Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Opening subject page...

Loading your content

Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

← Back to quizzes

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.

Question 1 / 20

0 of 20 answered

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 $g$ ?

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 described verbally as: “A parabola that opens upward with vertex at $(2,-5)$ .”

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

Which function has the lower minimum value?

Function $g$ , because its minimum value is $-3$ , which is lower than $-5$ .
Function $g$ , because it opens upward and upward-opening parabolas always have lower minimums.
Function $f$ , because its minimum value is $-5$ , which is lower than $-3$ . (correct answer)
They have the same minimum value because both have vertex x-coordinate $2$ .

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. For function f described as upward parabola with vertex (2,-5), minimum is -5; for function g given by g(x)=(x-2)^2-3, vertex at (2,-3) so minimum -3, compare -5 and -3. Choice B correctly identifies that function f has the lower minimum value because -5 < -3. If you chose choice A, remember lower means more negative, so -5 is below -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 (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 5

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

Function $g$ is given in the table.

Which function has the smaller minimum value?

Table for $g(x)$ :

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

Function $g$ , because its minimum value is $-1$ which is less than $-3$ .
Function $f$ , because its minimum value is $-3$ which is less than $1$ . (correct answer)
They have the same minimum value.
Function $g$ , because its minimum value is $1$ which is less than $-3$ .

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^2-4x+1, complete the square or use vertex formula x=-b/(2a)=2, then f(2)=-3 as minimum; for function g in the table, the lowest value is 1 at x=2, so compare -3 and 1. Choice A correctly identifies that function f has the smaller minimum value because -3 < 1. If you went with choice C, that's okay—just scan the table carefully for the smallest y-value. 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 6

Function $p$ is given by $p(x)=x^2-6x+8$ .

Function $q$ is described verbally as: “A parabola that opens upward with vertex at $(1,-3)$ .”

Which function has the smaller minimum value?

Function $p$ has the smaller minimum value.
Function $q$ has the smaller minimum value. (correct answer)
They have the same minimum value.
Not enough information to compare minimum values.

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 p(x) = x² - 6x + 8, complete the square: p(x) = (x-3)² - 9 + 8 = (x-3)² - 1, so the vertex is at (3, -1) and the minimum value is -1. Function q is described as having vertex at (1, -3), and since it opens upward, -3 is its minimum value. Since -3 < -1, function q has the smaller minimum. Choice B correctly identifies that function q has the smaller minimum value because -3 is less than -1. If you chose A, be careful with negative numbers—remember that -3 is smaller than -1 on the number line! Quick vertex trick: for ax² + bx + c, the x-coordinate of the vertex is -b/(2a), then substitute to find the y-coordinate. For verbal descriptions, the vertex coordinates are often given directly. Use each representation's strengths!

Question 7

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 8

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

Function $g$ is described verbally as: “A parabola that opens upward and has x-intercepts at $x=-1$ and $x=3$ .”

Which function has more x-intercepts?

Function $g$ has more x-intercepts.
They have the same number of x-intercepts. (correct answer)
Function $f$ has more x-intercepts.
Not enough information to determine the number of 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. For f(x) = (x+1)² - 4, expand to get f(x) = x² + 2x + 1 - 4 = x² + 2x - 3. Setting f(x) = 0: x² + 2x - 3 = 0, which factors as (x+3)(x-1) = 0, giving x-intercepts at x = -3 and x = 1 (two intercepts). Function g is described as having x-intercepts at x = -1 and x = 3 (also two intercepts). Both functions have exactly 2 x-intercepts! Choice C correctly identifies that they have the same number of x-intercepts (both have 2). If you chose differently, remember to count all x-intercepts—parabolas can have 0, 1, or 2 x-intercepts depending on how they're positioned! Know what each representation shows best: verbal descriptions often list x-intercepts directly, while for formulas you need to solve f(x) = 0. Quick tip: if a parabola is given in factored form like a(x-r)(x-s), the x-intercepts are at x = r and x = s!

Question 9

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 10

Function $h$ is described verbally as: “A linear function with slope $-2$ and y-intercept $5$ .”

Function $k$ is given by $k(x)=\tfrac{1}{2}x+1$ .

Which function has the greater slope?

Function $h$ has the greater slope (because $-2 > \tfrac{1}{2}$ ).
Function $k$ has the greater slope (because $\tfrac{1}{2} > -2$ ). (correct answer)
They have the same slope.
Not enough information to compare slopes.

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 h is described verbally as having slope -2, while function k is given by k(x) = ½x + 1, which is in slope-intercept form y = mx + b where m = ½ is the slope. Comparing slopes: -2 versus ½. Choice B correctly identifies that function k has the greater slope because ½ > -2 (positive numbers are always greater than negative numbers). If you picked A, remember that negative numbers are always less than positive numbers, regardless of their absolute values. 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, identify m in y = mx + b form), (3) Compare the extracted values. Know what each representation shows best: verbal descriptions summarize key features directly, while formulas in slope-intercept form make the slope immediately visible as the coefficient of x!

Question 11

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 12

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 13

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 14

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

Which function has the larger y-intercept?

Table for $g(x)$ :

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

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

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)=2x-3, the y-intercept is f(0)=-3; for function g in the table, when x=0, g(x)=1, so we compare -3 and 1 to see which is larger. Choice B correctly identifies that function g has the larger y-intercept because 1 > -3. Don't worry if you picked choice A—it's easy to mix up the signs, but remember that positive numbers are greater than negative ones. 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 $f$ is given by $f(x)=5-2x$ .

Function $g$ is given in the table.

At $x=3$ , which function has the greater output?

Table for $g(x)$ :

$g(1)=2$
$g(2)=1$
$g(3)=0$
$g(4)=-1$

Function $f$ , because $f(3)=-1$ and $-1>0$ .
Function $g$ , because $g(3)=0$ and $0>-1$ . (correct answer)
They are equal because both are decreasing.
Function $f$ , because $f(3)=1$ and $1>0$ .

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)=5-2x, at x=3, f(3)=-1; for function g in the table, g(3)=0, so compare -1 and 0 at that point. Choice B correctly identifies that function g has the greater output at x=3 because 0 > -1. If choice A was your pick, that's fine—recalculate f(3) to see it's 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! 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 16

Function $f$ is given by the equation $f(x)=-(x-1)^2+6$ .

Function $g$ is described as: “A parabola that opens downward with vertex at $(3,4)$ .”

Which function has the larger maximum value?

They have the same maximum value.
Function $g$ , because its maximum value is $3$ and $3>6$ .
Function $g$ , because its maximum value is $4$ and $4>6$ .
Function $f$ , because its maximum value is $6$ and $6>4$ . (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 given by f(x)=-(x-1)^2+6, it's a downward parabola with maximum at the vertex y=6; for function g described as a downward parabola with vertex at (3,4), the maximum is 4, so we compare 6 and 4. Choice A correctly identifies that function f has the larger maximum value because 6 > 4. If you selected choice C, no problem—double-check the vertex y-coordinate in the description to confirm the maximum. 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 17

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 18

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 19

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 20

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!

Opening subject page...

Loading your content

← Back to quizzes

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.

Question 1 / 20

0 of 20 answered

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 $g$ ?

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)

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 described verbally as: “A parabola that opens upward with vertex at $(2,-5)$ .”

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

Which function has the lower minimum value?

Function $g$ , because its minimum value is $-3$ , which is lower than $-5$ .
Function $g$ , because it opens upward and upward-opening parabolas always have lower minimums.
Function $f$ , because its minimum value is $-5$ , which is lower than $-3$ . (correct answer)
They have the same minimum value because both have vertex x-coordinate $2$ .

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. For function f described as upward parabola with vertex (2,-5), minimum is -5; for function g given by g(x)=(x-2)^2-3, vertex at (2,-3) so minimum -3, compare -5 and -3. Choice B correctly identifies that function f has the lower minimum value because -5 < -3. If you chose choice A, remember lower means more negative, so -5 is below -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 (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 5

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

Function $g$ is given in the table.

Which function has the smaller minimum value?

Table for $g(x)$ :

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

Function $g$ , because its minimum value is $-1$ which is less than $-3$ .
Function $f$ , because its minimum value is $-3$ which is less than $1$ . (correct answer)
They have the same minimum value.
Function $g$ , because its minimum value is $1$ which is less than $-3$ .

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^2-4x+1, complete the square or use vertex formula x=-b/(2a)=2, then f(2)=-3 as minimum; for function g in the table, the lowest value is 1 at x=2, so compare -3 and 1. Choice A correctly identifies that function f has the smaller minimum value because -3 < 1. If you went with choice C, that's okay—just scan the table carefully for the smallest y-value. 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 6

Function $p$ is given by $p(x)=x^2-6x+8$ .

Function $q$ is described verbally as: “A parabola that opens upward with vertex at $(1,-3)$ .”

Which function has the smaller minimum value?

Function $p$ has the smaller minimum value.
Function $q$ has the smaller minimum value. (correct answer)
They have the same minimum value.
Not enough information to compare minimum values.

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 p(x) = x² - 6x + 8, complete the square: p(x) = (x-3)² - 9 + 8 = (x-3)² - 1, so the vertex is at (3, -1) and the minimum value is -1. Function q is described as having vertex at (1, -3), and since it opens upward, -3 is its minimum value. Since -3 < -1, function q has the smaller minimum. Choice B correctly identifies that function q has the smaller minimum value because -3 is less than -1. If you chose A, be careful with negative numbers—remember that -3 is smaller than -1 on the number line! Quick vertex trick: for ax² + bx + c, the x-coordinate of the vertex is -b/(2a), then substitute to find the y-coordinate. For verbal descriptions, the vertex coordinates are often given directly. Use each representation's strengths!

Question 7

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 8

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

Function $g$ is described verbally as: “A parabola that opens upward and has x-intercepts at $x=-1$ and $x=3$ .”

Which function has more x-intercepts?

Function $g$ has more x-intercepts.
They have the same number of x-intercepts. (correct answer)
Function $f$ has more x-intercepts.
Not enough information to determine the number of 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. For f(x) = (x+1)² - 4, expand to get f(x) = x² + 2x + 1 - 4 = x² + 2x - 3. Setting f(x) = 0: x² + 2x - 3 = 0, which factors as (x+3)(x-1) = 0, giving x-intercepts at x = -3 and x = 1 (two intercepts). Function g is described as having x-intercepts at x = -1 and x = 3 (also two intercepts). Both functions have exactly 2 x-intercepts! Choice C correctly identifies that they have the same number of x-intercepts (both have 2). If you chose differently, remember to count all x-intercepts—parabolas can have 0, 1, or 2 x-intercepts depending on how they're positioned! Know what each representation shows best: verbal descriptions often list x-intercepts directly, while for formulas you need to solve f(x) = 0. Quick tip: if a parabola is given in factored form like a(x-r)(x-s), the x-intercepts are at x = r and x = s!

Question 9

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 10

Function $h$ is described verbally as: “A linear function with slope $-2$ and y-intercept $5$ .”

Function $k$ is given by $k(x)=\tfrac{1}{2}x+1$ .

Which function has the greater slope?

Function $h$ has the greater slope (because $-2 > \tfrac{1}{2}$ ).
Function $k$ has the greater slope (because $\tfrac{1}{2} > -2$ ). (correct answer)
They have the same slope.
Not enough information to compare slopes.

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 h is described verbally as having slope -2, while function k is given by k(x) = ½x + 1, which is in slope-intercept form y = mx + b where m = ½ is the slope. Comparing slopes: -2 versus ½. Choice B correctly identifies that function k has the greater slope because ½ > -2 (positive numbers are always greater than negative numbers). If you picked A, remember that negative numbers are always less than positive numbers, regardless of their absolute values. 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, identify m in y = mx + b form), (3) Compare the extracted values. Know what each representation shows best: verbal descriptions summarize key features directly, while formulas in slope-intercept form make the slope immediately visible as the coefficient of x!

Question 11

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 12

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 $p$

Question 13

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 14

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

Which function has the larger y-intercept?

Table for $g(x)$ :

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

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

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)=2x-3, the y-intercept is f(0)=-3; for function g in the table, when x=0, g(x)=1, so we compare -3 and 1 to see which is larger. Choice B correctly identifies that function g has the larger y-intercept because 1 > -3. Don't worry if you picked choice A—it's easy to mix up the signs, but remember that positive numbers are greater than negative ones. 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 $f$ is given by $f(x)=5-2x$ .

Function $g$ is given in the table.

At $x=3$ , which function has the greater output?

Table for $g(x)$ :

$g(1)=2$
$g(2)=1$
$g(3)=0$
$g(4)=-1$

Function $f$ , because $f(3)=-1$ and $-1>0$ .
Function $g$ , because $g(3)=0$ and $0>-1$ . (correct answer)
They are equal because both are decreasing.
Function $f$ , because $f(3)=1$ and $1>0$ .

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)=5-2x, at x=3, f(3)=-1; for function g in the table, g(3)=0, so compare -1 and 0 at that point. Choice B correctly identifies that function g has the greater output at x=3 because 0 > -1. If choice A was your pick, that's fine—recalculate f(3) to see it's 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! 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 16

Function $f$ is given by the equation $f(x)=-(x-1)^2+6$ .

Function $g$ is described as: “A parabola that opens downward with vertex at $(3,4)$ .”

Which function has the larger maximum value?

They have the same maximum value.
Function $g$ , because its maximum value is $3$ and $3>6$ .
Function $g$ , because its maximum value is $4$ and $4>6$ .
Function $f$ , because its maximum value is $6$ and $6>4$ . (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 given by f(x)=-(x-1)^2+6, it's a downward parabola with maximum at the vertex y=6; for function g described as a downward parabola with vertex at (3,4), the maximum is 4, so we compare 6 and 4. Choice A correctly identifies that function f has the larger maximum value because 6 > 4. If you selected choice C, no problem—double-check the vertex y-coordinate in the description to confirm the maximum. 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 17

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 18

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 19

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 20

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!