Practice Exercise 5 – Quadratic Regression – Maximum
5. Fertilizer is used to improve the yield of bushels per acre when growing corn
on a farm. As the amount of fertilizer is increased, the yield in bushels per acre
increases. However, at some point, too much fertilizer becomes
counterproductive. This is an example of what is known in the theory of
economics as the law of diminishing returns.
Enter the following X data into list number 1 (L1) and enter the following Y data
into list number 2 (L2) in your calculator via the STAT EDIT function. A quadratic
regression equation is to be determined to model this data.
Pounds of
Fertilizer
X
0
5
10
15
20
25
30
35
40
Actual
Bushels Per
Acre
Y
4
10
12
15
18
22
21
19
17
Plot these nine points on the following scatter plot diagram.
Bushels
24
You may GRAPH this
equation on your calculator
with WINDOW settings:
21
18
Xmin = 0
Xmax = 50
Xscl = 5
15
Ymin = 0
Ymax =30
Yscl = 3
12
9
6
3
5
10
15 20
25 30
35
40
Fertilizer
1
45
50
Practice Exercise 5 – Quadratic Regression – Maximum
Use the 2ND CATALOG function to turn DiagnosticsOn.
From the STAT CALC menu, execute the QuadReg function to determine the
quadratic regression equation for the given data in the table above. State the
2
following values for a, b, c and R as derived from execution of the QuadReg
function. Express these values of a, b, c and R2 with all of the digits of precision
to the right of the decimal point as displayed on your calculator when you
execute the QuadReg function: Y a X 2 b X c
a=
b=
c=
2
R =
Express the quadratic regression equation in the form: Y a X 2 b 
X c
In this following equation, express the values of a, b and c with three digits of
precision to the right of the decimal point:
Enter this quadratic regression equation into your calculator and use the
TABLESET and TABLE functions of your calculator to determine the predicted
values of Y in the following table. Express these values of Y with three digits of
precision to the right of the decimal point as displayed by the TABLE function on
your calculator:
Pounds of
Fertilizer
Predicted
Bushels Per Acre
X
0
5
10
15
20
25
30
35
40
Y
Plot these nine points and draw the quadratic regression curve superimposed on
the scatter plot diagram on the previous page.
2
Practice Exercise 5 – Quadratic Regression – Maximum
(c) Display the GRAPH of this equation on your calculator using the WINDOW
settings noted above. This will assist in drawing the curve above and it will set
the stage for the calculating the maximum point on the curve.
(d) After displaying the graph on your calculator, use the MAXIMUM function of
the 2ND CALC menu to determine the coordinates of the maximum point on the
graph. After the MAXIMUM function is activated, do the following:
Move the CURSOR to the left of the maximum point on the curve, that is,
somewhere in the region of X = 23. Then, hit ENTER.
Next, move the CURSOR to the right of the maximum point, that is, somewhere
in the region of X = 33. Then, hit ENTER.
Next, move the CURSOR to the vicinity of the maximum point and hit ENTER to
calculate the X and Y coordinates of the maximum point. Express X and Y with
all of the digits of precision to the right of the decimal point as displayed on your
calculator.
X=
Y=
3