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 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.

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 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.

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 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 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.

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 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.

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.