Accelerated Geometry CC
3.3 Notes Curve Fitting with Quadratic Models
Date ______
Explore: For each table find the pattern and complete the missing y-values. Identify the pattern of successive yvalues. How do the y-values change as the x-values increment by 1?
x
y
-3
-5
x
y
x
y
-2
-3
-1
-1
0
1
2
3
7
-3
0.125
-2
.25
-1
.5
0
1
1
2
2
3
-3
10
-2
5
-1
2
0
1
1
2
3
Each of the above tables is a common function. You can identify functions by examining the patterns of successive
ordered pairs.
Linear Functions- change by a constant difference.
Exponential Functions- change by a constant ratio.
Quadratic Functions- change by a constant ____________________ difference.
Real world occurrences often model mathematical functions. For each situation below explain what function would
best model the situation and why. (Linear, Exponential, or Quadratic.)
a) Water is leaking from a pipe at 2 ounces per minute.
__________________________________________________________________________________________
b) The area of a square and side length.
___________________________________________________________________________________________
c) Every hour the amount of bacteria triples.
___________________________________________________________________________________________
Examples: Identifying models based on a table. For each set determine whether the data could represent a linear,
exponential or quadratic function.
a) {(-4,
1
1
1
), (-2, ), (0, ), (2, 2), (4, 8)}
8
2
32
b) {(-4,5), (-2,2), (0,1), (2,2), (4,5)}
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__________________________________
c) {(-4,1), (-2,2), (0,3), (2,4), (4, 5)}
d) {(0, -1.5), (1, -0.5), (2, 2.5), (3,7.5)}
__________________________________
__________________________________
Example: Writing a Quadratic Function from data.
*Note: just as two points define a linear function, three noncollinear points define a quadratic function. You can
find the three coefficients a, b, and c of f(x) = ax2 + bx + c by using a system of three equations, one for each point.
The points do not need to have equally spaced x-values.
Write a quadratic function that fits the set of data points:
a) {(0, 5), (2, 1), and (3, 2).
b) {(0, -3), (1, 0), and (2, 1)
You try:
c) {(-2, 5), (0, -3), (3, 0)}
d) {(0, 1), (2,1), (3, -8)}
Example: Application.
1)A quadratic model is a quadratic function that represents a data set.
Models are useful for making estimates. The table shows approximate
times for 16mm films, given the diameter of the film reel. A statistical
method called quadratic regression, can be used to make a quadratic
model for the data set. The quadratic regression for this data set can be
modeled by t(d) = 0.397d2-3.12d +11.94, where t is the run time in
minutes and d is the film diameter in inches.
a) Use the model to estimate the run time for a reel of film with a
diameter of 15 in.
run
b) Find the diameter of a reel if the approximate run time is 84.41 minutes. Round your answer to the nearest inch.
2) The pizza prices for DeAngelo’s pizza parlor are shown at the right. A quadratic model for the price of a pizza
based upon the size is p( s)  0.125s 2  5.25s  35.05 .
a) Use the quadratic model to find the price of a pizza with an 18 in. diameter.
b) Does the function have a maximum point? If so find it and explain what this point
represents.
c) According to the model, how much should a 30 in. pizza cost? How much should an 8 in pizza cost?
d) Is the quadratic function a good model for the price of DeAngelo’s pizza? Explain your reasoning.