Middle School Math Quiz: Construct And Interpret Linear Functions
20 questions · exam conditions
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Construct And Interpret Linear FunctionsQuestion 1 of 20

A water tank initially contains 150 gallons of water. Water is being drained at a constant rate of 8 gallons per minute. Which function best represents the amount of water W(t)W(t) in the tank after tt minutes, and what does the slope represent in this context?

W(t)=150+8tW(t) = 150 + 8t; the slope represents the rate at which water is added to the tank
W(t)=150−8tW(t) = 150 - 8t; the slope represents the rate at which water is drained from the tank
W(t)=8t−150W(t) = 8t - 150; the slope represents the initial amount of water in the tank
W(t)=8−150tW(t) = 8 - 150t; the slope represents the time it takes to drain all the water
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Middle School Math Quiz

Middle School Math Quiz: Construct And Interpret Linear Functions

Practice Construct And Interpret Linear Functions 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 Construct And Interpret Linear Functions, 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.

All questions

Question 1

A water tank initially contains 150 gallons of water. Water is being drained at a constant rate of 8 gallons per minute. Which function best represents the amount of water W(t)W(t) in the tank after tt minutes, and what does the slope represent in this context?

  1. W(t)=150+8tW(t) = 150 + 8t; the slope represents the rate at which water is added to the tank
  2. W(t)=150−8tW(t) = 150 - 8t; the slope represents the rate at which water is drained from the tank (correct answer)
  3. W(t)=8t−150W(t) = 8t - 150; the slope represents the initial amount of water in the tank
  4. W(t)=8−150tW(t) = 8 - 150t; the slope represents the time it takes to drain all the water
Explanation: The correct answer is B. Since water is being drained at 8 gallons per minute, the amount decreases over time, so we subtract 8t from the initial 150 gallons. The slope -8 represents the rate of change (drainage rate). Choice A incorrectly adds water instead of subtracting. Choice C has the wrong initial value and would give negative amounts initially. Choice D incorrectly places the time coefficient with the initial value.

Question 2

A science lab cools a liquid. The temperature is 80°C80°C at time 0 minutes and 68°C68°C at time 4 minutes. Let tt be time (minutes) and let TT be temperature (°C°C). Which linear function models the temperature?

  1. T=−12t+4T=-12t+4
  2. T=3t+80T=3t+80
  3. T=−3t+80T=-3t+80 (correct answer)
  4. T=80t−3T=80t-3
Explanation: Using the two points (0,80)(0,80) and (4,68)(4,68), the slope is m=68−804−0=−124=−3m=\frac{68-80}{4-0}=\frac{-12}{4}=-3, in °C per minute. Since the temperature at t=0t=0 is 8080, the intercept is b=80b=80, giving T=−3t+80T=-3t+80. Checking: T(4)=−3(4)+80=68T(4)=-3(4)+80=68, which matches. Choice A is wrong because it uses the total temperature drop, −12-12, as the slope instead of the rate per minute. Choice B is wrong because a positive slope would mean the temperature is increasing, which contradicts the data. Choice D is wrong because it swaps the slope and intercept, using 8080 as the rate and −3-3 as the starting value.

Question 3

A phone plan charges a monthly fee plus a cost per gigabyte (GB) of data. The total cost is $50 when 4 GB are used, and $74 when 10 GB are used. Let xx be the number of GB and let yy be the total cost (in dollars).

Which linear function models the cost, and what do mm and bb mean?

  1. y=4x+50y=4x+50; m=4m=4 dollars per GB, b=50b=50 dollars monthly fee
  2. y=6x+26y=6x+26; m=6m=6 dollars per GB, b=26b=26 dollars monthly fee
  3. y=24x−46y=24x-46; m=24m=24 dollars per GB, b=−46b=-46 dollars monthly fee
  4. y=4x+34y=4x+34; m=4m=4 dollars per GB, b=34b=34 dollars monthly fee (correct answer)
Explanation: This question tests constructing a linear function y=mx+b from two cost points and interpreting m as cost per GB and b as monthly fee for the phone plan. Construction: from points (4,50) and (10,74), calculate m=(74-50)/(10-4)=24/6=4 dollars per GB, then b=50-44=50-16=34, forming y=4x+34. For example, in this plan, y=4x+34 means at 0 GB, cost is $34 (monthly fee), and each GB adds $4. The correct construction uses slope from Δy/Δx, intercept from b=y-mx, with interpretation including units like m=4 $/GB and b=34 $ monthly fee. A common error is wrong slope calculation or using the wrong point for b. Construction steps: (1) identify variables (x=GB, y=cost in dollars), (2) find m=(74-50)/(10-4)=4, (3) find b=50-44=34, (4) write y=4x+34, (5) verify with second point: 4*10+34=74. Interpretation: m=4 means 4 dollars per GB used, b=34 means 34 dollar base fee, units for clarity.

Question 4

The line on the coordinate plane passes through the points (0,7)(0,7) and (5,17)(5,17). Let xx be the input and yy be the output.

Which equation represents the line, and what do the slope and y-intercept mean?

  1. y=7x+2y=7x+2; slope 7 means yy increases by 7 for each 1 increase in xx, and the intercept 2 is the value of yy when x=0x=0
  2. y=2x+17y=2x+17; slope 2 means yy increases by 2 for each 1 increase in xx, and the intercept 17 is the value of yy when x=0x=0
  3. y=2x+7y=2x+7; slope 2 means yy increases by 2 for each 1 increase in xx, and the intercept 7 is the value of yy when x=0x=0 (correct answer)
  4. y=−2x+7y=-2x+7; slope -2 means yy decreases by 2 for each 1 increase in xx, and the intercept 7 is the value of yy when x=0x=0
Explanation: This question tests constructing a linear function y=mx+b from two points on a coordinate plane and interpreting m as rate of change and b as initial y-value. Construction: from points (0,7) and (5,17), calculate m=(17-7)/(5-0)=10/5=2, then b=7 (at x=0), forming y=2x+7. For example, the line y=2x+7 means when x=0, y=7, and y increases by 2 for each unit increase in x. The correct construction shows slope from Δy/Δx, intercept directly from (0,b), and general interpretation without specific units. A common error is inverting the slope or miscalculating b. Construction steps: (1) identify variables (x=input, y=output), (2) calculate m=(17-7)/(5-0)=2, (3) find b=7 from (0,7), (4) write y=2x+7, (5) verify with (5,17): 2*5+7=17. Interpretation: m=2 means y increases by 2 per unit x, b=7 means y=7 when x=0, avoiding sign errors like negative slope.

Question 5

A science lab heats water at a constant rate. The temperature is 18∘C18^\circ\text{C} at time 00 minutes and 38∘C38^\circ\text{C} at time 55 minutes. Let tt be time (minutes) and TT be temperature (in ∘C^\circ\text{C}).

Which function models TT as a function of tt, and what do mm and bb represent?

  1. T=5t+18T=5t+18; m=5 ∘C$/min,m=5\ ^\circ\text{C}$/min, b=18^\circ\text{C}atatt=0$
  2. T=14t+18T=\frac{1}{4}t+18; m=14 ∘C$/min,m=\frac{1}{4}\ ^\circ\text{C}$/min, b=18^\circ\text{C}atatt=0$
  3. T=4t+38T=4t+38; m=4 ∘C$/min,m=4\ ^\circ\text{C}$/min, b=38^\circ\text{C}atatt=0$
  4. T=4t+18T=4t+18; m=4 ∘C$/min,m=4\ ^\circ\text{C}$/min, b=18^\circ\text{C}atatt=0$ (correct answer)
Explanation: This question tests constructing T=mt+b from two temperature points over time, and interpreting m and b in heating context. Construction: from points (0,18) and (5,38), m=(38-18)/(5-0)=4 °C/min, b=18 °C, forming T=4t+18; interpretation: m is heating rate in °C/min, b is initial temperature in °C at t=0. For example, in a taxi scenario giving y=2x+3, interpret m=2 as $2 per mile rate, b=3 as $3 initial fee, function gives total cost y for x miles driven. The correct construction in choice A shows slope m=4 from change over time, intercept b=18 from start, with units and meanings. A common error is wrong m as in B or D, incorrect b as in C. Construction steps: (1) identify variables (t=minutes, T=temperature in °C), (2) find slope (m=(38-18)/(5-0)=4 °C/min), (3) find intercept (b=18 using (0,18)), (4) write function (T=4t+18), (5) verify (at t=5, T=4*5+18=38). Interpretation: state what m means (rate: 4 °C per minute), what b means (initial: 18 °C at 0 minutes), include units; errors: miscalculating slope, wrong b, forgetting units.

Question 6

A movie theater charges a $6 ticket fee plus $2 for each snack you buy. Let xx be the number of snacks and let yy be the total cost (in dollars).

Which equation models this situation, and what do mm and bb mean in context?

  1. y=2x+6y=2x+6; m=2m=2 dollars per snack, b=6b=6 dollars ticket fee (correct answer)
  2. y=2x−6y=2x-6; m=2m=2 dollars per snack, b=−6b=-6 dollars ticket fee
  3. y=12x+6y=\frac{1}{2}x+6; m=12m=\frac{1}{2} snacks per dollar, b=6b=6 dollars per snack
  4. y=6x+2y=6x+2; m=6m=6 dollars per snack, b=2b=2 dollars starting fee
Explanation: This question tests constructing a linear function y=mx+by=mx+b from a verbal description of a fixed fee and per-item cost, and interpreting mm and bb in the context of total movie theater expenses. Construction: from the description, extract the rate of 22 per snack which becomes slope m=2m=2 dollars per snack, and the initial 66 ticket fee which becomes intercept b=6b=6 dollars, giving y=2x+6y=2x+6; interpretation: mm is the rate of change in cost per snack with units dollars/snack, bb is the initial cost when x=0x=0 snacks. For example, in a taxi scenario giving y=2x+3y=2x+3, interpret m=2m=2 as 22 per mile rate, b=3b=3 as 33 initial fee, function gives total cost yy for xx miles driven. The correct construction in choice B shows slope m=2m=2 from the per-snack rate, intercept b=6b=6 from the ticket fee, and proper interpretation with units as dollars per snack and dollars for the fee. A common error is swapping mm and bb as in choice A, leading to incorrect equation and interpretations like m=6m=6 dollars per snack. Construction steps: (1) identify variables (x=x=snacks, y=y=total cost in dollars), (2) find slope (rate given: m=2m=2 dollars/snack), (3) find intercept (initial value: b=6b=6 dollars), (4) write function (y=2x+6y=2x+6), (5) verify (for x=0x=0, y=6y=6; for x=1x=1, y=8y=8). Interpretation: state what mm means (rate: 22 dollars per snack), what bb means (initial: 66 dollars ticket fee), include units (critical for context understanding); errors: reversing mm and bb meanings, inverting ratio for slope, omitting units.

Question 7

A swimmer's distance from the starting wall increases at a constant rate. At 2 seconds, the swimmer is 6 meters from the wall. At 8 seconds, the swimmer is 24 meters from the wall. Let tt be time (seconds) and let dd be distance (meters). Which linear function models this situation?

  1. d=18t+35d=18t+35; speed is 18 m/s and starts at 35 m
  2. d=2t+2d=2t+2; speed is 2 m/s and starts at 2 m
  3. d=3td=3t; speed is 3 m/s and starts at 0 m (correct answer)
  4. d=3t+6d=3t+6; speed is 3 m/s and starts at 6 m
Explanation: This question tests constructing a linear function d=mt+bd=mt+b from two points and interpreting mm as speed and bb as starting distance. From points (2,6)(2,6) and (8,24)(8,24): m=(24−6)/(8−2)=18/6=3m=(24-6)/(8-2)=18/6=3 m/s, then b=6−3(2)=0b=6-3(2)=0, giving d=3td=3t. Choice A does not match either data point. Choice B matches the first point but fails the second: 2(8)+2=182(8)+2=18, not 24. Choice D is incorrect because it assumes the swimmer's distance at 2 seconds (6 m) was also the starting distance at 0 seconds, without adjusting for the elapsed time; solving properly gives a starting distance of 0 m, not 6 m.

Question 8

A plant is 9 cm tall when it is measured at week 0 and 21 cm tall at week 4. Assume the plant grows at a constant rate. Let tt be weeks and let hh be height (cm). Which linear function models the height, and what do mm and bb represent?

  1. h=4t+9h=4t+9; m=4m=4 cm/week growth, b=9b=9 cm starting height
  2. h=3t+9h=3t+9; m=3m=3 cm/week growth, b=9b=9 cm starting height (correct answer)
  3. h=3t+21h=3t+21; m=3m=3 cm/week growth, b=21b=21 cm starting height
  4. h=13t+9h=\frac{1}{3}t+9; m=13m=\frac{1}{3} cm/week growth, b=9b=9 cm starting height
Explanation: This question tests constructing the linear function h=mt+b from data points of plant height over weeks (finding m from the growth rate between points and b as the starting height) and interpreting m and b in the context of height growth. Construction: from two points (t=0, h=9) and (t=4, h=21), calculate m=(21-9)/(4-0)=3, then b=9 since at t=0; forming h=3t+9; from a table, one would find slope Δh/Δt and intercept at t=0. Interpretation: m is the rate of change with units (3 cm/week), b is the initial value when t=0 (9 cm starting height). For example, in a taxi scenario giving y=2x+3, interpret m=2 as $2 per mile rate, b=3 as $3 initial fee, function gives total cost y for x miles driven. The correct construction shows slope from the rate between points (m=3 cm per week), intercept from the initial point (b=9 cm), and proper interpretation with units as growth rate and starting height. A common error is inverting the slope calculation as Δt/Δh, leading to m=1/3, or miscalculating b by not using t=0 properly. Construction steps: (1) identify variables (t=weeks, h=height in cm), (2) find slope (calculate from points: m=(21-9)/(4-0)=3 cm/week), (3) find intercept (using point at t=0: b=9, or calculate b=h-mt), (4) write function (h=3t+9), (5) verify (check at t=4: 3*4+9=21). Interpretation: state what m means (rate: 3 cm per week of growth), what b means (initial: 9 cm height at week 0), include units (critical for context understanding); errors include swapping rate and initial meanings, or omitting units.

Question 9

A runner starts 2 miles from home and runs toward home at a constant rate of 0.5 mile per minute. Let tt be time in minutes and let dd be the runner's distance from home (miles). Which function models dd, and what do mm and bb mean?

  1. d=0.5t+2d=0.5t+2; m=0.5m=0.5 miles/min, b=2b=2 miles from home at start
  2. d=−2t+0.5d=-2t+0.5; m=−2m=-2 miles/min, b=0.5b=0.5 miles from home at start
  3. d=2t−0.5d=2t-0.5; m=2m=2 miles/min, b=−0.5b=-0.5 miles from home at start
  4. d=−0.5t+2d=-0.5t+2; m=−0.5m=-0.5 miles/min (distance decreases), b=2b=2 miles from home at start (correct answer)
Explanation: This question tests constructing the linear function d=mt+b from a description of a runner's distance decreasing over time (finding m as the negative rate and b as the starting distance) and interpreting m and b in the context of distance from home. Construction: from the verbal description, extract the rate of -0.5 miles per minute (decreasing) which becomes slope m=-0.5, and initial 2 miles which becomes intercept b=2, giving d=-0.5t+2; from points, calculate m=(d₂-d₁)/(t₂-t₁). Interpretation: m is the rate of change with units (-0.5 miles/min, decreasing), b is the initial value when t=0 (2 miles from home). For example, in a taxi scenario giving y=2x+3, interpret m=2 as $2 per mile rate, b=3 as $3 initial fee, function gives total cost y for x miles driven. The correct construction shows slope from the decreasing rate (m=-0.5 miles per minute), intercept from the initial distance (b=2 miles), and proper interpretation with units as speed toward home and starting position. A common error is using a positive slope instead of negative, or inverting the ratio to get m=-2. Construction steps: (1) identify variables (t=minutes, d=distance in miles), (2) find slope (rate given: m=-0.5 miles/min since decreasing), (3) find intercept (initial value: b=2), (4) write function (d=-0.5t+2), (5) verify (e.g., at t=0, d=2). Interpretation: state what m means (rate: -0.5 miles per minute, distance decreases), what b means (initial: 2 miles from home at start), include units (critical for context understanding); errors include forgetting the negative sign or reversing m and b meanings.

Question 10

A temperature sensor records data showing a linear relationship. At 2 PM, the temperature was 68°F. At 6 PM, the temperature was 76°F. Assuming this trend continues, what function models temperature T(h)T(h) where hh is hours after noon, and what was the temperature at noon?

  1. T(h)=2h+64T(h) = 2h + 64; temperature at noon was 64°F (correct answer)
  2. T(h)=4h+60T(h) = 4h + 60; temperature at noon was 60°F
  3. T(h)=2h+68T(h) = 2h + 68; temperature at noon was 68°F
  4. T(h)=h+66T(h) = h + 66; temperature at noon was 66°F
Explanation: The correct answer is A. At 2 PM (h=2), T=68°F and at 6 PM (h=6), T=76°F. The rate of change is (76-68)/(6-2) = 8/4 = 2°F per hour. Using point (2,68): 68 = 2(2) + b, so b = 64. The function is T(h) = 2h + 64. At noon (h=0), T(0) = 64°F. Choice B has incorrect slope and y-intercept. Choice C incorrectly uses the 2 PM temperature as the y-intercept. Choice D has incorrect slope.

Question 11

Examine the coordinate plane showing the cost of a gym membership over time. What is the initial value of this linear function, and what does the slope tell us about the membership structure?

  1. Initial value is $60; slope of 20 means there is a $20 monthly fee added
  2. Initial value is $40; slope of 15 means there is a $15 monthly fee added
  3. Initial value is $40; slope of 20 means there is a $20 enrollment fee at signup
  4. Initial value is $40; slope of 20 means there is a $20 monthly fee added (correct answer)
Explanation: The correct answer is D. From the graph, the y-intercept (initial value) is $40, representing the enrollment fee. The line passes through (0,40) and (3,100), so the slope is (100-40)/(3-0) = 60/3 = 20. This slope represents the monthly fee of $20. Choice A has the wrong initial value. Choice B has incorrect slope. Choice C misinterprets what the slope represents.

Question 12

A plant is 12 cm tall when it is measured at week 0. It is 30 cm tall at week 6. Let tt be the number of weeks since week 0, and let hh be the height (in cm). Which linear function models the plant's height?

  1. h=12t+3h=12t+3
  2. h=3t+30h=3t+30
  3. h=3t+12h=3t+12 (correct answer)
  4. h=6t+12h=6t+12
Explanation: Using the two points (0,12)(0,12) and (6,30)(6,30), the slope is m=30−126−0=186=3m=\frac{30-12}{6-0}=\frac{18}{6}=3 cm per week, and since the height at t=0t=0 is 12 cm, the intercept is b=12b=12. This gives h=3t+12h=3t+12. Choice A is wrong because it swaps the slope and intercept, using 12 as the rate and 3 as the starting height. Choice B is wrong because it uses the ending height, 30, as the intercept instead of the starting height. Choice D is wrong because it uses an incorrect slope of 6 instead of the correct rate of 3 cm per week.

Question 13

A video game charges a one-time download fee and then a monthly subscription. The total cost yy (in dollars) is shown in the table, where xx is the number of months.

Which equation fits the table, and what do mm and bb mean in this context?

  1. y=15x+5y=15x+5; m=15m=15 dollars per month, b=5b=5 dollars download fee
  2. y=5x+15y=5x+15; m=5m=5 dollars per month, b=15b=15 dollars download fee (correct answer)
  3. y=10x+5y=10x+5; m=10m=10 dollars per month, b=5b=5 dollars download fee
  4. y=5x+10y=5x+10; m=5m=5 dollars per month, b=10b=10 dollars download fee
Explanation: This question tests constructing the linear function y=mx+b from a table of video game costs over months (finding m as the monthly rate and b as the download fee by calculating slope and intercept) and interpreting m and b in the context of total cost. Construction: from a table, find slope Δy/Δx as the monthly increase which becomes m, and the y-value at x=0 as b; alternatively, from points calculate m=(y₂-y₁)/(x₂-x₁) and b=y-mx. Interpretation: m is the rate of change with units (5permonth,forexample),bistheinitialvaluewhenx=0(5 per month, for example), b is the initial value when x=0 (15 download fee). For example, in a taxi scenario giving y=2x+3, interpret m=2 as $2 per mile rate, b=3 as $3 initial fee, function gives total cost y for x miles driven. The correct construction shows slope from the rate in the table (m=5 dollars per month), intercept from the initial cost (b=15 dollars), and proper interpretation with units as subscription rate and download fee. A common error is misreading the table to swap m and b values, or omitting units in interpretation like stating slope as '5' without dollars per month. Construction steps: (1) identify variables (x=months, y=total cost in dollars), (2) find slope (rate from table: Δy/Δx=5 dollars per month), (3) find intercept (value at x=0 from table or calculation: b=15), (4) write function (y=5x+15), (5) verify (check against table points). Interpretation: state what m means (rate: 5 dollars per month subscribed), what b means (initial: 15 dollars for download), include units (critical for context understanding); errors include calculating slope as Δx/Δy, or using a non-zero point for b without adjustment.

Question 14

A student has $35 in a fundraiser account and then collects $6 each week. Let ww be weeks and let AA be the amount of money (in dollars). Which equation models the situation, and what do mm and bb represent?

  1. A=6w+35A=6w+35; m=6m=6 dollars/week, b=35b=35 dollars starting amount (correct answer)
  2. A=35w+6A=35w+6; m=35m=35 dollars/week, b=6b=6 dollars starting amount
  3. w=6A+35w=6A+35; m=6m=6 weeks/dollar, b=35b=35 weeks
  4. A=6w−35A=6w-35; m=6m=6 dollars/week, b=−35b=-35 dollars starting amount
Explanation: This question tests constructing the linear function A/mw+b from a description of fundraising (finding m as the weekly collection rate and b as the starting amount) and interpreting m and b in the context of money over weeks. Construction: from the verbal description, extract the rate of $6 per week which becomes slope m=6, and initial 35whichbecomesinterceptb=35,givingA=6w+35;frompoints,calculatemandbaccordingly.Interpretation:mistherateofchangewithunits(35 which becomes intercept b=35, giving A=6w+35; from points, calculate m and b accordingly. Interpretation: m is the rate of change with units (6 per week), b is the initial value when w=0 ($35 starting amount). For example, in a taxi scenario giving y=2x+3, interpret m=2 as $2 per mile rate, b=3 as $3 initial fee, function gives total cost y for x miles driven. The correct construction shows slope from the weekly rate (m=6 dollars per week), intercept from the initial amount (b=35 dollars), and proper interpretation with units as collection rate and starting balance. A common error is swapping m and b, like setting m=35, or writing the equation with w as dependent variable. Construction steps: (1) identify variables (w=weeks, A=amount in dollars), (2) find slope (rate given: m=6 dollars/week), (3) find intercept (initial value: b=35), (4) write function (A=6w+35), (5) verify (e.g., at w=0, A=35). Interpretation: state what m means (rate: 6 dollars per week collected), what b means (initial: 35 dollars at week 0), include units (critical for context understanding); errors include calculating slope incorrectly or omitting units.

Question 15

A plant grows according to a linear pattern. The data shows that after 4 weeks, the plant is 15 cm tall, and after 9 weeks, it is 27.5 cm tall. What was the plant's height when it was first measured (at week 0), and what does the rate of change represent?

  1. Initial height was 5 cm; rate of change of 2.5 cm/week represents weekly growth (correct answer)
  2. Initial height was 7 cm; rate of change of 3 cm/week represents weekly growth
  3. Initial height was 10 cm; rate of change of 2.5 cm/week represents weekly growth
  4. Initial height was 5 cm; rate of change of 1.25 cm/week represents weekly growth
Explanation: The correct answer is A. The rate of change is (27.5-15)/(9-4) = 12.5/5 = 2.5 cm per week. Using point (4,15): 15 = 2.5(4) + b, so 15 = 10 + b, thus b = 5 cm. The initial height was 5 cm and the plant grows 2.5 cm per week. Choice B has incorrect rate and initial value. Choice C has incorrect initial height. Choice D has incorrect rate of change.

Question 16

The temperature of a cup of soup decreases at a steady rate. The temperature is 160∘F160^\circ\text{F} at time t=0t=0 minutes and 148∘F148^\circ\text{F} at time t=4t=4 minutes.

Which function models the temperature TT (in ∘F^\circ\text{F}) as a function of time tt (minutes), and what do mm and bb represent?

  1. T=−4t+148T=-4t+148; m=−4 ∘F$/min,m=-4\ ^\circ\text{F}$/min, b=148^\circ\text{F}$ starting temperature
  2. T=−3t+148T=-3t+148; m=−3 ∘F$/min,m=-3\ ^\circ\text{F}$/min, b=148^\circ\text{F}$ starting temperature
  3. T=−3t+160T=-3t+160; m=−3 ∘F$/min,m=-3\ ^\circ\text{F}$/min, b=160^\circ\text{F}$ starting temperature (correct answer)
  4. T=3t+160T=3t+160; m=3 ∘F$/min,m=3\ ^\circ\text{F}$/min, b=160^\circ\text{F}$ starting temperature
Explanation: This question tests constructing the linear function T=mt+b from temperature data points over time (finding m as the cooling rate and b as the starting temperature) and interpreting m and b in the context of temperature decrease. Construction: from points (t=0, T=160) and (t=4, T=148), calculate m=(148-160)/(4-0)=-3, then b=160, forming T=-3t+160; from table, find ΔT/Δt and intercept. Interpretation: m is the rate of change with units (-3 °F/min), b is the initial value when t=0 (160 °F starting temperature). For example, in a taxi scenario giving y=2x+3, interpret m=2 as $2 per mile rate, b=3 as $3 initial fee, function gives total cost y for x miles driven. The correct construction shows slope from the rate (m=-3 °F per minute), intercept from initial (b=160 °F), and proper interpretation with units as cooling rate and starting temperature. A common error is forgetting the negative sign for decrease, or miscalculating m as -4 by wrong delta. Construction steps: (1) identify variables (t=minutes, T=temperature in °F), (2) find slope (from points: m=(148-160)/(4-0)=-3 °F/min), (3) find intercept (b=160 at t=0), (4) write function (T=-3t+160), (5) verify (at t=4: -3*4+160=148). Interpretation: state what m means (rate: -3 °F per minute decrease), what b means (initial: 160 °F at t=0), include units (critical for context understanding); errors include positive slope or omitting units.

Question 17

Based on the table shown, what is the rate of change of the linear function, and what does it represent in this context?

  1. Rate of change is 3; it represents the cost per hour of renting the equipment
  2. Rate of change is 25; it represents the initial rental fee before any hourly charges
  3. Rate of change is 15; it represents the cost per hour of renting the equipment (correct answer)
  4. Rate of change is 40; it represents the total cost after 2 hours of rental time
Explanation: The correct answer is C. The rate of change is calculated as (40-25)/(2-1) = 15/1 = 15, representing the hourly rental rate. Choice A uses an incorrect calculation. Choice B confuses the rate of change with the y-intercept (initial fee). Choice D incorrectly identifies a data point as the rate of change.

Question 18

A student saves money each week. She starts with $40 and saves $15 per week. Let ww be the number of weeks and let SS be the total savings (in dollars).

Which function models this situation, and what do mm and bb mean?

  1. S=15+w+40S=15+w+40; m=1m=1 dollar/week saved, b=55b=55 dollars at week 0
  2. S=40w+15S=40w+15; m=40m=40 dollars/week saved, b=15b=15 dollars to start
  3. S=15w+40S=15w+40; m=15m=15 dollars/week saved, b=40b=40 dollars at week 0 (correct answer)
  4. S=4015wS=\frac{40}{15}w; m=4015m=\frac{40}{15} dollars/week saved, b=0b=0 dollars at week 0
Explanation: This question tests constructing a linear function S=mw+bS=mw+b from description of initial amount and weekly saving rate, and interpreting mm and bb in savings context. Construction: from description, rate 1515 per week as m=15m=15, initial 4040 as b=40b=40, giving S=15w+40S=15w+40; interpretation: mm is saving rate in dollars/week, bb is starting savings in dollars at w=0w=0. For example, in a taxi scenario giving y=2x+3y=2x+3, interpret m=2m=2 as 22 per mile rate, b=3b=3 as 33 initial fee, function gives total cost yy for xx miles driven. The correct construction in choice B shows slope m=15m=15 from weekly rate, intercept b=40b=40 from initial, with proper units and meanings. A common error is swapping mm and bb as in A, incorrect equation as in C or D. Construction steps: (1) identify variables (ww=weeks, SS=savings in dollars), (2) find slope (rate: m=15m=15 dollars/week), (3) find intercept (initial: b=40b=40 dollars), (4) write function (S=15w+40S=15w+40), (5) verify (at w=0w=0, S=40S=40; w=1w=1, S=55S=55). Interpretation: state what mm means (rate: 15 dollars per week), what bb means (initial: 40 dollars at week 0), include units; errors: reversing mm and bb, omitting intercept, forgetting units.

Question 19

A runner's distance from the start increases at a constant rate. The runner is 0.50.5 miles from the start at t=0t=0 minutes and 2.02.0 miles from the start at t=30t=30 minutes. Let tt be time (minutes) and dd be distance (miles).

Which linear function models dd as a function of tt, and what do mm and bb mean?

  1. d=0.5t+0.05d=0.5t+0.05; m=0.5m=0.5 miles/minute, b=0.05b=0.05 miles at t=0t=0
  2. d=301.5t+0.5d=\frac{30}{1.5}t+0.5; m=20m=20 miles/minute, b=0.5b=0.5 miles at t=0t=0
  3. d=0.05t+0.5d=0.05t+0.5; m=0.05m=0.05 miles/minute, b=0.5b=0.5 miles at t=0t=0 (correct answer)
  4. d=0.05t+2.0d=0.05t+2.0; m=0.05m=0.05 miles/minute, b=2.0b=2.0 miles at t=0t=0
Explanation: This question tests constructing a linear function d=mt+b from two points of runner's distance over time, and interpreting m and b in context. Construction: from points (0,0.5) and (30,2.0), calculate m=(2.0-0.5)/(30-0)=0.05 miles/minute, b=0.5 miles, forming d=0.05t+0.5; interpretation: m is speed in miles/minute, b is starting distance in miles at t=0. For example, in a taxi scenario giving y=2x+3, interpret m=2 as $2 per mile rate, b=3 as $3 initial fee, function gives total cost y for x miles driven. The correct construction in choice A shows slope m=0.05 from change in distance over time, intercept b=0.5 from initial, with proper units and meanings. A common error is swapping m and b as in B, wrong b as in C, or incorrect slope calculation as in D. Construction steps: (1) identify variables (t=minutes, d=distance in miles), (2) find slope (m=(2.0-0.5)/(30-0)=0.05 miles/minute), (3) find intercept (b=0.5 using (0,0.5)), (4) write function (d=0.05t+0.5), (5) verify (at t=30, d=0.05*30+0.5=2.0). Interpretation: state what m means (rate: 0.05 miles per minute), what b means (initial: 0.5 miles at 0 minutes), include units; errors: inverting ratio, wrong calculation, omitting units.

Question 20

A plant is 1212 cm tall at week 00 and 2727 cm tall at week 33. Let tt be the number of weeks and let hh be the height (in cm).

Which linear function models the plant's height, and what do mm and bb represent?

  1. h=3t+12h=3t+12; m=3m=3 cm/week growth rate, b=12b=12 cm starting height
  2. h=5t+27h=5t+27; m=5m=5 cm/week growth rate, b=27b=27 cm starting height
  3. h=5t+12h=5t+12; m=5m=5 cm/week growth rate, b=12b=12 cm starting height (correct answer)
  4. h=12t+5h=12t+5; m=12m=12 cm/week growth rate, b=5b=5 cm starting height
Explanation: This question tests constructing a linear function y=mx+b from two points representing plant height over time, and interpreting m and b in the context of growth. Construction: from points (0,12) and (3,27), calculate m=(27-12)/(3-0)=5 cm/week, then b=12 - 50=12 cm, forming h=5t+12; interpretation: m is the growth rate in cm/week, b is the starting height in cm when t=0. For example, in a taxi scenario giving y=2x+3, interpret m=2 as $2 per mile rate, b=3 as $3 initial fee, function gives total cost y for x miles driven. The correct construction in choice A shows slope m=5 from change in height over weeks, intercept b=12 from initial height, and proper interpretation with units as cm/week and cm. A common error is miscalculating slope as 3 like in B, or swapping m and b as in C, or wrong b as in D. Construction steps: (1) identify variables (t=weeks, h=height in cm), (2) find slope (calculate from points: m=(27-12)/(3-0)=5 cm/week), (3) find intercept (b=12 using point (0,12)), (4) write function (h=5t+12), (5) verify (at t=3, h=53+12=27). Interpretation: state what m means (rate: 5 cm per week), what b means (initial: 12 cm at week 0), include units; errors: calculating slope as Δx/Δy, using wrong point for b, forgetting units.