# 5.7 Describe and Compare Function Characteristics

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5.7 Describe and Compare Function Characteristics
How do you analyze and compare functions? How do you sketch a graph from a verbal description?

Increasing over an interval Decreasing over an interval
The graphs of π¦=π₯ and π¦= 2 π₯ rise from left to right for all real numbers. These functions are always increasing. The graphs of π¦=βπ₯ and π¦= 0.5 π₯ fall from left to right for all real numbers. These functions are always decreasing. The function π¦= π₯ 2 is decreasing for π₯β€0, but increasing for π₯β₯0. Increasing over an interval Decreasing over an interval Values of π(π₯) π π₯ 1 <π π₯ 2 whenever π₯ 1 < π₯ 2 π π₯ 1 >π π₯ 2 whenever π₯ 1 < π₯ 2 Graph of π(π₯) 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 a growth rate a little higher than at the beginning of the period.

Average Rate of Change For a linear function, the rate of change is constantβ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.

For the function f(x) = 2x β 2 + 1, find the average rate of change over the intervals [β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)? SOLUTION The average rate of change over an interval [x1, x2] is . f(x2) β f(x1) x 2 β x1 f(x) = 2x β 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 βdraftβ players for their teams
FOOTBALL Fantasy football participants βdraftβ players for their teams. Below are models predicting fantasy points for two positions as a function of the playerβ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.

Tight ends: (1, 175) (10, 90) (20, 60) (30, 45) (40, 35) (50, 25)
Running Backs 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) 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.

Even & Odd Functions, page 360
A function π is an even function if π βπ₯ =π(π₯) for all x in its domain. The graph of an even function is symmetric about the y-axis. A function π is an odd functions if π βπ₯ =βπ(π₯) 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 180Β° rotation about the origin.

Even Function Odd Function

Determine whether the function is even, odd, or neither.
a f(x) = x3 β 7x b. g(x) = 6 x + 1 a. Replace x with βx in the equation for the function, and then simplify. f(βx) = (βx)3 β 7(βx) = βx3 + 7x = β(x3 β 7x) = β f(x) Because f(βx) = β f(x), the function is odd. 6 (βx) + 1 6 βx + 1 b. g(βx) = = 6 βx + 1 6 x + 1 6 βx + 1 6 x + 1 Because β  and β  β , g(x) is neither even nor odd.

5.7 Assignment Page 361, 3, 4, 6, 8, all