Construct and Interpret Linear Functions - 8th Grade Math

1. 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 $t$ be time (seconds) and let $d$ be distance (meters). Which linear function models this situation?
   Answer: $d=3t$; speed is 3 m/s and starts at 0 m This question tests constructing a linear function $d=mt+b$ from two points and interpreting $m$ as speed and $b$ as starting distance. From points $(2,6)$ and $(8,24)$: $m=(24-6)/(8-2)=18/6=3$ m/s, then $b=6-3(2)=0$, giving $d=3t$. Choice A does not match either data point. Choice B matches the first point but fails the second: $2(8)+2=18$, 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.
2. A student saves money each week. She starts with $40 and saves $15 per week. Let $w$ be the number of weeks and let $S$ be the total savings (in dollars). Which function models this situation, and what do $m$ and $b$ mean?
   Answer: $S=15w+40$; $m=15$ dollars/week saved, $b=40$ dollars at week 0 This question tests constructing a linear function $S=mw+b$ from description of initial amount and weekly saving rate, and interpreting $m$ and $b$ in savings context. Construction: from description, rate $15$ per week as $m=15$, initial $40$ as $b=40$, giving $S=15w+40$; interpretation: $m$ is saving rate in dollars/week, $b$ is starting savings in dollars at $w=0$. 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 B shows slope $m=15$ from weekly rate, intercept $b=40$ from initial, with proper units and meanings. A common error is swapping $m$ and $b$ as in A, incorrect equation as in C or D. Construction steps: (1) identify variables ($w$=weeks, $S$=savings in dollars), (2) find slope (rate: $m=15$ dollars/week), (3) find intercept (initial: $b=40$ dollars), (4) write function ($S=15w+40$), (5) verify (at $w=0$, $S=40$; $w=1$, $S=55$). Interpretation: state what $m$ means (rate: 15 dollars per week), what $b$ means (initial: 40 dollars at week 0), include units; errors: reversing $m$ and $b$, omitting intercept, forgetting units.
3. A runner's distance from the start increases at a constant rate. The runner is $0.5$ miles from the start at $t=0$ minutes and $2.0$ miles from the start at $t=30$ minutes. Let $t$ be time (minutes) and $d$ be distance (miles). Which linear function models $d$ as a function of $t$, and what do $m$ and $b$ mean?
   Answer: $d=0.05t+0.5$; $m=0.05$ miles/minute, $b=0.5$ miles at $t=0$ 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.
4. A video game charges a one-time download fee and then a monthly subscription. The total cost $y$ (in dollars) is shown in the table, where $x$ is the number of months. Which equation fits the table, and what do $m$ and $b$ mean in this context?
   Answer: $y=5x+15$; $m=5$ dollars per month, $b=15$ dollars download fee 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 ($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.
5. A school fundraiser sells custom water bottles. The club spends $60 upfront to make a banner, and then earns $8 profit for each bottle sold. Let $x$ be the number of bottles sold and let $P$ be the total profit (in dollars). Which equation models the profit, and what do $m$ and $b$ mean?
   Answer: $P=8x-60$; $m=8$ dollars per bottle, $b=-60$ dollars profit at $x=0$ because of the upfront cost This question tests constructing a linear function $P=mx+b$ from a description with upfront cost and per-item profit, leading to negative b, and interpreting m as profit per bottle and b as initial profit (negative due to cost). From the description, rate is $8$ profit per bottle so $m=8$, initial is -$60$ due to banner cost, giving $P=8x-60$ where x=bottles, P=profit. For example, in this fundraiser, $P=8x-60$ means at 0 bottles, profit is -60 (loss), and each bottle adds $8$ profit. The correct construction treats the upfront as negative intercept, with interpretation including units and explaining negative b as loss from cost. A common error is making b positive or reversing signs. Construction steps: (1) identify variables (x=bottles, P=dollars), (2) find $m=8$ from per-bottle profit, (3) find $b=-60$ from upfront cost, (4) write $P=8x-60$, (5) verify: for $x=0$, $P=-60$. Interpretation: $m=8$ means 8 dollars profit per bottle, $b=-60$ means -60 dollar profit (loss) at 0 bottles due to cost, units for context.
6. A movie theater charges a $6 ticket fee plus $2 for each snack you buy. Let $x$ be the number of snacks and let $y$ be the total cost (in dollars). Which equation models this situation, and what do $m$ and $b$ mean in context?
   Answer: $y=2x+6$; $m=2$ dollars per snack, $b=6$ dollars ticket fee This question tests constructing a linear function $y=mx+b$ from a verbal description of a fixed fee and per-item cost, and interpreting $m$ and $b$ in the context of total movie theater expenses. Construction: from the description, extract the rate of $2$ per snack which becomes slope $m=2$ dollars per snack, and the initial $6$ ticket fee which becomes intercept $b=6$ dollars, giving $y=2x+6$; interpretation: $m$ is the rate of change in cost per snack with units dollars/snack, $b$ is the initial cost when $x=0$ snacks. 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 B shows slope $m=2$ from the per-snack rate, intercept $b=6$ from the ticket fee, and proper interpretation with units as dollars per snack and dollars for the fee. A common error is swapping $m$ and $b$ as in choice A, leading to incorrect equation and interpretations like $m=6$ dollars per snack. Construction steps: (1) identify variables ($x=$snacks, $y=$total cost in dollars), (2) find slope (rate given: $m=2$ dollars/snack), (3) find intercept (initial value: $b=6$ dollars), (4) write function ($y=2x+6$), (5) verify (for $x=0$, $y=6$; for $x=1$, $y=8$). Interpretation: state what $m$ means (rate: $2$ dollars per snack), what $b$ means (initial: $6$ dollars ticket fee), include units (critical for context understanding); errors: reversing $m$ and $b$ meanings, inverting ratio for slope, omitting units.
7. 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 $x$ be the number of GB and let $y$ be the total cost (in dollars). Which linear function models the cost, and what do $m$ and $b$ mean?
   Answer: $y=4x+34$; $m=4$ dollars per GB, $b=34$ dollars monthly fee 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=50-4*4=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=50-4*4=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.
8. The line on the coordinate plane passes through the points $(0,7)$ and $(5,17)$. Let $x$ be the input and $y$ be the output. Which equation represents the line, and what do the slope and y-intercept mean?
   Answer: $y=2x+7$; slope 2 means $y$ increases by 2 for each 1 increase in $x$, and the intercept 7 is the value of $y$ when $x=0$ 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.
9. 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 $T(h)$ where $h$ is hours after noon, and what was the temperature at noon?
   Answer: $T(h) = 2h + 64$; temperature at noon was 64°F 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.
10. A gym charges a $12 sign-up fee plus $8 per month. Let $x$ be the number of months and let $y$ be the total cost (in dollars). Which equation models this situation, and what do $m$ and $b$ mean?
    Answer: $y=8x+12$; $m=8$ dollars per month, $b=12$ dollars sign-up fee This question tests constructing the linear function y=mx+b from a verbal description of a gym membership (finding m as the monthly rate and b as the sign-up fee) and interpreting m and b in the context of total cost over months. Construction: from the verbal description, extract the rate of $8 per month which becomes slope m=8, and the initial $12 sign-up fee which becomes intercept b=12, giving y=8x+12; alternatively, if points were given, calculate m=(y₂-y₁)/(x₂-x₁) and then b=y-mx. Interpretation: m is the rate of change with units ($8 per month), b is the initial value when x=0 ($12 sign-up 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 monthly rate (m=8 dollars per month), intercept from the initial fee (b=12 dollars), and proper interpretation with units as the monthly charge and one-time fee. A common error is swapping m and b, like interpreting the sign-up as the rate or inverting the slope as months per dollar. Construction steps: (1) identify variables (x=months, y=total cost in dollars), (2) find slope (rate given: m=8 dollars per month), (3) find intercept (initial value: b=12), (4) write function (y=8x+12), (5) verify by checking for x=0, y=12, and for x=1, y=20. Interpretation: state what m means (rate: 8 dollars per month of membership), what b means (initial: 12 dollars charged at sign-up), include units (critical for context understanding); errors include reversing m and b meanings, or forgetting units making interpretation vague.
11. A student has $35 in a fundraiser account and then collects $6 each week. Let $w$ be weeks and let $A$ be the amount of money (in dollars). Which equation models the situation, and what do $m$ and $b$ represent?
    Answer: $A=6w+35$; $m=6$ dollars/week, $b=35$ dollars starting amount 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 $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.
12. A plant is 12 cm tall when it is measured at week 0. It is 30 cm tall at week 6. Let $t$ be the number of weeks since week 0, and let $h$ be the height (in cm). Which linear function models the plant's height?
    Answer: $h=3t+12$ Using the two points $(0,12)$ and $(6,30)$, the slope is $m=\frac{30-12}{6-0}=\frac{18}{6}=3$ cm per week, and since the height at $t=0$ is 12 cm, the intercept is $b=12$. This gives $h=3t+12$. 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.
13. A line passes through the points $(2,9)$ and $(6,21)$. Which equation is the line in the form $y=mx+b$, and what do $m$ and $b$ represent (in general) for this line?
    Answer: $y=3x+3$; $m=3$ is the change in $y$ per 1 unit of $x$, $b=3$ is the value of $y$ when $x=0$ This question tests constructing y=mx+b from two points on a line, and generally interpreting m and b without specific context. Construction: from points (2,9) and (6,21), calculate m=(21-9)/(6-2)=3, then b=9-3*2=3, forming y=3x+3; interpretation: m is rate of change in y per unit x, b is y-value when x=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=3 from Δy/Δx, intercept b=3 calculated properly, with general interpretations. A common error is inverting slope as in B, wrong m or b as in C or D. Construction steps: (1) identify variables (x and y abstract), (2) find slope (m=(21-9)/(6-2)=3), (3) find intercept (b=9-3*2=3), (4) write function (y=3x+3), (5) verify (at x=6, y=3*6+3=21). Interpretation: state what m means (change in y per x), what b means (y at x=0), no units here; errors: Δx/Δy for slope, wrong b calculation, swapping meanings.
14. A runner starts 2 miles from home and runs toward home at a constant rate of 0.5 mile per minute. Let $t$ be time in minutes and let $d$ be the runner's distance from home (miles). Which function models $d$, and what do $m$ and $b$ mean?
    Answer: $d=-0.5t+2$; $m=-0.5$ miles/min (distance decreases), $b=2$ miles from home at start 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.
15. Based on the table shown, what is the rate of change of the linear function, and what does it represent in this context?
    Answer: Rate of change is 15; it represents the cost per hour of renting the equipment 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.
16. A science lab cools a liquid. The temperature is $80°C$ at time 0 minutes and $68°C$ at time 4 minutes. Let $t$ be time (minutes) and let $T$ be temperature ($°C$). Which linear function models the temperature?
    Answer: $T=-3t+80$ Using the two points $(0,80)$ and $(4,68)$, the slope is $m=\frac{68-80}{4-0}=\frac{-12}{4}=-3$, in °C per minute. Since the temperature at $t=0$ is $80$, the intercept is $b=80$, giving $T=-3t+80$. Checking: $T(4)=-3(4)+80=68$, which matches. Choice A is wrong because it uses the total temperature drop, $-12$, 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 $80$ as the rate and $-3$ as the starting value.
17. A science lab heats water at a constant rate. The temperature is $18^\circ\text{C}$ at time $0$ minutes and $38^\circ\text{C}$ at time $5$ minutes. Let $t$ be time (minutes) and $T$ be temperature (in $^\circ\text{C}$). Which function models $T$ as a function of $t$, and what do $m$ and $b$ represent?
    Answer: $T=4t+18$; $m=4\ ^\circ\text{C}$/min, $b=18^\circ\text{C}$ at $t=0$ 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.
18. 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?
    Answer: Initial value is $40; slope of 20 means there is a $20 monthly fee added 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.
19. 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 $t$ be weeks and let $h$ be height (cm). Which linear function models the height, and what do $m$ and $b$ represent?
    Answer: $h=3t+9$; $m=3$ cm/week growth, $b=9$ cm starting height 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.
20. The temperature of a cup of soup decreases at a steady rate. The temperature is $160^\circ\text{F}$ at time $t=0$ minutes and $148^\circ\text{F}$ at time $t=4$ minutes. Which function models the temperature $T$ (in $^\circ\text{F}$) as a function of time $t$ (minutes), and what do $m$ and $b$ represent?
    Answer: $T=-3t+160$; $m=-3\ ^\circ\text{F}$/min, $b=160^\circ\text{F}$ starting temperature 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.

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