7.2 Polynomial Functions and Their Graphs7.2 Polynomial Functions and Their Graphs7.2 Polynomial Functions and Their Graphs7.2 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

a > 0 a < 0

left right left right

n is even

n is odd

rise rise

fall rise

fall fall

rise fall

Rise or Fall???

Example 1Describe the end behavior of each function.

a) V(x) = x3 – 2x2 – 5x + 3

b) R(x) = 1 + x – x2 – x3 + 2x4

falls on the left and rises on the right

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 2Graph P(x) = -2x3 – x2 + 5x + 6.a) Approximate any local maxima or minima to the nearest tenth.

b) Find the intervals over which the function is increasing and decreasing.

minimum: (-1.1,2.0)

maximum: (0.8,8.3)

increasing:

x > -1.1 and x < 0.8decreasin

g: x < -1.1 or x > 0.8

Example 3The 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.

a) Find a quartic regression model for the number of students who participated in the ACT program during the given years

x y

10 714

15 822

20 836

25 739

30 817

35 945

b) Use the regression model to estimate the number of students who participated in the ACT program in 1985.

A(x) = -0.004x4 + 0.44x3 – 17.96x2 + 293.83x - 836

estimate using model is about 767,000

Homework

Lesson 7.2 exercises 29-36