We may do this by looking at the graphs of the functions, by comparing tables of values of the functions or by examining the algebraic representations of the functions.

Questions we may wish to answer in comparing two functions include such things as

Whether one function is always greater than or less than the other,
Whether the graphs intersect and if so how often,
Whether one or both functions are strictly increasing or decreasing on an interval,
How the function values compare at selected points in the domain,
How the slopes differ, and
How the maximum or minimum values compare.

Practice questions

QUESTION 1

Consider the line $P$P given by the equation $y=-12+\frac{x}{10}$y=−12+x10, and the table of values for parabola $Q$Q.

Parabola $Q$Q:

$x$`x`	$-8$−8	$-7$−7	$-6$−6	$-5$−5	$-4$−4
$y$`y`	$-1$−1	$-4$−4	$-5$−5	$-4$−4	$-1$−1

Line $P$P is shown below. Draw the graph of the parabola $Q$Q.

Loading Graph...
How many times do $P$P and $Q$Q intersect?
Twice
A
Three times
B
Zero times
C
Once
D
Which function has the higher function value at the $y$y-intercept?
$Q$Q
A
$P$P
B
Which function has the higher function value at $x=-1$x=−1?
$Q$Q
A
$P$P
B

QUESTION 2

The parabola $P$P is given by $y=\left(x-5\right)^2+5$y=(x−5)2+5 and the parabola $Q$Q is given by $y=-2\left(x-7\right)\left(x-3\right)$y=−2(x−7)(x−3).

Find the vertex of parabola $Q$Q.

$\left(\editable{},\editable{}\right)$(,)
The graph of parabola $P$P is shown below. Graph the parabola $Q$Q.

Loading Graph...
By referring to the graphs, which function has a maximum at $y=8$y=8?
$P$P
A
$Q$Q
B
$P$P and $Q$Q
C
Neither $P$P or $Q$Q
D
By referring to the graphs, which function has a minimum at $y=5$y=5?
$P$P
A
$Q$Q
B
$P$P and $Q$Q
C
Neither $P$P or $Q$Q
D
Complete the following statements:

Parabola $P$P has $\editable{}$ zero(s).

Parabola $Q$Q has $\editable{}$ zero(s).
How many times do the parabolas intersect?
Once
A
Zero times
B
Three times
C
Twice
D

QUESTION 3

The graph of the exponential function $P$P, given by $y=4^{-x}$y=4−x is shown below.

Graph the line $Q$Q given by the equation $y=8x+12$y=8x+12.

Loading Graph...
How many times do $P$P and $Q$Q intersect?
Zero times
A
Twice
B
Three times
C
Once
D
Which function has the higher function value at the $y$y-intercept?
$P$P
A
$Q$Q
B
Which function has the higher function value at $x=1$x=1?
$P$P
A
$Q$Q
B

Outcomes

III.F.IF.9

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

11.07 Comparing function types