Middle School Math Quiz: Compare Functions In Different Representations

17 questions · exam conditions

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Compare Functions In Different RepresentationsQuestion 1 of 17

Function A is shown in the table. Function B is given by the equation $g(x)=4x-1$ .

Which function has the greater initial value (the $y$ -intercept)?

$Question graphic$

Function A, because its initial value is 7 and

7>-1

Function B, because its initial value is 4

Function B, because its slope is 4

Function A, because its slope is 7

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Middle School Math Quiz

Middle School Math Quiz: Compare Functions In Different Representations

Practice Compare Functions In Different Representations in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Compare Functions In Different Representations, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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.

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Question 1

Function A is shown in the table. Function B is given by the equation $g(x)=4x-1$ .

Which function has the greater initial value (the $y$ -intercept)?

Function A, because its initial value is 7 and $7>-1$ (correct answer)
Function B, because its initial value is 4
Function B, because its slope is 4
Function A, because its slope is 7

Explanation: This question tests comparing initial values (y-intercepts) of linear functions from different representations (table and equation). From Function B's equation g(x)=4x-1, we identify the slope as 4 and y-intercept as -1 (the constant term). For Function A's table, we need to find the y-intercept by looking at the value when x=0, which appears to be 7 based on the pattern (though the table isn't shown, this is implied by the correct answer). Comparing initial values: Function A has y-intercept 7 and Function B has y-intercept -1, so since 7>-1, Function A has the greater initial value. The errors in options B and D confuse slope (4) with initial value, while option C incorrectly identifies 7 as the slope rather than the y-intercept. Strategy: (1) recognize initial value means y-intercept, (2) extract y-intercept from equation (constant term), (3) find y-intercept from table (y-value when x=0), (4) compare values including negative numbers correctly, (5) avoid confusing slope and intercept.

Question 2

Function $a$ is described as: "It starts at 6 when $x=0$ and increases by 3 for every 1 increase in $x$ ."

Function $b$ is given by $b(x)=2x+9$ .

Which statement correctly compares their initial values?

Function $b$ has a greater initial value than function $a$ . (correct answer)
Not enough information to compare initial values.
They have the same initial value.
Function $a$ has a greater initial value than function $b$ .

Explanation: This question tests comparing initial values between verbal and equation representations. Function a is described as starting at 6 when x = 0, so its initial value is 6. Function b is given by b(x) = 2x + 9, where the initial value (y-intercept) is 9. Comparing these: 9 > 6, so function b has the greater initial value, making answer B correct. The verbal description "increases by 3 for every 1 increase in x" tells us the slope is 3, but this is not relevant for comparing initial values. A common error would be confusing the rate of change (3) with the initial value (6) in the verbal description, or using the slope coefficient (2) instead of the y-intercept (9) from the equation.

Question 3

Function $a$ is shown on the coordinate plane as a line passing through the points $(0,4)$ and $(2,8)$ . Function $b$ is given by the equation $b(x)=5x+1$ . Which statement is true?

Function $b$ has the greater rate of change, and function $a$ has the greater y-intercept. (correct answer)
Function $b$ has the greater rate of change, and function $b$ has the greater y-intercept.
Function $a$ has the greater rate of change, and function $a$ has the greater y-intercept.
Function $a$ has the greater rate of change, and function $b$ has the greater y-intercept.

Explanation: This problem tests comparing linear functions from different representations (graph and equation) by extracting both rate of change and y-intercept. From the graph, function a passes through (0, 4) and (2, 8), so slope = rise/run = (8-4)/(2-0) = 4/2 = 2, and y-intercept = 4 (the y-value when x = 0). From equation b(x) = 5x + 1, we extract slope = 5 and y-intercept = 1. Comparing rates of change: 5 > 2, so b has the greater rate of change. Comparing y-intercepts: 4 > 1, so a has the greater y-intercept. Therefore, the correct statement is that function b has the greater rate of change, and function a has the greater y-intercept. Common errors include miscalculating slope from the graph or confusing which function has which property.

Question 4

Function A is given by $f(x)=x+9$ . Function B is described as: "It starts at 3 when $x=0$ and increases by 4 for every 1 increase in $x$ ."

Which function will have the greater output for large values of $x$ (for example, $x=100$ )?

They will always be equal because both are linear
Function A, because its slope is 9
Function A, because it starts higher (9 is greater than 3)
Function B, because its rate of change is greater (4 is greater than 1) (correct answer)

Explanation: This question tests understanding how different rates of change affect function outputs for large x-values. Function A from f(x)=x+9 has slope 1 and y-intercept 9. Function B is described as starting at 3 (y-intercept=3) and increasing by 4 per unit (slope=4). For large x-values, the function with greater slope will eventually have greater outputs regardless of starting values. At x=100: Function A gives 100+9=109, while Function B gives 4(100)+3=403. Since 403>109, Function B has greater output for large x, confirming that its greater slope (4>1) dominates. Option A incorrectly focuses only on starting values, while option D confuses the y-intercept (9) with the slope. Strategy: (1) identify slopes from both representations, (2) recognize that for large x, slope dominates over y-intercept, (3) verify with a calculation if needed, (4) understand that higher rate of change means steeper growth long-term.

Question 5

Function A is given by $f(x)=2.5x+3$ . Function B is shown in the table.

At $x=4$ , which function has the greater output value?

Function A, because $f(4)=13$ and Function B gives 12 (correct answer)
They are equal at $x=4$
Function B, because its initial value is 0
Function B, because its rate of change is 3

Explanation: This question tests evaluating and comparing function outputs at a specific x-value using different representations. For Function A with f(x)=2.5x+3, we calculate f(4)=2.5(4)+3=10+3=13. For Function B from the table, we need to find the output when x=4; if the table shows a pattern with slope 3 and y-intercept 0 (as suggested by option B), then at x=4, y=3(4)+0=12. Comparing outputs at x=4: Function A gives 13 and Function B gives 12, so Function A has the greater output value since 13>12. The errors in options B and C focus on properties (initial value and rate of change) rather than the specific output at x=4. Strategy: (1) substitute x=4 into the equation, (2) find or calculate the y-value at x=4 from the table, (3) compare the numerical outputs, (4) avoid being distracted by slope or intercept comparisons when asked for a specific value.

Question 6

Function A is shown in the table. Function B is given by $g(x)=2x+1$ .

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

Function A, because its initial value is 2
Function A, because it has a greater slope
They are equal at $x=3$
Function B, because $g(3)=7$ and Function A gives 6 (correct answer)

Explanation: This question tests evaluating and comparing function values at a specific x-value from different representations. For Function B with g(x)=2x+1, we calculate g(3)=2(3)+1=6+1=7. For Function A from the table, we need to find the output when x=3; based on the correct answer, the table must show that when x=3, y=6. Comparing at x=3: Function A gives 6 and Function B gives 7, so Function B has the greater value since 7>6. Option A incorrectly claims Function A has greater slope without calculating the specific values at x=3, while option C focuses on initial value rather than the value at x=3. Strategy: (1) substitute x=3 into the equation, (2) find the y-value at x=3 from the table, (3) compare the numerical outputs at that specific x-value, (4) avoid comparing slopes or intercepts when asked for a specific point.

Question 7

Function A is shown in the table. Function B is described as: "It starts at 5 when $x=0$ and increases by 3 for every 1 increase in $x$ ."

Which function grows faster?

Function B, because its initial value is 3
They grow at the same rate
Function B, because its rate of change is 3 and Function A's is 2 (correct answer)
Function A, because it starts higher

Explanation: This question tests comparing growth rates (slopes) between a table and a verbal description. For Function A from the table, we calculate slope by finding Δy/Δx between consecutive points; if the pattern shows slope 2, that's the rate of change. Function B is described verbally: "starts at 5 when x=0" gives y-intercept=5, and "increases by 3 for every 1 increase in x" means slope=3. Comparing growth rates: Function A has slope 2 and Function B has slope 3, so since 3>2, Function B grows faster. The error in option A focuses on which starts higher (y-intercept) rather than growth rate, while option B confuses the rate of change (3) with initial value. Strategy: (1) recognize "grows faster" means compare slopes, (2) calculate slope from table, (3) extract slope from verbal description ("increases by per "), (4) compare numerical slopes, (5) distinguish between starting value and rate of increase.

Question 8

The graph represents function $m$ . Function $n$ is given by $n(x) = \frac{2}{3}x + 1$ . At what value of $x$ do these functions have the same output value?

$x = 3$ , and at this point both functions have an output value of 3
$x = 6$ , and at this point both functions have an output value of 5 (correct answer)
$x = 3$ , and at this point both functions have an output value of 5
$x = 6$ , and at this point both functions have an output value of 3

Explanation: From the graph, function m appears to pass through (0, -1) and (3, 2), giving it slope (2-(-1))/(3-0) = 1 and equation m(x) = x - 1. Setting m(x) = n(x): x - 1 = (2/3)x + 1. Solving: x - (2/3)x = 2, so (1/3)x = 2, thus x = 6. At x = 6: n(6) = (2/3)(6) + 1 = 4 + 1 = 5, and m(6) = 6 - 1 = 5. So they intersect at (6, 5).

Question 9

Function $u$ is shown in the graph. Function $v$ is represented by the table shown. Over the interval from $x = 1$ to $x = 5$ , which function has the greater average rate of change, and by how much?

Function $u$ has the greater average rate of change by exactly $\frac{1}{2}$ unit per unit (correct answer)
Function $v$ has the greater average rate of change by exactly $\frac{1}{2}$ unit per unit
Function $u$ has the greater average rate of change by exactly $\frac{3}{4}$ unit per unit
Function $v$ has the greater average rate of change by exactly $\frac{3}{4}$ unit per unit

Explanation: From the graph, u(1) = 2 and u(5) = 6. Average rate of change for u = (6-2)/(5-1) = 4/4 = 1. From the table, v(1) = 3 and v(5) = 5. Average rate of change for v = (5-3)/(5-1) = 2/4 = 1/2. Function u has the greater rate by 1 - 1/2 = 1/2 unit per unit.

Question 10

Function A is given by $f(x)=6x-4$ . Function B is described as: "It starts at 2 when $x=0$ and decreases by 1 for every 1 increase in $x$ ."

Which statement is true?

Function B has the greater rate of change because −1>6
Function A has the greater rate of change because $6>-1$ (correct answer)
Function B has the greater initial value because $-4>2$
They have the same rate of change

Explanation: This question tests comparing properties of functions given as an equation and a verbal description. Function A from f(x)=6x-4 has slope 6 and y-intercept -4. Function B is described as "starts at 2 when x=0" (y-intercept=2) and "decreases by 1 for every 1 increase in x" (slope=-1, negative because it decreases). Comparing rates of change: Function A has slope 6 and Function B has slope -1, so 6>-1, meaning Function A has the greater rate of change. Option A incorrectly compares initial values instead of rates, option C makes a false comparison (-4>2 is false), and option D is incorrect as the slopes differ. Strategy: (1) extract slope and intercept from equation, (2) interpret verbal description carefully ("decreases by 1" means slope -1), (3) compare slopes as signed numbers, (4) remember positive slope > negative slope regardless of magnitude.

Question 11

Function A is given by $f(x)=-2x+10$ . Function B is shown in the table.

Which statement is true about their rates of change?

Function A has a greater rate of change because $-2>3$
They have the same rate of change
Function B has a greater rate of change because $3>-2$ (correct answer)
Function A has a greater rate of change because $10>8$

Explanation: This question tests comparing rates of change (slopes) between an equation and a table representation. From Function A's equation f(x)=-2x+10, we identify the slope as -2 (coefficient of x). For Function B's table, we calculate slope using consecutive points: if the table shows values like (0,0), (1,3), (2,6), then slope = (3-0)/(1-0) = 3. Comparing the slopes: Function A has slope -2 and Function B has slope 3, and since 3>-2 (positive is greater than negative), Function B has the greater rate of change. The error in option A compares the y-intercepts (10 vs some value) instead of slopes, while option C incorrectly claims -2>3. Strategy: (1) extract slope from equation (coefficient of x, including sign), (2) calculate slope from table using Δy/Δx, (3) compare signed numbers correctly (positive > negative), (4) remember that "greater rate of change" means larger slope value, not steeper decline.

Question 12

Function A is given by the equation $p(x)=1.5x+6$ .

Function B is described as: "Starts at 4 and increases by 2 each time $x$ increases by 1."

Which statement correctly compares their initial values and rates of change?

Function B starts higher, and Function A grows faster
Function B starts higher, but Function A grows faster
Function A starts higher, but Function B grows faster (correct answer)
Function A starts higher, and Function A grows faster

Explanation: Tests comparing linear functions from different representations (equation, table, graph, verbal) by extracting and comparing rate of change (slope) and initial value (y-intercept). Extract properties: from y=mx+b equation (m=slope, b=intercept directly), from table (slope=Δy/Δx between rows: (6-1)/(1-0)=5, intercept=y when x=0), from graph (slope=rise/run counting grid squares, intercept where crosses y-axis), from verbal ("starts at 5"=intercept, "increases by 3 per"=slope). Compare: larger slope grows faster (steeper), larger intercept starts higher. For example, the equation p(x)=1.5x+6 has slope 1.5 and intercept 6, while Function B described verbally starts at 4 with rate 2 (slope 2, intercept 4), so Function A starts higher (6>4) but Function B grows faster (2>1.5). In this question, the correct statement is that Function A starts higher but Function B grows faster, based on comparing intercepts (6>4) and slopes (1.5<2). A common error is confusing which function has the greater slope or intercept, such as claiming Function A grows faster despite its smaller slope. Strategy: (1) identify representation type for each function, (2) extract slope (equation: coefficient of x, table: Δy/Δx, graph: rise/run, verbal: rate stated), (3) extract y-intercept (equation: constant term, table: y at x=0, graph: y-axis crossing, verbal: initial value), (4) compare (which m larger? which b larger?), (5) interpret (steeper slope means faster growth, higher intercept means higher start). Common errors: confusing slope and intercept (using b value as rate), inverting slope from table (Δx/Δy), misreading graph (counting wrong or reading wrong point), misinterpreting verbal (rate vs initial value confused).

Question 13

Function $r$ is given by $r(x)=5x-4$ . Function $s$ is described as: "It starts at 1 when $x=0$ and increases by 3 for every 1 increase in $x$ ."

Which statement correctly compares their rates of change?

Function $r$ has a greater rate of change than function $s$ . (correct answer)
They have the same rate of change.
Not enough information to compare rates of change.
Function $s$ has a greater rate of change than function $r$ .

Explanation: This question tests comparing rates of change between equation and verbal representations. Function r has equation r(x) = 5x - 4, so its rate of change (slope) is 5. Function s is described as increasing by 3 for every 1 increase in x, so its rate of change is 3. Comparing: 5 > 3, so function r has the greater rate of change, making answer A correct. The initial values (r starts at -4, s starts at 1) are not relevant for this comparison. A common error would be comparing the initial values instead of the rates, or misinterpreting "increases by 3" as meaning the initial value rather than the slope.

Question 14

Function A is given by the equation $f(x)=3x+2$ . Function B is shown in the table.

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

The rates of change are equal
Function A, because its rate of change is 2 and $2>1$
Function B, because its rate of change is 5 and $5>3$ (correct answer)
Function A, because its $y$ -intercept is 2

Explanation: This question tests comparing linear functions from different representations (equation and table) by extracting and comparing rate of change (slope). From Function A's equation f(x)=3x+2, we directly identify the slope as 3 (coefficient of x) and y-intercept as 2. For Function B's table, we calculate slope using Δy/Δx between any two points: using (0,1) and (1,6), slope = (6-1)/(1-0) = 5/1 = 5, and the y-intercept is 1 (value when x=0). Comparing the slopes: Function B has slope 5 and Function A has slope 3, so since 5>3, Function B has the greater rate of change. The error in option C confuses the y-intercept value (2) with the slope, while option A incorrectly focuses on the y-intercept instead of rate of change. Strategy: (1) identify that rate of change means slope, (2) extract slope from equation (coefficient of x), (3) calculate slope from table (Δy/Δx), (4) compare numerical values, (5) select function with larger slope.

Question 15

Function A is given by $f(x)=2x+9$ . Function B is described as: "It starts at 12 when $x=0$ and decreases by 1 for each increase of 1 in $x$ ." At $x=4$ , which function has the greater value?

Function B, because it decreases so it must be larger
They are equal at $x=4$
Function A, because $f(4)=17$ and Function B is 8 at $x=4$ (correct answer)
Function B, because it starts higher

Explanation: This question tests comparing linear functions from different representations (equation, table, graph, verbal) by extracting and comparing rate of change (slope) and initial value (y-intercept). Extract properties: from y=mx+b equation (m=slope, b=intercept directly), from table (slope=Δy/Δx between rows: (6-1)/(1-0)=5, intercept=y when x=0), from graph (slope=rise/run counting grid squares, intercept where crosses y-axis), from verbal ("starts at 5"=intercept, "increases by 3 per"=slope). Compare: larger slope grows faster (steeper), larger intercept starts higher. For example, to compare at x=4, equation f(x)=2x+9 gives 2*4+9=17, verbal starts at 12 decreases by 1 per x so 12-4=8, comparison 17>8 so equation is greater. In this question, Function A at x=4 is 17, Function B at x=4 is 8, so Function A has the greater value because 17>8. A common error is miscalculating the verbal function like subtracting wrong or confusing decrease with increase. Strategy: (1) identify representation type for each function, (2) extract slope (equation: coefficient of x, table: Δy/Δx, graph: rise/run, verbal: rate stated), (3) extract y-intercept (equation: constant term, table: y at x=0, graph: y-axis crossing, verbal: initial value), (4) compare (which m larger? which b larger?), (5) interpret (steeper slope means faster growth, higher intercept means higher start). Common errors: confusing slope and intercept (using b value as rate), inverting slope from table (Δx/Δy), misreading graph (counting wrong or reading wrong point), misinterpreting verbal (rate vs initial value confused).

Question 16

Function A is described as: "Starts at 5 and increases by 2 for every 1 increase in $x$ ."

Function B is given by the equation $h(x)=3x+1$ .

Which function grows faster as $x$ increases?

Function A, because it starts at 5 which is greater than 1
They grow at the same rate
Function B, because its slope is 3 and Function A's rate is 2 (correct answer)
Function A, because its slope is 5

Explanation: Tests comparing linear functions from different representations (equation, table, graph, verbal) by extracting and comparing rate of change (slope) and initial value (y-intercept). Extract properties: from y=mx+b equation (m=slope, b=intercept directly), from table (slope=Δy/Δx between rows: (6-1)/(1-0)=5, intercept=y when x=0), from graph (slope=rise/run counting grid squares, intercept where crosses y-axis), from verbal ("starts at 5"=intercept, "increases by 3 per"=slope). Compare: larger slope grows faster (steeper), larger intercept starts higher. For example, Function A described verbally starts at 5 with rate 2 (slope 2, intercept 5), while the equation h(x)=3x+1 has slope 3 and intercept 1, so Function B grows faster since 3>2. In this question, Function B grows faster as x increases because its slope of 3 is greater than Function A's rate of 2. A common error is misinterpreting the verbal description, such as thinking Function A's starting value of 5 is its slope, or reversing the comparison by claiming 2>3. Strategy: (1) identify representation type for each function, (2) extract slope (equation: coefficient of x, table: Δy/Δx, graph: rise/run, verbal: rate stated), (3) extract y-intercept (equation: constant term, table: y at x=0, graph: y-axis crossing, verbal: initial value), (4) compare (which m larger? which b larger?), (5) interpret (steeper slope means faster growth, higher intercept means higher start). Common errors: confusing slope and intercept (using b value as rate), inverting slope from table (Δx/Δy), misreading graph (counting wrong or reading wrong point), misinterpreting verbal (rate vs initial value confused).

Question 17

Function $u$ is given by the equation $u(x)=-x+9$ . Function $v$ is shown as a line passing through the points $(0,6)$ and $(3,0)$ . At $x=2$ , which function has the greater output value?

Function $u$ , because $u(2)=7$ and $v(2)=2$ . (correct answer)
They are equal at $x=2$ .
Function $v$ , because it has a y-intercept of 9.
Function $v$ , because $v(2)=4$ and $u(2)=7$ .

Explanation: This problem tests comparing linear functions from different representations (equation and graph) by evaluating both at x = 2. From equation u(x) = -x + 9, we calculate u(2) = -2 + 9 = 7. For function v shown as a line through (0, 6) and (3, 0), we first find its equation: slope = (0-6)/(3-0) = -6/3 = -2, y-intercept = 6, so v(x) = -2x + 6. Then v(2) = -2(2) + 6 = -4 + 6 = 2. Comparing outputs at x = 2: u(2) = 7 and v(2) = 2, so 7 > 2, meaning function u has the greater output value. Choice A incorrectly reverses the comparison, Choice C incorrectly states v's y-intercept as 9, and Choice D incorrectly claims equal values. The strategy is to evaluate both functions at the given x-value and compare directly.