Graphing Scatter Plots and Mathematical Models on the TI-83
Enter x and y data points in a data table:
To create the scatter plot of the data: GRAPH.
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.
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=ax2+bx+c. (shape: parabola)
b. CubicReg gives a cubic model in the form y=ax3+bx2+cx+d. (shape: “s” curve)
c. QuartReg gives a quartic model in the form y=ax4+ bx3+cx2+ . . . . (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*bx. (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 R2 or r2 and r.
R2 (or r2) is called the coefficient of determination; r is called the correlation coefficient.
These give helpful information about how well the model fits the data.
The closer R2 (or r2) is to 1, and the closer r is to 1 or –1, the better the fit.
If neither of these diagnostics appear, go to 2nd CATALOG,
and scroll down through the alphabetical list and select DiagnosticOn.
To display the graph of the model you have just calculated on the scatter plot,
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.