Algebra Quiz: Comparing Functions Represented In Different Ways

20 questions · exam conditions

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Comparing Functions Represented In Different WaysQuestion 1 of 20

Function $f$ is given by $f(x)=-x^2+2x+3$ .

Function $g$ is shown on the coordinate plane as a parabola with x-intercepts at $x=-1$ and $x=3$ .

Which statement correctly compares the x-intercepts of $f$ and $g$ ?

f

and

g

have the same x-intercepts.

f

has x-intercepts

-1

and

3

, while

g

has x-intercepts

-3

and

1

f

has no real x-intercepts, but

g

has two.

f

has x-intercepts

-1

and

3

, and

g

has x-intercepts

-1

and

3

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Algebra Quiz

Algebra Quiz: Comparing Functions Represented In Different Ways

Practice Comparing Functions Represented In Different Ways in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Comparing Functions Represented In Different Ways, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Function $f$ is given by $f(x)=-x^2+2x+3$ .

Function $g$ is shown on the coordinate plane as a parabola with x-intercepts at $x=-1$ and $x=3$ .

Which statement correctly compares the x-intercepts of $f$ and $g$ ?

$f$ and $g$ have the same x-intercepts.
$f$ has x-intercepts $-1$ and $3$ , while $g$ has x-intercepts $-3$ and $1$ .
$f$ has no real x-intercepts, but $g$ has two.
$f$ has x-intercepts $-1$ and $3$ , and $g$ has x-intercepts $-1$ and $3$ . (correct answer)

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f given by f(x)=-x^2+2x+3, solve for x-intercepts by setting to zero: roots at x=-1 and x=3; for function g graphed with x-intercepts at -1 and 3, they match exactly. Choice C correctly identifies that both have x-intercepts at -1 and 3. If you picked choice B, that's understandable—check the quadratic formula or factoring to confirm f's roots. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Quick y-intercept trick: in a formula, set x = 0 and calculate. On a graph, see where it crosses the y-axis. In a table, find the y-value when x = 0. Three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y = mx + b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.

Question 2

Function $f$ is given algebraically by $f(x)=2x-3$ . Function $g$ is shown by the table below.

Table for $g$ :

when $x=0$ , $g(x)=1$
when $x=2$ , $g(x)=5$

Which function has the larger y-intercept?

Function $g$ , because its y-intercept is $1$ and $1>-3$ . (correct answer)
Function $f$ , because its y-intercept is $3$ and $3>1$ .
They have the same y-intercept, because both include the point $(2,5)$ .
Function $f$ , because its y-intercept is $-3$ and $-3>1$ .

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare y-intercepts, calculate f(0) = 2*0 - 3 = -3 from the formula for f, and read g(0) = 1 directly from the table for g, showing that 1 > -3. Choice B correctly identifies that function g has the larger y-intercept because 1 > -3. Don't worry if you mixed up the intercept with another point like (2,5)—just remember the y-intercept is always at x=0, so double-check that value in each representation. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values—for example, to compare y-intercepts, find where x=0 in the formula, look where the graph crosses the y-axis, or find y when x=0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features—use each representation's strengths!

Question 3

Function $h$ is described verbally as: "a linear function with slope $-4$ and y-intercept $2$ ."

Function $k$ is given by the table:

$x$ : $-1,\ 0,\ 1$

$k(x)$ : $5,\ 3,\ 1$

Which function has the steeper slope (greater slope magnitude)?

Function $h$ , because $|-4|>|-2|$ . (correct answer)
Function $k$ , because its slope is $-1$ .
Function $k$ , because its slope is $-2$ and $|-2|>|-4|$ .
They are equally steep, because both slopes are negative.

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare slope magnitudes, note the verbal description gives h a slope of -4 (magnitude 4); for k's table, calculate slope as (3-5)/(0-(-1)) = -2 or (1-3)/(1-0) = -2 (magnitude 2), so 4 > 2. Choice A correctly identifies that function h has the steeper slope because |-4| > |-2|. If you thought k's slope was -1, that's okay—just practice calculating change in y over change in x from table points to get more comfortable. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.

Question 4

Function $f$ is given by $f(x)=3x+6$ .

Function $g$ is shown by the table:

$x$ : $-2,\ -1,\ 0,\ 1$

$g(x)$ : $4,\ 2,\ 0,\ -2$

Which function has the larger x-intercept?

Function $f$ , because its x-intercept is $-2$ , which is larger than $0$ .
Function $f$ , because its x-intercept is $2$ , which is larger than $0$ .
Function $g$ , because its x-intercept is $0$ , which is larger than $-2$ . (correct answer)
They have the same x-intercept.

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare x-intercepts, solve 3x + 6 = 0 for f to get x = -2; for g's table, note g(0) = 0, indicating x-intercept at 0 (and table suggests it's linear, crossing once), so 0 > -2. Choice B correctly identifies that function g has the larger x-intercept because 0 > -2. You might have thought f's intercept was 2 from mis-solving, but always set y=0 and solve for x carefully—you've got this! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.

Question 5

Function $f$ is described verbally as: "A linear function with slope $-4$ and y-intercept $2$ ."

Function $g$ is given algebraically by $g(x)=x+2$ .

Which function has the greater rate of change (slope)?

They have the same slope because both have y-intercept $2$ .
Function $f$ , because $-4>1$ .
Function $f$ , because it decreases and decreasing means a greater slope.
Function $g$ , because $1>-4$ . (correct answer)

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f described verbally with slope -4, that's the rate of change; for function g given by g(x)=x+2, the slope is 1 from the coefficient of x, so we compare -4 and 1 to see which is greater. Choice B correctly identifies that function g has the greater rate of change because 1 > -4. If you chose choice A, that's understandable—negative slopes can be tricky, but greater means larger algebraically, so positive beats negative. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Quick y-intercept trick: in a formula, set x = 0 and calculate. On a graph, see where it crosses the y-axis. In a table, find the y-value when x = 0. Three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y = mx + b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.

Question 6

Function $A$ is given by $A(t)=20(1.05)^t$ .

Function $B$ is described verbally as: "An exponential function with initial value 20 that multiplies by 1.08 each time $t$ increases by 1."

Which function has the faster growth rate?

Not enough information to compare growth rates.
They grow at the same rate.
Function $A$ grows faster.
Function $B$ grows faster. (correct answer)

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. Function A has the form A(t) = 20(1.05)^t, where the growth factor is 1.05. Function B multiplies by 1.08 each time t increases by 1, so its growth factor is 1.08. Since 1.08 > 1.05, function B grows faster—it multiplies by a larger number each time period! Choice B correctly identifies that function B grows faster because its growth factor (1.08) is larger than A's growth factor (1.05). If you chose A, remember that in exponential functions f(t) = a(b)^t, the base b determines the growth rate—larger b means faster growth! Know what each representation shows best: formulas show the growth factor as the base of the exponential, while verbal descriptions often state the multiplier directly. For exponential growth, compare the bases (growth factors) to determine which grows faster!

Question 7

Function $p$ is described as follows: "The function starts at a y-value of 5 when $x = 0$ , decreases linearly to a minimum of -3 when $x = 4$ , then increases linearly back to 5 when $x = 8$ ." Function $q(x) = |x - 3| + 1$ . Which comparison of their ranges is correct?

Both functions have the same minimum value, but $p$ has a smaller maximum value than $q$
Function $p$ has a smaller minimum value and the same maximum value as $q$
Function $p$ has a smaller minimum value, but $q$ has no maximum value (correct answer)
Function $q$ has a smaller minimum value and no maximum value compared to $p$

Explanation: Function p has a minimum of -3 and maximum of 5, so its range is [-3, 5]. Function q(x) = |x - 3| + 1 has a minimum value of 1 (when x = 3) and no maximum since absolute value functions extend to positive infinity. Therefore, p has a smaller minimum (-3 < 1), but q has no maximum value.

Question 8

Plan A costs $15 plus $4 per hour. This can be modeled by $A(h)=4h+15$ .

Plan B is shown in the table.

Which plan is cheaper for $h=5$ hours?

Table for $B(h)$ :

$B(1)=18$
$B(3)=24$
$B(5)=30$

Plan A, because $A(5)=35$ and $35<30$ .
Plan B, because $B(5)=30$ and $30<35$ . (correct answer)
They cost the same for 5 hours.
Plan A, because its hourly rate is higher.

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For plan A given by A(h)=4h+15, at h=5, A(5)=35; for plan B in the table, B(5)=30, so compare 35 and 30 to see which is cheaper. Choice B correctly identifies that plan B is cheaper for 5 hours because 30 < 35. If you selected choice A, that's okay—just plug in h=5 carefully into the formula. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features. Use each representation's strengths!

Question 9

Function $A$ is given by $A(t)=50(1.10)^t$ .

Function $B$ is described verbally as: "an exponential function with initial value $50$ and growth factor $1.05$ per unit of $t$ ."

Which function has the faster growth rate?

Function $B$ , because $1.05>1.10$ .
Function $A$ , because $1.10>1.05$ . (correct answer)
They grow at the same rate, because they have the same initial value $50$ .
Function $B$ , because $50(1.05)^t$ will always be larger than $50(1.10)^t$ .

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare growth rates, note A's formula has growth factor 1.10; B's verbal description gives 1.05, so 1.10 > 1.05 means A grows faster. Choice B correctly identifies that function A has the faster growth rate because 1.10 > 1.05. It's common to confuse which factor is larger, but remember the bigger the base >1, the faster the exponential growth—keep practicing! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features—use each representation's strengths! Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property!

Question 10

Function $m$ is given by $m(x)=-2x+5$ .

Function $n$ is shown by the table:

$x$ : $0,\ 1,\ 2$

$n(x)$ : $5,\ 4,\ 3$

What is the difference between the y-intercepts of $m$ and $n$ (that is, $m(0)-n(0)$ )?

$-5$
$0$ (correct answer)
$5$
$2$

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To find the difference in y-intercepts, calculate m(0)=-2*0+5=5 from the formula; read n(0)=5 from the table, so 5-5=0. Choice B correctly identifies that the difference is 0. It's possible to miscalculate m's intercept as -5 from the slope, but always plug in x=0 directly— you're doing awesome! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.

Question 11

Function $f$ is given by $f(x)=x^2+2x-8$ .

Function $g$ is described verbally as: "a parabola with x-intercepts at $x=-1$ and $x=5$ ."

Which statement correctly compares the number of x-intercepts of $f$ and $g$ ?

$f$ has 0 x-intercepts and $g$ has 2 x-intercepts.
$f$ has 2 x-intercepts and $g$ has 2 x-intercepts. (correct answer)
$f$ has 2 x-intercepts and $g$ has 1 x-intercept.
$f$ has 1 x-intercept and $g$ has 2 x-intercepts.

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare number of x-intercepts, factor f(x)=x^2+2x-8=(x+4)(x-2) for roots at -4 and 2 (two intercepts); g's verbal description gives intercepts at -1 and 5 (two intercepts), so they have the same number. Choice B correctly identifies that f has 2 x-intercepts and g has 2 x-intercepts. If you thought f had no intercepts, perhaps from not solving the quadratic, but using the quadratic formula or factoring reveals them—great job trying! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values—for example, to compare y-intercepts, find where x=0 in the formula, look where the graph crosses the y-axis, or find y when x=0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features—use each representation's strengths!

Question 12

Function $p$ is given algebraically by $p(x)=x^2-4x+1$ .

Function $q$ is described verbally as: "a parabola that opens up with vertex at $(1,-2)$ ."

Which function has the smaller minimum value?

Function $p$ , because its minimum value is $-3$ , which is smaller than $-2$ . (correct answer)
Function $q$ , because its minimum value is $-1$ , which is smaller than $-3$ .
Function $q$ , because its minimum value is $-2$ , which is smaller than $-1$ .
They have the same minimum value.

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare minimum values, complete the square or use vertex formula for p(x) = x^2 - 4x + 1 to find vertex at x=2, p(2)=-3; q's verbal description gives minimum at vertex y=-2, so -3 < -2. Choice A correctly identifies that function p has the smaller minimum value because -3 is smaller than -2. If you calculated p's minimum as -1, no problem—just remember the vertex x = -b/(2a) helps find it accurately. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values—for example, to compare y-intercepts, find where x=0 in the formula, look where the graph crosses the y-axis, or find y when x=0 in the table! Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property!

Question 13

A bike rental shop offers two pricing functions.

Plan $F$ : $F(h)=12+4h$ dollars, where $h$ is the number of hours.

Plan $G$ is shown in the table:

$h$ : $0,\ 1,\ 2,\ 3$

$G(h)$ : $8,\ 14,\ 20,\ 26$

Which plan is cheaper for 2 hours?

They cost the same for 2 hours. (correct answer)
Plan $F$ , because $F(2)=16$ and $G(2)=20$ .
Plan $G$ , because $G(2)=20$ and $F(2)=26$ .
Plan $F$ , because $F(2)=20$ and $G(2)=26$ .

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare costs at h=2, calculate F(2)=12+4*2=20 from the formula; read G(2)=20 from the table, so they are equal. Choice D correctly identifies that they cost the same for 2 hours. You might have misread the table and thought G(2)=26 (which is actually G(3)), but always match the exact input value—you're improving every time! When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.

Question 14

Function $f$ is described as: "An exponential function with initial value $40$ that grows by 25% each time $t$ increases by 1."

Function $g$ is given algebraically by $g(t)=40(1.1)^t$ .

Which function increases faster?

Function $g$ , because it is written as a formula and $f$ is only described.
Function $f$ , because its growth factor is $1.25$ and $1.25>1.1$ . (correct answer)
They increase at the same rate because both start at $40$ .
Function $g$ , because $1.1>1.25$ .

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. For function f described with 25% growth, that's a factor of 1.25; for function g given by g(t)=40(1.1)^t, the growth factor is 1.1, so compare 1.25 and 1.1 to see which increases faster. Choice B correctly identifies that function f increases faster because 1.25 > 1.1. If choice A seemed right, no worries—percent growth translates to the multiplier, so 25% is 1.25, bigger than 1.1. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Example: to compare y-intercepts, find where x = 0 in the formula, look where the graph crosses the y-axis, or find y when x = 0 in the table! Know what each representation shows best: formulas are great for calculating specific values and seeing patterns in the equation; graphs excel at showing maximums, minimums, and overall shape; tables are perfect for finding exact values at specific points; descriptions summarize key features. Use each representation's strengths!

Question 15

Function $A$ is described as: "An exponential function with initial value $40$ that grows by $20\%$ each time $t$ increases by 1."

Function $B$ is given by $B(t)=30(1.3)^t$ .

Which function has the higher initial value (value when $t=0$ )?

Function $A$ , because $A(0)=40$ and $B(0)=30$ . (correct answer)
Function $B$ , because $B(0)=1.3$ and $A(0)=40$ .
Function $B$ , because $B(0)=39$ and $A(0)=40$ .
They have the same initial value.

b^0

= 1), not the base 'b' = 1.3. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (read from verbal description, evaluate at t = 0 for formulas), (3) Compare the extracted values. For exponential functions, the initial value is always the coefficient in front of the exponential term!

Question 16

Function $f$ is defined by the equation $f(x) = -2x^2 + 8x - 3$ . Function $g$ is represented by the table shown. Which statement correctly compares the y-intercepts of these functions?

The y-intercept of $f$ is 3 units greater than the y-intercept of $g$
The y-intercept of $g$ is 2 units greater than the y-intercept of $f$ (correct answer)
The y-intercept of $f$ is 2 units greater than the y-intercept of $g$
The y-intercepts of $f$ and $g$ are equal in value

Explanation: The y-intercept of f is f(0) = -2(0)¬≤ + 8(0) - 3 = -3. From the table, when x = 0, g(x) = -1, so the y-intercept of g is -1. Since -1 - (-3) = 2, the y-intercept of g is 2 units greater than the y-intercept of f.

Question 17

Function $u$ can be described as: "A linear function that passes through points $(0, 8)$ and $(5, 3)$ ." The table represents function $v$ . At what x-value do these functions have the same output?

$x = 2$ , where both functions equal 6 (correct answer)
$x = 3$ , where both functions equal 5
$x = 4$ , where both functions equal 4
$x = 1$ , where both functions equal 7

Explanation: Function u has slope (3-8)/(5-0) = -1 and y-intercept 8, so u(x) = -x + 8. Checking the intersection: u(2) = -2 + 8 = 6, and from the table v(2) = 6. Therefore, both functions equal 6 when x = 2.

Question 18

Function $a$ is defined by the equation $a(x) = x^2 - 6x + 5$ . The table shows values for function $b$ . Which function has the greater minimum value, and by how much?

Function $a$ has a minimum 3 units greater than the minimum of $b$
Function $b$ has a minimum 3 units greater than the minimum of $a$
Function $a$ has a minimum 1 unit greater than the minimum of $b$
Function $b$ has a minimum 1 unit greater than the minimum of $a$ (correct answer)

Explanation: For a(x) = x¬≤ - 6x + 5, completing the square: a(x) = (x-3)¬≤ - 4, so the minimum is -4. From the table, the minimum value of b appears to be -3 (at x = 2). Since -3 - (-4) = 1, function b has a minimum that is 1 unit greater than the minimum of a.

Question 19

Function $m$ is represented in the table. Function $n(x) = 3x - 7$ . Over the interval $[1, 4]$ , which statement about their average rates of change is true?

Function $m$ has an average rate of change that is 1 unit per interval greater than $n$
Function $n$ has an average rate of change that is 1 unit per interval greater than $m$ (correct answer)
Function $m$ has an average rate of change that is 2 units per interval greater than $n$
Both functions have the same average rate of change over this interval

Explanation: For function m over [1,4]: average rate = (11-5)/(4-1) = 6/3 = 2. For function n over [1,4]: n(1) = 3(1)-7 = -4, n(4) = 3(4)-7 = 5, so average rate = (5-(-4))/(4-1) = 9/3 = 3. Since 3 - 2 = 1, function n has an average rate of change that is 1 unit per interval greater than m.

Question 20

The graph shows function $r(x)$ , which has x-intercepts at $x = -2$ and $x = 4$ . Function $s(x) = (x + 1)(x - 3)$ . Which statement correctly compares the x-intercepts of these functions?

The sum of the x-intercepts of $r$ is equal to the sum of the x-intercepts of $s$
The product of the x-intercepts of $r$ is greater than the product of the x-intercepts of $s$
The distance between the x-intercepts of $s$ is 2 units less than for $r$ (correct answer)
Function $r$ has x-intercepts that are each 1 unit farther from the origin than those of $s$

Explanation: Function r has x-intercepts at -2 and 4, so the distance between them is 4-(-2) = 6 units. Function s(x) = (x+1)(x-3) has x-intercepts at x = -1 and x = 3, so the distance between them is 3-(-1) = 4 units. Since 6-4 = 2, the distance between s's x-intercepts is 2 units less than r's.