Instructor S Supplement

MAT135 Supplement

MAT135

Instructor’s Supplement

Table of Contents

Page

Graphing Calculator Basics 3

Supplemental Material Section 10.411

Other Class Assignments

A Room of One’s Own 16

Recycling18

The Great Quadratic19

Cell Phone Plans 22

Soccer Club23

Rita’s Problem25

Completing the Square28

Don’t Bug Me29

Getting to Motown33

The Graph of a Function37

I. Graphing Calculator Basics

A. MODE

For most of the calculations you perform, you will

use the MODE window shown in the figure at the right.

To change the mode, move the cursor over the selection

to be used, and then press . To return to the home

screen, press .

1. Normal Sci Eng: For most of your calculations you will leave the mode in Normal. This expresses the number with digits to both the left and right of the decimal point. If the number is too large to be shown in normal mode, the calculator will automatically switch to scientific Sci (scientific) and Eng (engineering) notation will express the answer as a power of 10.

2. Float 0123456789: This will set the number of digits to be displayed.

3. Radian Degree: These are mode settings used in trigonometry.

4. Func Par Pol Seq: These are graphing modes. Func (function) mode is the mode you will use most often. It is the mode for any function that is to be graphed with rectangular coordinates (y is a function of x).

5. Connected Dot: This determines whether a graph is plotted with the points connected to produce a smooth curve or if just each point is plotted.

6. Sequential Simul: If you have more than one function entered to graph you can select Sequential to have the graphs rendered one at a time or Simul (simultaneous) to have all the functions graphed at one time.

7. : Real mode, rectangular mode, complex mode, and polar complex mode.

8. Full Horiz G-T: These determine the view that you see on the screen.

Full / Horiz / G-T

B. MATH

1: FRAC (display as a fraction) displays an

answer as a fraction. You can use FRAC

with numbers, expressions, lists, sequences,

matrices, etc. If the denominator has more than

three digits or cannot be simplified, the number

is given as a decimal. To access FRAC hit and then 1 after you enter the expression.

Calculate an expression. /
You can change an answer
to a fraction. Just
enter 1 after
the calculation is done. /
Some expressions cannot
be expressed as a fraction. /
Fractions can be
evaluated as decimals. /

2:DEC displays the answer as a decimal. It is used just like FRAC. This is the default setting for answers. It is accessed by entering and

then 2.

You can cube a single
number, the numbers
in a sequence, etc. /

3: 3cubes the expression. It is accessed by entering a value, then 3. You can cube an expression, list, matrix, etc.

You take the cube
root of a single number,
a list of numbers, etc. /

4:evaluates the cube root of a number or

expression. Enter and then 4 and then

the expression. Close the parentheses.

You must enter the
root first and
then the root key. /

5:evaluates any root you select. To use this key,

you must enter the root first, then the function and

then the expression. To use this, press 5.

You may use { } to find the root of several numbers at once.

C. ALPHA

Accessing the ALPHA key will allow you to enter alpha

characters, which are in green above the keys. The TI-83

and the TI-84 will enter only capital letters. If you enter

you can access A-LOCK (alpha lock). This will allow

you to enter as many characters as you need. Alpha characters

are used in programs and you can use any alpha character to store a value.

You can store a value in any variable if you use the key. For

example, to store the value 15 in the variable N,

you would press: 15 (above the ON button) (the variable to use, say N).

D. Cursors

The cursor keys on the TI-83 and the TI-84 are used much like cursor keys on a computer. You move the cursor up, down, left and right to manipulate your position on the screen.

The button is used to access all of the

2nd functions - those that are printed above the

keys and are printed in yellow. Each of these

can be accessed by first pressing and then

the desired function.

1. Changing the Contrast: Darken and Lighten the Screen.

When batteries get weak, the screen will get lighter. You can darken your screen by using repeatedly until the screen is as dark as you like. To lighten your screen, use

. As you lighten and darken your screen, you will see numbers in the upper right hand part of the home screen. The maximum number is 9. You should replace your batteries before you need to have the contrast set this high.

2. Move from one end of a line to another.

You can move to the beginning of a line by pressing and you can move to the end of a line by pressing . This will allow you to edit a line without having to scroll all the way to the end or beginning of the line. These keys work like the "Home" and "End" keys do on a computer.

3. Recall previous entries.

You can recall previous entries by pressing, which brings up a storage area called ENTRY. You can press these keys and recall numerous entries, up to a total of 128 bytes. Since there is a storage limit, the number of entries you can recall will vary. Once you have recalled an entry you can then edit it or just re-evaluate it.

E. Variable Key

The graphing mode selected will

determine the variable entered.

Mode / Variable Key Entry
Func (Function) / X
Par (Parametric) / T
Pol (Polar) /
Seq (Sequence) / n

F. Exponential and Logarithmic Keys

These keys are used for reciprocals,

logarithms, radical, and exponential

expressions.

H. Notation Keys

There are several notation keys that you

will use when entering expression. Their locations

are shown in the figure to the right.

I. Graphing

Access the graph menu by pressing and then enter the function. To type the variable you can use the variable key . For exponents, refer to the exponential keys.

When a function has radical coefficients, the parentheses must be closed before the variable is entered. Failure to do so will result in an incorrect graph. For example, this is the correct way to enter.

When entering a radical function, you must put the entire expression within parentheses.

Setting a WINDOW: You can enter the dimensions

of the graph by accessing (next to ).

When you use this button you will see a window like the

one to the right. To change the values in the window,

just scroll down using the cursor and enter the values

that you want.

Using ZOOM: The ZOOM menu offers you several preset windows. The ZOOM key is next to the key.

1: ZBox / This allows you to box in a particular area of the graph.
2: Zoom In / Zooms in with the center at the point of the cursor. The default factor is 4.
3: Zoom Out / Zooms out with the center at the point of the cursor. The default factor is 4.
4: ZDecimal / This is one of the most useful windows. It is also called a friendly window the TRACE key will move in increments of 0.1 with this window.
5: ZSquare / Adjusts Xmin and Xmax so that the ticks are the same distance apart on both axes.
6: ZStandard / This is also a useful window. The window size is X:[-10, 10] and Y:[-10, 10]. However, TRACE does not move in increments of 0.1.
7: ZTrig / Sets the window to.
8: ZInteger / Redefines the window to Xscl=10 and Yscl=10.
9: ZoomStat / Gives a window that shows all statistical data points.
10: ZoomFit / Sets a window that shows the maximum and minimum value of Y with the given X window.

TRACE:

When using TRACE, the X-values are restricted

to the interval [Xmin, Xmax]. For values outside

of that interval you will have to change the

window. You can use the left and right cursors to

trace the curve or you can enter a value of x by hitting the

the TRACE key and then a numerical value. If you

use the cursors to trace along the curve, the graph will

be redrawn as you move beyond the set window and

the window will change accordingly.

TABLE: TABLE is not restricted to any range of values.

You can use a full screen for TABLE or you can use

a split screen by changing the MODE.

Enter the function. / Graph with ZOOM4. / Evaluate with
the TABLE.

CALC MENU: The CALC menus is accessed by

Pressing TRACE. The menu is shown

at the right.

When you see this
screen, enter a value. / Press ENTER and a
value will be shown. / You can repeat
for other values.

1. value: This key can be used to evaluate the function at any value within the range [Xmin, Xmax].

After entering the function to be graphed and setting the window, access the CALCULATE menu, 1:value,

2. zero: After entering the function and setting the window, access the CALCULATE

Menu, 2:zero. After setting the boundaries by moving the cursor using TRACE, the two set boundaries should have the zero between them.

Enter a Left Bound
of x=-2. / Enter a Right Bound
of x=-1.5. / Ignore "Guess?" and
press enter. / This is the zero (approx.).

3. minimum: After entering the function and setting the window, access the CALCULATE Menu, 3: minimum. After setting the interval to be searched by moving the cursor using TRACE, the two set boundaries should have the minimum between them.

Graph the function. / Access "minimum". / Enter the left bound.
Enter the right bound. / Ignore "Guess" and
press enter. / Minimum is at
the point (-1,-4).

4,maximum: works the same way as minimum.

5. intersection: After entering the function and setting the window, access the CALCULATE Menu, 5: intersection. If you have more than two graphs on the screen at once, it is suggested that you turn one of them off before using the intersect option. To turn off a function, move thecursor over the = sign and press enter.

After graphing the functions, access the CALCULATE menu, 5: intersection. Boundaries are set near the point of intersection similar to what was done to find a minimum or maximum.

Piecewise Functions

Piecewise functions require careful entry. Graph the function. Access the TEST menu by pressing . Enter each part of the function in conjunction with the TEST menu. The parentheses are important.

TEST Menu
(2nd, MATH) / Enter the function.
Put in DOT mode, if desired. / This is the graph using
ZDecimal. / This is the graph using
ZStandard.

Some images in this tutorial have been taken from various sites.

References:

(free download of some video training clips)

Supplement to Section 10.4, Algebra for College Students,

Dugopolski, Sixth Edition

I. Polynomial functions are functions that have the form:

The value of n must be a nonnegative integer.

The coefficientsare. These are real numbers.

Notice that the second to the last term in this form actually has x raised to an exponent of 1, and that the last term in this form actually has x raised to an exponent of 0, as in

We do not show these exponents. Of course, x raised to a power of 0 makes it equal to 1, which is why the last term is just a constant.

/ This is NOT
a polynomial term... / ...because the variable has a negative exponent, which means the variable is in the denominator.
/ This is NOT
a polynomial term... / ...because the variable is in the denominator, which means the exponent is negative.
/ This is NOT
a polynomial term... / ...because the variable is inside a radical.
4x2 / This IS a polynomial term... / ...because it obeys all the rules stated above.

The degree of the polynomial function is the highest value for nwhere

Example 1: the degree of is 5 .

Which of the following is a polynomial?

a. c.

b. d.

II. The x-intercept(s) and y-intercept of a graph:

The x-intercept (root, solution, zero) is that value of xwhere the graph crosses or touches the x-axis. At the x-intercept -- on the x-axis -- y = 0. To find the x-intercept(s), let

The y-intercept is that value of y where the graph crosses the y-axis. At the y-intercept,

x = 0.

Example 2: Find the roots of this polynomial:

f(x) = (x + 4)(x + 2)(x− 1)

The x-intercepts are the roots. There are three real roots:
The polynomial is of degree 3. (Note that there are three x-intercepts).
The y-intercept is. Why?

For the polynomial shown above, there are three distinct real roots. The number of times a root repeats is called its multiplicity. Each of the roots above has a multiplicity of one.

Example 3: Find the roots and the y-intercept of this polynomial:

f(x) = (x + 2)(x + 2)(x − 1)

Note that there is a repeating root. If the roots are. The root at has a multiplicity of two. It is sufficient to say that the polynomial has roots at .

The multiplicity of a root gives information about the shape of the graph.

If the root has odd multiplicity, the polynomial will cross the x-axis at the root.
If the root has even multiplicity, the polynomial will “bounce off” the x-axis at the root: that is, it will not cross the x-axis at the root.

Example 4:

y = (x + 6)(x – 7) / y = (x + 6)(x – 7)2
x = –6 once
x = 7 once / x = – 6 once
x = 7 twice

Roots at Roots at

The graph crosses the x-axis The graph crosses the x-axis

at both roots. at but not at .

y = (x + 6)2 (x – 7) / y = (x + 6)2(x – 7)2
x = – 6 twice
x = 7 once / x = – 6 twice
x = 7 twice

Roots at Roots at

The graph crosses the x-axis The graph does not crosses the x-axis

at but not at . at either of the roots.

All four graphs have the same zeroes (roots, solutions, x-intercepts) but the multiplicity of the root determines whether the graph crosses the x-axis at that zero or if it instead turns back the way it came.

Find the zeros of the polynomials below, the multiplicity of each, and describe the behavior of the polynomial at each of the roots. Then find the y-intercept and the degree of the polynomial.

III. Symmetry

x / f(x)
-3
-2
-1
0
1
2
3

Note that,,, and, for any value of x,

This is true of functions that are called even functions. The graph of these functions is symmetric with respect to the y-axis.

x / f(x)
-3
-2
-1
0
1
2
3

Note that,,, and, for any value of x,

This is true of functions that are called odd functions. The graph of these functions is symmetric with respect to the origin.

A function may be neither odd nor even, and so may not have any symmetry.

Test the following functions to determine if there is any symmetry.

a. f.

b. g.

c. h.

d.i.

e. j.

A Room of One’s Own

1. A student finds an apartment for $650.00 per month. In order to rent the apartment, the student must pay the first month’s rent up front, and a security deposit equal to the monthly rent. Assuming that this student has $100.00 saved in the bank, and can save $70.00 per month, how long will it be before the student can move to this apartment?

2. One way to move out sooner is to save more money. How long will it take to move, assuming the student saves x dollars per month?

3. How much would the student have to save per month to be able to move in three months?

4. Suppose the student cannot save more than $70.00 per month. How long would it be before the student can move if the student decides to take a roommate who will pay half of the first month’s rent and security deposit?

5. How long before the student could move if the student had 2 roommates?

6. How many roommates would the student need to be able to move out in three months?

Do you think this apartment would be big enough?

A Room of One’s Own (Solutions)1

so about 18 months

2. One way to move out sooner is to save more money. How long will it take to move, assuming the student saves x dollars per month?

Moving out time = (technically)

3. How much would the student have to save per month to be able to move in three months?

so $400.00 per month