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Algebra Help: Comparing Functions Represented In Different Ways

Review real example questions for Comparing Functions Represented In Different Ways in Algebra.

Question 1 / 10

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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 $g$ ?

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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.