Back to Exercise: Compare function properties

Exercises: Compare Function Properties

Work through each section in order. When comparing functions, always cite specific values from each representation to support your conclusion.

Grade 9·19 problems·~30 min·Common Core Math - HS Functions·group·hsf-if-c-9

Work through problems with immediate feedback

Recall / Warm-Up

The function $f(x) = -(x-2)^2 + 9$ is in vertex form. What is its maximum value?

$-2$

$2$

$9$

$x = 2$

The table below shows values for $g(x)$ .

$x$	$g(x)$
0	5
1	2
2	1
3	2
4	5

What is the minimum value of $g(x)$ shown in the table?

$0$

$1$

$2$

$x = 2$

The average rate of change of $h(x)$ from $x = 1$ to $x = 4$ is computed as $\frac{h(4) - h(1)}{4 - 1}$ . If $h(1) = 7$ and $h(4) = 16$ , what is the average rate of change?

$9$

$3$

$\frac{16}{7}$

$\frac{7}{16}$

Fluency Practice

Function A (graph): a downward-opening parabola with vertex at approximately $(3, 6)$ and $y$ -intercept at $(0, -3)$ .

Function B (algebraic): $g(x) = -(x - 1)^2 + 8$ .

Which function has the larger maximum value?

Function A, with a maximum of approximately 6.

Function B, with a maximum of 8.

They have equal maxima.

Function A, because it reaches its maximum at $x = 3$ which is farther right.

Function A (algebraic): $f(x) = x^2 - 4$ .

Function B (table):

$x$	$g(x)$
$-3$	$5$
$-1$	$-3$
$0$	$-4$
$1$	$-3$
$3$	$5$

Which function has the larger $y$ -intercept?

Function A, with $y$ -intercept $-4$ .

Function B, with $y$ -intercept $-4$ .

They have equal $y$ -intercepts of $-4$ .

Function A, because $f(x) = x^2 - 4$ has a subtracted $4$ , making it larger.

Function A (table):

$x$	$f(x)$
$0$	$2$
$2$	$6$
$4$	$10$

Function B (algebraic): $g(x) = 3x - 1$ .

Compare the average rates of change of both functions over $[0, 4]$ . Which function has the greater average rate of change?

Function A, with average rate of change $2$ .

Function B, with average rate of change $3$ .

They have equal average rates of change.

They cannot be compared because they are in different representations.

Function A (verbal): "A linear function that starts at 4 when $x = 0$ and increases by 5 for each unit increase in $x$ ."

Function B (algebraic): $g(x) = 6x - 2$ .

What is the $y$ -intercept of Function A minus the $y$ -intercept of Function B? (Compute A's $y$ -intercept minus B's $y$ -intercept.)

Function A (algebraic): $f(x) = x^2 - 6x + 5$ .

Function B (verbal): "A quadratic that opens upward with vertex at $(4, -3)$ ."

Which function has the smaller minimum value?

Function A, with minimum $-4$ .

Function B, with minimum $-3$ .

They have equal minima.

Function A, because it has smaller coefficients.

Varied Practice

Two parabolas are graphed. Parabola A has vertex at approximately $(1, 7)$ and opens downward. Parabola B has vertex at approximately $(4, 7)$ and also opens downward.

What can you conclude?

Function A has a larger maximum than Function B.

Function B has a larger maximum than Function A.

Both functions appear to have the same maximum value of approximately 7.

The maxima cannot be compared from a graph.

Function A (table):

$x$	$f(x)$
$1$	$3$
$3$	$7$
$5$	$11$

Function B (algebraic): $g(x) = 3x - 2$ .

A student claims: "I can tell from the table that Function A has a constant rate of change, and it's the same as Function B's rate." Is the student's claim valid?

No — tables only show specific values and you cannot conclude the rate is constant for all $x$ .

Yes — the table shows a constant rate of change of 2 per unit, but Function B's rate is 3, so the rates are not equal.

Yes — the table shows a constant rate of change of 2 per unit, and Function B's rate is also 2. The rates are equal over the shown interval, but Function B's rate is also 3 (contradiction — the student should check more carefully).

Partially valid — the table does show a constant rate of 2 per unit ( $\frac{7-3}{3-1} = 2$ ), suggesting linear behavior. But Function B has rate 3, not 2. So the rates are not equal.

Function A (verbal): "A linear function that passes through $(0, 8)$ and decreases by 3 for every unit increase in $x$ ."

Function B (algebraic): $g(x) = -2x + 5$ .

Which function has the larger $x$ -intercept?

Function A, with $x$ -intercept at $\frac{8}{3}$ .

Function B, with $x$ -intercept at $2.5$ .

They have equal $x$ -intercepts.

Function A, because it has a larger $y$ -intercept.

Function A (algebraic): $f(x) = x^2 - 4x$ .

Function B (table):

$x$	$g(x)$
$-1$	$9$
$0$	$4$
$2$	$0$
$4$	$4$
$5$	$9$

Write two comparison statements, each comparing the same property for both functions and citing specific values as evidence.

Word Problems

Two companies model their weekly revenue (in thousands of dollars) as functions of price $p$ (in dollars):
Company A (algebraic): $R_A(p) = -2p^2 + 40p$
Company B (table, based on market research):

$p$	$R_B(p)$
$5$	$75$
$8$	$96$
$10$	$100$
$12$	$96$
$15$	$75$

Which company achieves the higher maximum revenue, and what is that maximum?

Company A, with maximum revenue of 200 thousand dollars.

Company B, with maximum revenue of 100 thousand dollars.

They have equal maximum revenues of 100 thousand dollars.

Company A, with maximum revenue of 40 thousand dollars.

At what price does Company A reach its maximum revenue?

$p = 5$

$p = 10$

$p = 20$

$p = 40$

Function A (verbal): "A population that starts at 5,000 and grows by 200 people per year."

Function B (algebraic): $Q(t) = 4000 + 300t$ , where $Q$ is population and $t$ is years.

After how many years will the two populations be equal? Enter the number of years.

Error Analysis

A student compared two functions:
Function A: maximum value of 12 over the domain $[0, 10]$ .
Function B: maximum value of 10 over the domain $[0, 6]$ .

The student concluded: "Function A has the larger maximum, so Function A is the better function."

What is wrong with the student's comparison?

The student correctly compared the maxima. The conclusion is valid.

The student compared maxima over different domains. Function A's domain extends further, which may explain its higher maximum. The comparison is not fair unless the domains are the same.

The student should have compared the minima, not the maxima.

Maxima cannot be compared between functions without knowing the algebraic expressions.

A student compared two functions from a graph and wrote:

"Function A has an exact maximum of 6.4 at $x = 2.3$ . Function B has an exact maximum of 6.1 at $x = 3.7$ . Therefore, Function A has the larger exact maximum."

What is the error in the student's reasoning?

The student used the wrong formula for the maximum.

Graph readings are approximate, not exact. The student cannot claim "exact" values from a graph — the difference (6.4 vs 6.1) may fall within graph-reading error.

The student should have compared the minima instead.

The student's conclusion is correct — graph maxima can always be read exactly.

Challenge / Extension

Function A (graph): a linear function passing through $(0, 10)$ and $(5, 0)$ .
Function B (table): starts with $B(0) = 1$ and doubles every unit ( $B(1)=2$ , $B(2)=4$ , $B(3)=8$ , ...).

Write three comparison statements for these functions. For each, name the property, state the value for both functions, and make a relational judgment. Include at least one comparison about long-term behavior.

Function A is a quadratic with vertex $(2, 8)$ that opens downward. Function B is a linear function with $y$ -intercept 6 and slope 3.

At $x = 0$ , which function has the larger value?

Function A, because it has a maximum at $(2, 8)$ which is larger than 6.

Function B, with $B(0) = 6$ .

Function A — its value at $x = 0$ is greater than 6, but we need more information to know by how much.

We cannot determine which is larger at $x = 0$ without knowing the quadratic's leading coefficient or its value at $x = 0$ .

0 of 19 answered