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.

This question tests constructing a linear function T=mt+b from two points and interpreting m as rate of temperature change (negative for decrease) and b as initial temperature. Construction: from (0,80) and (4,68), m=(68-80)/(4-0)=-12/4=-3 °C per minute, b=80, forming T=-3t+80. For example, in this cooling scenario, T=-3t+80 means starting at 80°C, decreasing by 3°C each minute. The correct construction captures the negative slope from data, intercept at t=0, with interpretation including units and direction like decreases 3°C/min, starts at 80°C. A common error is ignoring the negative sign or swapping points in slope calculation. Construction steps: (1) identify variables (t=minutes, T=°C), (2) calculate m=(68-80)/(4-0)=-3, (3) find b=80, (4) write T=-3t+80, (5) verify: -3*4+80=68. Interpretation: m=-3 means temperature decreases by 3°C per minute, b=80 means initial temperature of 80°C, units important.

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.

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.

This question tests interpreting the slope m and y-intercept b in the linear function y=$3x+10$ from the context of popcorn costs (understanding m as the cost per refill and b as the initial bucket cost). Construction: from verbal description, extract rate of $3 per refill as m=$3$, initial $10 as b=$10$, giving y=$3x+10$; from points, calculate m and b similarly. Interpretation: m is the rate of change with units ($3 per refill), b is the initial value when x=0 ($10 starting cost). 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 interpretation states slope as the increase of $3 per refill and intercept as $10 when no refills, with proper units. A common error is swapping m and b meanings, like saying slope is $10 per refill, or interpreting slope as refills per dollar. Construction steps: (1) identify variables (x=refills, y=total spent in dollars), (2) find slope (rate given: m=$3$ dollars/refill), (3) find intercept (initial: b=$10$), (4) write function (y=$3x+10$), (5) verify (e.g., at x=0, y=$10$). Interpretation: state what m means (rate: total increases $3 per refill), what b means (initial: $10 for the bucket at 0 refills), include units (critical for context understanding); errors include reversing meanings or forgetting units.

This question tests identifying and interpreting the slope m and y-intercept b from a verbal description of bike rental costs. Construction: from description, extract rate $3 per hour as m=3, initial $8 as b=8, though not building equation here; interpretation: m is rate of change in dollars/hour, b is initial cost in dollars at 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 identification in choice B shows m=3 from hourly increase, b=8 from starting cost, with proper explanations including context. A common error is swapping m and b as in A, inverting units as in C, or negative slope as in D. Construction steps: (1) identify variables (x=hours, y=cost in dollars), (2) find slope (rate: m=3 dollars/hour), (3) find intercept (initial: b=8 dollars), (4) imply function y=3x+8, (5) verify (at x=0, y=8; x=1, y=11). Interpretation: state what m means (rate: 3 dollars per hour), what b means (initial: 8 dollars at 0 hours), include units; errors: reversing m and b, wrong ratio, forgetting context.

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.

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=9-32=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=9-32=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.

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.

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.