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7.2 Polynomial Functions and Their Graphs
Objectives:
•Identify and describe the important features of the
graph of a polynomial function
•Use a polynomial function to model real-world data
Exploring End Behavior of f(x) = axn
Graph each function separately. For each function,
answer parts a-c.
1) y = x2
2) y = x4
3) y = 2x2
4) y = 2x4
5) y = x3
6) y = x5
7) y = 2x3
8) y = 2x5
9) y = -x2
10) y = -x4
11) y = -2x2
12) y = -2x4
13) y = -x3
14) y = -x5
15) y = -2x3
16) y = -2x5
a. Is the degree of the function even or odd?
b. Is the leading coefficient positive or negative?
c. Does the graph rise or fall on the left? on the right?
Exploring End Behavior of f(x) = axn
Rise or Fall???
a > 0
left
a < 0
right
left
right
n is even
rise
rise
fall
fall
n is odd
fall
rise
rise
fall
Example 1
Describe the end behavior of each function.
a) V(x) = x3 – 2x2 – 5x + 3
falls on the left and rises on the right
b) R(x) = 1 + x – x2 – x3 + 2x4
rises on the left and the right
Graphs of Polynomial Functions
f(a) is a local maximum if there is an interval around
a such that f(a) > f(x) for all values of x in the
interval, where x = a.
f(a) is a local minimum if there is an interval around
a such that f(a) < f(x) for all values of x in the
interval, where x = a.
Increasing and Decreasing Functions
Let x1 and x2 be numbers in the domain of a function, f.
The function f is increasing over an open interval if for
every x1 < x2 in the interval, f(x1) < f(x2).
The function f is decreasing over an open interval if for
every x1 < x2 in the interval, f(x1) > f(x2).
Example 2
Graph P(x) = -2x3 – x2 + 5x + 6.
a) Approximate any local maxima or minima to the
nearest tenth.
minimum: (-1.1,2.0)
maximum: (0.8,8.3)
b) Find the intervals over which the function is increasing
and decreasing.
increasing: x > -1.1 and x < 0.8
decreasing: x < -1.1 or x > 0.8
Example 3
The table below gives the number of students who
participated in the ACT program during selected years
from 1970 to 1995. The variable x represents the
number of years since 1960, and y represents the
number of participants in thousands.
x
y
a) Find a quartic regression model for the
10
714
number of students who participated in
15
822
the ACT program during the given years
20 836
A(x) = -0.004x4 + 0.44x3 – 17.96x2 + 293.83x - 836
25 739
b) Use the regression model to estimate
30
817
the number of students who participated
in the ACT program in 1985.
35
estimate using model is about 767,000
945
Homework
Lesson 7.2 exercises 29-36
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