**Graphing Scatter Plots and Mathematical
Models on the TI-83**

**Enter x and y data points in a data table:**

- STAT ® EDIT
menu ®
1:Edit.
- Enter the x- and y-values in two separate
columns. Normally you would enter the x-values in the L
_{1}column, then press 4 to go to L_{2}and enter the y-values. Make sure that each point has its x- and y-coordinates in the same row. - To clear lists of unwanted data sets: STAT ® EDIT
menu ® 4:ClrList, then specify which list(s) you want to clear,
such as 2nd L1 ® , ® 2nd L2.

**To create the
scatter plot of the data:**
GRAPH.

- ZOOM ® ZOOM
menu ® 9:ZoomStat is a great shortcut for setting the viewing
window to fit the data points.
- If you get an error message, check 2
^{nd}STATPLOT for correct setup:

ü Plot1
should be On.

ü Type
should be scatterplot.

ü Make sure
the Xlist and Ylist are correct.

ü
All other Plots should be Off.

- Press Y= and make sure all functions are either cleared out or
turned off. (They are On if the “=” before the function is highlighted in
black, for example \Y
_{1}=2x+5. Either clear them or, to turn them off without clearing them, use the 3 key to select = and press ENTER.

In
mathematical modeling, we seek to find a mathematical function whose graph has
a good fit to the data in the scatter plot.
Such a function is called a mathematical model. Several
functions are available, from straight lines (linear models) to curves
(non-linear models).

1. Linear models. Two linear models
are available on the TI-83:

a.
The median-median line (Med-Med)
is given in the form y=ax+b, where a is the slope of
the line, and b is the y-intercept.

b. The least squares linear regression line. As a linear model, this minimizes the sum of
the squares of the difference between each point’s actual y-value and the
y-value the model would predict for that x.
The TI-83 offers two formats which yield the same line:

(1) LinReg(ax+b), where a is the
slope of the line, and b is the y-intercept. We will use this.

(2) LinReg(a+bx), where a is the
y-intercept, and b is the slope of the line.

2. Non-linear models (curves):

a.
QuadReg gives a quadratic model in
the form y=ax^{2}+bx+c.
(shape: parabola)

b.
CubicReg gives a cubic model in
the form y=ax^{3}+bx^{2}+cx+d.
(shape: “s” curve)

c.
QuartReg gives a quartic model in
the form y=ax^{4}+ bx^{3}+cx^{2}+ . . . . (shape: “u” or “w” curve)

d.
LnReg gives a logarithmic model in
the form y=a+b*ln(x). (shape:
logarithmic curve)

e.
ExpReg gives an exponential model in the form y=a*b^{x}. (shape:
exponential curve)

f.
PwrReg gives a power model in the
form y=a*x^b. (shape: parabola, “u”, or “s”
curve)

**To get the equation of the
function for a mathematical model for your data set,**

press STAT ® CALC
menu, then choose the desired type of function for your model.

It will calculate, store, and report the values of the parameters of the
function: a, b, etc.

Depending on the type of function
chosen and whether or not the diagnostics are turned on,

CALC may also report the value(s) of *R ^{2}* or

R

These give helpful information about how well the model fits the data.

The closer R

If neither of these diagnostics appear, go to 2

and scroll down through the alphabetical list and select DiagnosticOn.

go to Y= and insert the equation as follows:

VARS ® VARS menu ® 5:Statistics ® EQ menu ® 1:RegEQ.

GRAPH should now show both the data points and mathematical model.

Once you
have found the model that appears to fit the data best, you will use it to make
judgments about the nature of the data.
(Depending on the nature of the data, some
of these questions will be reasonable to ask, and others would make no sense.)

1.
What value of the function (y)
would you predict for a given value of x?

2. What value of x would you expect to produce a zero value of the
function (y)? (x-intercept)

3. What value of x would you expect to produce the maximum (or minimum)
value for y?

And, what is the maximum (or minimum) value of the function?

4. Where is the function increasing?
Decreasing?

5. At what value of x does the function (y) reach a certain value?

These questions can be answered using a variety of methods, both
algebraic methods and graphical methods.
Don’t rely solely on algebraic or graphic methods; let them complement
one another. Use common sense and all
the tools at your disposal. Use
graphical methods, either when you are weak in the algebraic method, or for
greater speed than algebraic methods, or to check your algebra if you prefer
algebraic methods.

You can use WINDOW, ZOOM, and TRACE to navigate around the graph to get a good idea of what is going
on. However, for accurate answers to
questions such as those above, use the function. It will enable you to calculate these with
much greater accuracy.

1.
To predict the value of the
function (y) for a given x, use CALC ®
1:value. You supply the x value, and it
calculates the y value. Or you can go to
the TABLE and find the x value and read off the y value.

2.
To find x-intercepts, that is,
x-values that make y zero, use CALC ® 2:zero (some calculators call this
“root”, since the x-values that make y zero are called the zeroes or the roots
of the function). You are asked to give
left and right bounds for x in the region where the function crosses the
x-axis. Then the calculator uses
numerical methods to home in on the x-intercept in the range you gave it.

3.
To find the location of a maximum
or minimum value, or to find that value, use CALC ® 3:
minimum or 4:maximum. You are asked for
left and right bounds for x in the region where the max or min occurs. Then the calculator uses numerical methods to
home in on the point in the range you gave it.
The x-value of that point is where the max or min occurs; the y value is
the max or min.

4.
To determine where a function is
increasing/decreasing, the location of max/ min helps, then zooming out to get
the big picture when needed. More about
this later.

5.
To find where the function reaches
a certain value, or to find where two functions intersect, first enter the
value or second function enter in Y = , then use CALC ®
5:intersect and choose your two functions and give a guess at the x value where
the intersection occurs. Then the
calculator uses numerical methods to find the x value.

Get good at using the features your calculator has to help you analyze
functions better. Your graphing
calculator is a wonderful tool for use in analyzing functions and modeling real
world data. In this course, we will be
doing both as we work at sharpen algebraic skills. The goal is good algebraic and calculator
skills as tools for data analysis and understanding functions that model real
world phenomena.