5.7 Describe and Compare Function Characteristics

How do you analyze and compare functions?How do you sketch a graph from a verbal

description?

• The graphs of rise from left to right for all real numbers. These functions are always increasing.

• The graphs of fall from left to right for all real numbers. These functions are always decreasing.

• The function is decreasing for , but increasing for .

Increasing over an interval Decreasing over an interval

Values of

Graph of Rises from left to right Falls from left to right

FLYING Sketch a graph of a(t) where a represents altitude and t represents time for the situation below. Label key information such as local minima and maxima and intervals where a(t) is increasing or decreasing.

A small plane flying at a constant cruising altitude is caught in a storm. Air currents carry the plane up before pushing it down rapidly below its original altitude. The plane exits the storm and returns gradually to its cruising altitude.

SOLUTION

1. Sketch a graph for the situation described below. Label key information.

The US gross domestic product (GDP) growth rate over a 2 year period can be modeled by a polynomial that began slightly positive before dropping to significantly negative, swinging back to significantly positive, then falling to agrowth rate a little higher than at the beginning of the period.

Average Rate of Change

• For a linear function, the rate of change is constant—the slope.

• For a nonlinear function, you can find the average rate of change over any interval by finding the slope of the segment that connect the points on the graph determined by the endpoints of the interval.

SOLUTION

For the function f(x) = 2x – 2 + 1, find the average rate of change over the intervals [–2, 0], [0, 2], [2, 4], [4, 6], and [6, 8]. What happens to the average rate of change as x increases? What does this mean for the graph of f(x)?

The average rate of change over an interval [x1, x2] is

.f(x2) – f(x1) x 2 – x1

f(x) = 2x – 2 + 1

The average rate of change is always positive, so the graph always rises. The average rate of change increases over successive intervals of the domain(by a factor of 4 for each increase of 2 in x). Because the average rate of change over each interval is equivalent to the slope of the segment from(x1, f(x1)) to (x2, f(x2)), the graph rises more and more steeply as x increases.

FOOTBALL Fantasy football participants “draft” players for their teams. Below are models predicting fantasy points for two positions as a function of the player’s rank. Compare fantasy points as a function of rank for the models.

Tight Ends Running Backs

Comparing ordered pairs, the points for running backs are always greater than for tight ends for each rank over the domain shown, but for running backs, the points decrease more rapidly as rank changes than for tight ends.

Comparing ordered pairs, the points for running backs are always greater than for tight ends for each rank over the domain shown, but for running backs, the points decrease more rapidly as rank changes than for tight ends.

Tight ends: (1, 175) (10, 90) (20, 60) (30, 45) (40, 35) (50, 25)Running backs: (1, 400) (10, 200) (20, 140) (30, 110) (40, 85) (50, 65)

Tight Ends Running Backs

Even & Odd Functions, page 360• A function is an even function if for all x in its

domain.• The graph of an even function is symmetric about

the y-axis.• A function is an odd functions if for all x in its

domain.• The graph of an odd function is symmetric about

the origin.• A graph that is symmetric about the origin looks

the same after a 180° rotation about the origin.

Even Function Odd Function

Determine whether the function is even, odd, or neither.

a. f(x) = x3 – 7x

b. g(x) = 6x + 1

Replace x with –x in the equation for the function, and then simplify.

a.

f(–x) = (–x)3 – 7(–x) = –x3 + 7x = –(x3 – 7x) = – f(x)

Because f(–x) = – f(x), the function is odd.

b. g(–x) = =6(–x) + 1

6–x + 1

Because ≠ and ≠ – , g(x) is neither even nor odd.

6–x + 1

6x + 1

6–x + 1

6x + 1

5.7 Assignment

Page 361, 3, 4, 6, 8, 15-21 all