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 𝑦 = 𝑥 and 𝑦 = 2𝑥 rise from left
to right for all real numbers. These functions
are always increasing.
• The graphs of 𝑦 = −𝑥 and 𝑦 = 0.5𝑥 fall from
left to right for all real numbers. These
functions are always decreasing.
• The function 𝑦 = 𝑥 2 is decreasing for 𝑥 ≤ 0,
but increasing for 𝑥 ≥ 0.
Values of 𝑓(𝑥)
Graph of 𝑓(𝑥)
Increasing over an interval
Decreasing over an interval
𝑓 𝑥1 < 𝑓 𝑥2 whenever 𝑥1 < 𝑥2
𝑓 𝑥1 > 𝑓 𝑥2 whenever 𝑥1 < 𝑥2
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
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.
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)?
SOLUTION
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.
Tight Ends
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 𝑓 is an even function if 𝑓 −𝑥 = 𝑓(𝑥)
for all x in its domain.
• The graph of an even function is symmetric about
the y-axis.
• A function 𝑓 is an odd functions if 𝑓 −𝑥 = −𝑓(𝑥)
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.
6
g(x) = x + 1
a. Replace x with –x in the equation for the function, and
then simplify.
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 –x +6 1 ≠ x +61 and –x +6 1 ≠ – x +61 , g(x) is
neither even nor odd.
5.7 Assignment
Page 361, 3, 4, 6, 8, 15-21 all