The expression is assumed equal to zero, so rewriting the equation as y1-y2 will set the equation equal to 0.  If the equation editor is empty, use Y1-Y2 for the equation.  If the equation editor is not empty, press the up arrow, this will allow you to change the equation.

To input Y1-Y2, press the v key (4D), from the Y-VARS menu choose 1:Function..., then 1:Y1, repeat using 2:Y2.  Press e.  The equation will appear at the top of the screen and the unknowns will be listed under it.  Place the cursor on X, then press [SOLVE] (a, e, (10E)).  Notice a solid square appears next to the solved variable and indicates the equation is balanced. 

Solve Command

Problem:  What is the height of a can with a radius of 3 inches, if the volume must be 100 cubic inches?

Input the equation V = pR²H into the Solver and the appropriate values in V and R, then solve for H.

FINANCIAL

Problem:  What is the interest on an investment of $2000, compounded quarterly at 6% for 5 years?

Press the A key (4B), choose 1: Finance key to get to the time-value-money menu.  Select the TVM Solver.  Notice the Present Value is entered as a negative number (payment); the N can be entered as the number of payments or as the number of years, provided that the number of Compounded Periods is indicated as 4.

Problem:  How much money should be deposited in a savings and loan association paying 6% compounded quarterly in order to have $3000 in 5 years?

Problem:  How long will it take a dollar to double at 8% compounded semiannually?

It takes 8.84 years compounded semiannually and 17.67 compounding periods, that is 17.67 half years.

Problem:  A mathematics teacher deposits $1000 in his savings and loan at the end of each quarter for 10 years.  How much money does he have at the end of 10 years if the savings and loan pays 8% interest compounded quarterly?

 Notice the difference if the money is deposited at the beginning of the period.

Problem:  A family decides to make monthly deposits into a college-education fund for a daughter so that she will have $20,000 at the end of 8 years.  They locate a bond fund that pays 12% compounded monthly.  How much must the family deposit each month?

Notice the difference if the payments are at the beginning of the period.

Problem:  You purchase a house for $100,000, pay 20% down and amortize your debt with monthly payments for 30 years.  What is your monthly payment if your loan charges 9% compounded monthly?

You can look at the amortization table by using Y1, Y2 and Y3 to store the principal, payments and balance and then use the TABLE in the calculator.

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SHADE

To graph x + y ³ 5 and -2x + y £ 2, change the inequalities to y1 ³ 5 - x and y2 £ 2 + 2x.  Press ! key and input Y1=5-X and Y2=2+2X.  Cycle through the styles of graphs and choose “shade above” for Y1 and “shade below” for Y2.  Press the M key to change from split screen to Full screen.  Choose the standard viewing window. 

Another method is to use the Inequal APPS.  Go to the A key (row r, column B), choose Inequalz.  This will be covered in detail in the Finite Mathematics Class.

A third method is to use the Shade command. 

Press the ! key.  Place the cursor on the = sign and press e key,  this will deselect the function.  Go to the home screen [QUIT].  Press the [DRAW] key (`, p (4C)), choose 7:Shade( from the menu.  The syntax for the Shade command:  Shade(lowerfunction, upperfunction, Xleft, Xright, shading patterns, shading resolution).  This will Shade area above lowerfunc, below upperfunc, to the right of X=Xleft, to the left of X=Xright, with shading patterns 1 to 4, and with shading resolution between 1 and 8.  Eight is the least amount of shading, one is the most.  From the home screen input Shade(Y1,Y2,-10,10,1,6), press e.  Notice not all the options need to be used for the Shade function.  Use ClrDraw to clear the drawing.  Try entering the following two Shade functions with different resolutions.

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MATRICES

Up to ten matrices can be stored.  A matrix may have up to 99 rows or columns.  The Fraction command works with matrices and there is a random matrix.

Solve the following system of equations:

 x -  y +  z = 0

2x      - 3z = -1

-x - y + 2z = -1

using the following methods:

· matrix operations (inverse times constant vector)

· determinants (Cramer’s Rule)

· row reduction by using elementary row operations

· row reduction automatically

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Inverse times constant vector method

Input the coefficient matrix into A.  Press the [MATRX] key (` i (5A)), choose the EDIT menu, then 1:[A], e.  Make the dimension 3 X 3 by using the arrow keys.  Enter the coefficients, pressing e after each entry.  Press [QUIT] (`, M (2B)) to return to the home screen.  Repeat the process to enter the 3 X 1 matrix of constants.  Press the [MATRX], choose the EDIT menu, then 2:[B], e.  Make the dimension 3 X 1 by using the arrow key.  Enter the coefficients, pressing e after each entry.  Return to the home screen.  To find the solution, input [A]^-1*[B].   Press the [MATRX] key, choose the NAMES menu, then 1:[A] 3X3, e, the inverse key i (5A), the [MATRX] key, NAMES, 2:[B] 3X1, e.

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Cramer’s Rule - Determinants

Store Matrix A in Matrix C, so two copies of the coefficient matrix exist.  Press [MATRX], choose NAMES menu, 1:[A] 3X3, press = key (9A), [MATRX], NAMES, 3:[C] 3X3.  Find the determinant of matrix A which is the denominator in using Cramer’s Rule.  Press [MATRX], MATH, 1:det, [MATRX], NAMES, 1:[A] 3X3, e.

To find the value of y, substitute the constants in the second column of matrix C and divide the determinant of C by the determinant of A.  Press [MATRX], EDIT, 3:[C] 3X3.  Using the arrow keys and the e key replace column 2 with 0, -1, -1.  Go to the home screen by using [QUIT].  Press [MATRX], MATH, 1:det, [MATRX], NAMES, 3:[C] 3X3.  Notice if you divide det [C] by det [A], namely -8 by -4, the solution for y is 2, which is what we arrived by using the inverse method.  To find the values for x and z, continue to store matrix A in matrix C and then change matrix C.

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Elementary Row Operations

To form the augmented matrix for the given system of equations, press [MATRX], MATH, 7:augment(,[MATRX], NAMES, 1:[A] 3X3, , (comma(6B)), [MATRX], NAMES, 2:[B] 3X1, ) (6D), =, [MATRX], NAMES, 4:[D] 3X4.

Matrix D is the augmented matrix which will be row reduced.  Press [MATRX], choose MATH, F:*row+(, then -2, [D], 1, 2), e.  The result of the last operation is stored in ANS.  To change the element in the (3, 1) position, press [MATRX], choose MATH, D:row+(, ANS, 1, 2).  To change the element in the (2, 2) position to a 1, choose E:*row(, from the MATH menu under [MATRX] key: *row(.5,ANS,2).

The remaining columns can be reduced in a similar fashion.  The syntax for the row operations:

rowSwap(matrix,rowA,rowB)	Returns a matrix with rowA of matrix swapped with rowB
row+(matrix,rowA,rowB)	Returns a matrix with rowA of matrix added to rowB and stored in rowB
*row(value,matrix,row)	Returns a matrix with row of matrix multiplied by value and stored in row.
*row+(value,matrix,rowA,rowB	Returns a matrix with rowA of matrix multiplied by value, added to rowB, and stored in rowB

Row Reduced Echelon Form

This same problem can be solved by using the rref( command.  The rref( command returns the reduced row-echelon form of a real matrix.  The number of columns must be greater than the number of rows.  Recall that matrix D is the augmented matrix of the coefficient matrix A and the matrix of constants B.  From the home screen, press the [MATRX] key, go to the MATH submenu, choose the B:rref( command, then choose the matrix D from the NAMES submenu.  The ref( command returns the row-echelon form of the matrix. 

STATISTICS

Students were asked how many people came to their Thanksgiving Day Dinner.  The following data represents the number of guests the students reported:

3, 5, 4, 8, 9, 15, 4, 7, 12, 11, 3, and 6.

On the home screen, input the data into a List, L1.  Use the braces [ { ] (`, ( (6C)), (comma key (6B)), the data, close the list, the = key (9A), the [L1] key (9B), then press e.  To calculate the mean, the standard deviation, etc., press S (3C), select CALC, then 1:1-Var Stats, press [L1], then e.  Notice that there are two screens of descriptive statistics; press the down arrow to scroll down to the second screen.

To sort the data, press S key (3C), choose EDIT, 2:SortA(, then press [L1], ), e.  Press the right arrow key to scroll to the right.

To plot this data, first press ! key (1A) to be sure all functions are turned off.  To turn off a function, move the cursor on the equal sign and press e.  When you move away from the equal sign the equal sign will no longer be highlighted.  Press [STAT PLOT] (`,! (1A)), choose 1:Plot1..., under type choose a histogram or a box and whiskers or a modified box and whiskers, because the data is one variable statistics.  At Xlist:  choose L1, under Freq:  choose1.  Press the # key (1C) and choose 9:ZoomStat.  Press the % key (1E).  The viewing window is not the best choice for this data.  ZoomStat redefines the viewing window so that all statistical data points are displayed.  For one-variable plots, only Xmin and Xmax are adjusted.  Go back, press the @ key and make the following changes:  Xmin=0, Xmax=18, Xscl=3, Ymin=-1, Ymax=6, Yscl=1, Xres=1.

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Two Variable Statistics

A manager of a small textile plant wishes to study the relationship between the time required to complete a certain task and the noise level at the work station.  He collects data for a random sample of five employees and finds:

Noise Level	.5	1	1.5	2	2.5
Time required to complete task	1	1.8	2.9	3.6	4.8

To enter the data, press S, choose EDIT, 1:Edit.  If there is data in L1, highlight L1 at the top of the list and press C (4E).  When you move the cursor down the list, you will see that the list is now empty.  Another method is to use the ClrList command located on the submenu of the EDIT under the STAT key.  Enter the x values in the L1 column, the corresponding values in L2.  Press the [STAT PLOT] key, choose 1:Plot1, On.  Under Type, choose the scatterplot, under Xlist, L1, under Ylist, L2, under Mark,ð.  Press @, input the following values:

Xmin=-1, Xmax=4, Xscl=1, Ymin=-1, Ymax=6, Yscl=1, Xres=1.

To find and graph the equation of the line of regression, press S key, Calc, 4:LinReg(ax+b), [L1], [L2], Y1, e.  To find Y1, press the v key (4E), from the Y-VARS submenu, choose 1:Function.., then choose 1:Y1.  This will paste the equation of the line which was just computed into Y1.  Press % key.  The regression line and the actual data points are on the same graph.  Press the up and down arrows to switch from the Stat plot to the Y1 plot.  To look at the coefficient of correlation and r², press the [CATALOG] key (`,0 (10B)), press D to move down the catalog list, choose DiagnosticOn.  Press e.  The calculator screen will show Done.  Repeat the LinReg command and notice that the values for r and r² for the line are displayed.

To see if a natural logarithm gives a better fit of the data, press S, CALC, 9:LnReg, [L1], [L2], Y2, e.  Notice, that the LnReg is not as good a fit of the data, however, you expected that because r, the correlation coefficient, was not as close to 1 as it was when the LinReg was used.

There are other options for fitting a model to data, namely:  quadratic, cubic, quartic, logarithmic, exponential, power, logistic, and sinusoidal.

Turn all the plots off.  Press the [STAT PLOT] key, choose 4:PlotsOff, press e. 

New Statistic Commands on TI-84 Plus

Find T values given probabilities and degree of freedom:  invT

To find the critical t-value for α = .05 left tailed hypothesis test with sample size of 12, use invT command found in the [DISTR] (`, v keys (4D)).  Remember that sample size of 12 means 11 degrees of freedom.

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To find the critical t-value for α = .05 right tailed hypothesis test with sample size of 12, use InvT command found in the [DISTR] (`, v keys (4D)).

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Goodness of Fit

Problem from Triola Textbook:

Mars, Inc., claims that its M&M plain candies are distributed with the color percentages of 30% brown, 20% yellow, 20% red, 10% orange, 10% green and 10% blue.  A random sample showed the following:

Color	Observed Frequency
Brown	33
Yellow	26
Red	21
Orange	8
Green	7
Blue	5

Test the claim that the color distribution is as claimed by Mars, Inc.

Calculate the Expected Frequencies based on the claims made by Mars.

Color	Observed Frequency	Expected E = n*p
Brown	33	.3(100) = 30
Yellow	26	.2(100) = 20
Red	21	.2(100) = 20
Orange	8	.1(100) = 10
Green	7	.1(100) = 10
Blue	5	.1(100) = 10
S	100	100

Input the Observed frequencies in L1 and the Expected values in L2.

Press the S key (3C) go to the Tests drop down menu,

choose D:chi-square GOF-Test.  Input the following information, and then press Calculate.  Notice that the last line gives the individual calculations for each of the 5 categories. 

Go back to the same menu, but this time press Draw to see the visual representation as well.

With a p-value of .3111 which is greater than the alpha of .05, we fail to reject the null hypothesis.  There is not sufficient evidence to warrant rejection of the claim that the colors are distributed with the percentages given by Mars, Inc.

DERIVATIVES & INTEGRALS

Find the derivative of 2x² - 4x +1 at x = 1.5.

You can find the derivative at a point on the graph using the dy/dx option under the [CALC] key.

Input Y1= 2X²-4X+1, use the viewing window as [-2, 2.7] by [-1, 2].  Press the [CALC] key, choose 6:dy/dx from the menu.  Move the cursor to x = 1.5, y = -.5.  Press e.  The output on the bottom of the screen is dy/dx=2.  To see the tangent line drawn and the equation of the tangent line at that point, press [DRAW] (`,p (4C)), choose 5:Tangent(, then e. 

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Find the definite integral for x-1 from x = 1 to x = 2.

Input Y2= X-1, and turn off Y1.  Press the % key.  Press the [CALC] key and choose 7:òf(x)dx from the menu.  In response to “Lower Limit?”, either move the cursor to x=1, y = 0, or press 1, then press e, “Upper Limit?”, either move the cursor to x=2, y=1, or press 2, then press e.  The integrated area will be shaded and on the bottom of the screen òf(x) dx=.5 will be printed.

The numerical derivative and the integral could have been found on the home screen without using the graph.  Go to the home screen.

To find the derivative of Y1 at x = 1.5, press m key, choose 8:nDeriv(, input Y1, X, 1.5), press e.  Output will be 2.000.

To find the integral of Y2 between x = 1 and x = 2, select 9:fnInt( under the m] key.  On the home screen, the input should look like: fnInt(Y2,X,1,2).  Output will be .500.

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Find the maximum for Y=2X³+X²-13X+6.

One method of finding the maximum is to use the graph of the function and the [CALC] key.

Input the function in Y1.  Use a viewing window of [-4.7, 4.7] scale 1 by [-20, 42] scale 10.  Press the % key to see the function on the screen.  Press the [CALC] key (1D), choose 4:maximum.  In response to “Lower Bound?”, move the cursor to approximately x = -2.5, y = 13.5, e, then for “Upper Bound?”, move the cursor to x = -.6, y = 13.728, e, “Guess?”, move the cursor to x = -1.3, y = 20.196, e.  The calculator will respond with Maximum x = -1.64803, y = 21.188284.

The maximum can be found on the home screen without using the graph.  From the home screen, press the m] key, choose 7:fMax( from the MATH menu.  The syntax for this command is fMax(expression, variable, lower, upper).  Input Y1 as the function, x is the variable, -10 as an x value to the left of the maximum, 10 as an x value to the right of the maximum.  Notice this command gives the x coordinate of the maximum. This value is presently stored in x.  To find the y coordinate, use Y1(X) on the home screen.

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Graph the function, the first derivative and the second derivative of

Y=2X³+X²-13X+6.

Let Y2= nDeriv(Y1,X,X) and Y2= nDeriv(Y2,X,X).

With the cursor on the right side of the = sign of Y2, press the m key, choose 8:nDeriv( under the MATH option.  The syntax is nDeriv(expression, variable, value).  Repeat the process for Y3.  Choose a different “style” for each of the functions.  Press the % key.  Notice that you see the graph of the original function, the graph of its first derivative and the graph of its second derivative.  By using the root command under [CALC] key for Y2, one could find the maximum and minimum.  Similarly, by using the root command for Y3, one could find the inflection point.  More accurate results could be found by using the ZOOM command, etc.  Numerical values can be viewed by setting the table minimum to -1 and the increment to .1.  More accurate results can be found by changing the table increment.  To split the screen, press the M key (2B), choose G-T in the last row.  Notice the corresponding table values as the cursor moves along the functions.

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OTHER GRAPHING TECHNIQUES

Piecewise Functions:

a)  The greatest integer function (step function)

Turn off or erase all the Y functions.  Let Y1 = int (X).  Press the m key (4A), use the right arrow key to go to the NUM menu, choose 5:int(.  Then press X.  Press the # key (1C), choose 4:Zdecimal for the viewing window.  All the “steps” are connected!!!  Go back to the Y= screen and choose the dot “style” for the graph of Y1. Notice that the graph looks like the “real thing.”

b)  Graph

Press the ! key.  Let Y1 = (.5x-1)(x<-2) + (x²)(x³-2).  Press the [TEST] key (`,m (4A)) for the inequalities symbols.

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SPLIT SCREEN

Press M key (2B), highlight Horiz, press e, press the % key, the screen is now split, with the graphics on the top.  Press ! key, and the y functions will be displayed in the lower half of the split screen, with the graphs in the upper screen.  Notice the step function has been entered as two different functions and the dot “style” has not been used in this version.

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