MAT135 Supplement
MAT 135
Instructor’s Supplement
Table of Contents
Page
Graphing Calculator Basics 3
Supplemental Material Section 10.4 11
Other Class Assignments
A Room of One’s Own 16
Recycling 18
The Great Quadratic 19
Cell Phone Plans 22
Soccer Club 23
Rita’s Problem 25
Completing the Square 28
Don’t Bug Me 29
Getting to Motown 33
The Graph of a Function 37
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-TB. 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 singlenumber, the numbers
in a sequence, etc. /
3: 3 cubes the expression. It is accessed by entering a value, then 3. You can cube an expression, list, matrix, etc.
You take the cuberoot 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 theroot 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 EntryFunc (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 withthe TABLE.
CALC MENU: The CALC menus is accessed by
Pressing TRACE. The menu is shown
at the right.
When you see thisscreen, 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 Boundof 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 the cursor 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:
http://hotmath.com/graphing_calculators/ti84_movie_index.html
http://www.watchmeware.com/ti83ptutor.html (free download of some video training clips)
http://www.atomiclearning.com/ti_84 (free download of some video training clips)
http://mtl.math.uiuc.edu/tutorials.php
http://calculator.maconstate.edu/calc_topics.html
http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/ti.html#WINDOW
Supplement to Section 10.4, Algebra for College Students,
Dugopolski, Fifth Edition
I. Polynomial functions are functions that have the form:
.
The value of n must be a nonnegative integer.
The coefficients are. 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 NOTa 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 n where
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 x where 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)