Quadratic Regression

Report
= “change”

In the 2 scenarios below, find the change in x and the
change in y.
X
1800
1850
1900
1950
2000
2050
3000
Y
125
150
172
198
223
247
272
X
1800
1810
1820
1830
1840
1850
1860
Y
100
150
175
180
172
149
102
X
Y
X
Y

What conclusions can you draw? What are the similarities
& differences? How would you model this data?
So far, the data we have worked with
has a linear relationship
 We have discussed 3 forms of linear
modeling:

› Best Fit line
› Least Squares Regression Line
› Median-Median line
The data we have used has a linear
relationship. This means that the rate of
change (the slope) is constant.
 For linear data, every increase in the
independent variable (X) has a constant
increase in the dependent variable (Y).

Slope =
y
= constant
x
Data does not always follow a linear model.
 Data may increase sharply, reach a
maximum, then decline. Or, it may
decrease sharply, reach a minimum, then
increase again.
 The data in the graph at
the right is shaped like a
parabola – so it would follow
a quadratic model instead
of a linear model.


Recall that parabolas are graphs of
quadratic equations. They follow the
model of y = ax2 + bx +c.
› If a > 0, the parabola opens up (smiling)
› If a < 0, the parabola opens down (frowning)

Real-life examples of data that follows a
quadratic pattern include:
› Stock market (Peaks and Valleys)
› Disease outbreaks (Black Plague, Polio, AIDS)
› Particle motion (Ball trajectory, Draining water)
Data can also follow an exponential
model.
 Exponential data either

› Increases exponentially, where the change
in y continues to increase for each change
in x OR
› Decreases exponentially, where the change
in y continues to decrease for each change
in x.

Examples of Real-Life data that follows an
exponential model include:
› Population growth (increasing)
› National Debt (increasing)
› Radioactive Decay (decreasing)

Exponential equations
500
follow the model
y = a(b)x
300
(where a and b are constants)
Population
Rabbits
100
-100 0
2
4
Months
6

To determine if data is linear, quadratic
or exponential
› Create a Scatterplot of the data and look
for the overall pattern
› Evaluate the change in Y for each change
in X
IF
y
x
= Constant
= Increasing, 0, then Decreasing or
Decreasing, 0, then Increasing
= Increasing exponentially
(e.g., doubles every time) or
Decreases exponentially
(e.g., halves every time)
LINEAR
QUADRATIC
EXPONENTIAL
Year (x)
Widgets (y)
1850
34
1875
29
1900
25
1925
21
1950
16
1975
11
Time (x)
Widgets (y)
0
0
1
10
2
14
3
13
4
9
5
6
Time (x)
Widgets (y)
0
0
1
2
2
4
3
16
4
256
5
65,536
2000
6
The table below lists the total estimated
numbers of AIDS cases, by year of
diagnosis from 1999 to 2003 in the United
States
AIDS
 Notice the data peaks
Year
Cases
in 2001, then drops off.
1999
41,356
 This is a good indicator
2000
41,267
that Quadratic Regression
2001
40,833
will provide the most
2002
41,289
accurate model of the data
2003
43,171

(Source: US Dept. of Health and Human Services, Centers for Disease Control and Prevention, HIV/AIDS Surveillance, 2003.)
1). Plot the data, letting x = 0 correspond to the year 1998.
2). Find a quadratic function that models the data.
• Using your calculator, enter the Year as L1 and #of Cases as L2
• Use the QuadReg function on your calculator to calculate the
regression equation
3). Plot the function on the graph with the data and determine how
well the graph fits the data,
4). Use the model (equation) to predict the cumulative number of AIDS
cases for the year 2006.
Year
AIDS
Cases
1999
41,356
2000
41,267
2001
40,833
2002
41,289
2003
43,171

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