This question tests your ability to construct linear functions from given information like points. To construct a linear function from two points, we find the slope using m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept using one of the points: substitute the point and slope into y = mx + b and solve for b. Once you have m and b, you've got your function y = mx + b! Here, the points are like (0,8) and (4,20): m=(20-8)/(4-0)=12/4=3, and b=8 (given at x=0), so f(x)=3x+8. Choice B correctly constructs the linear function by finding the slope as 3 and using the given intercept 8, giving the function f(x)=3x+8. Perfect! If you misread f(4) or calculate slope wrong, you might pick another, but always use the formula step-by-step. The two-point linear function recipe: (1) Find slope: m = (y₂ - y₁)/(x₂ - x₁) using your two points, (2) Find y-intercept: pick either point, substitute x, y, and m into y = mx + b, solve for b, (3) Write it: y = [your m]x + [your b]. To check, verify both original points work in your function!

This question tests your ability to construct exponential functions from given information like descriptions. From context descriptions, listen for the clues: 'per,' 'each,' or 'constant rate' suggest linear (use that rate as slope), while 'percent growth,' 'doubles,' or 'halves' suggest exponential (convert to growth/decay factor). 'Starting with' or 'initial' tells you the y-intercept (linear) or initial value (exponential). The culture 'starts with 200' (initial a=200) and 'grows by 10% each hour' (growth factor 1+0.10=1.10), so $P(t)=200(1.10)^t$. Choice A correctly constructs the exponential function by using the starting amount as a=200 and converting the percent growth to the factor 1.10, giving the function $P(t)=200(1.10)^t$. Perfect! A common mix-up is using addition instead of multiplication for growth, but 'percent' signals exponential—keep practicing to spot it! Context tip: if the problem says 'starts at [value],' that's your y-intercept (linear) or initial value (exponential). If it says 'increases by [number] each time,' that's slope (linear). If it says 'increases by [percent] each time' or 'multiplies by [number],' that's exponential with that as your rate or factor. The language tells you exactly what you need!

This question tests your ability to construct exponential functions from given information like points. For exponential functions, we need the initial value a (the y-value when x = 0) and the growth factor b. We're given (0, 2), so a = 2. To find b, we use the other point (3, 54). Since y = a · $b^x$, we have 54 = 2 · b³. Solving: b³ = 54/2 = 27, so b = ∛27 = 3. Therefore, y = 2 · $3^x$. Choice A correctly constructs y = 2 · $3^x$ by using the y-intercept point for a = 2 and solving for b = 3 from the second point. Perfect! Choice B with b = 9 would give 2 · 9³ = 2 · 729 = 1458 when x = 3, not 54 - always verify your second point! For exponential from two points: (1) If one point has x = 0, that y-value is your a, (2) Use the other point to find b: substitute into y = a · $b^x$ and solve, (3) Write your function. Check by verifying both points work!

This question tests your ability to construct linear functions from given information like points. To construct a linear function from two points, we find the slope using m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept using one of the points: substitute the point and slope into y = mx + b and solve for b. Once you have m and b, you've got your function y = mx + b! For points (-1,4) and (3,-4): m=(-4-4)/(3-(-1))=-8/4=-2; using (-1,4): 4=-2*(-1)+b, so b=2, giving y=-2x+2. Choice B correctly constructs the linear function by finding the slope as -2 and the intercept as 2, giving the function y=-2x+2. Perfect! It's common to slip on negative signs or point selection, but checking both points in the final equation ensures it's spot on. The two-point linear function recipe: (1) Find slope: m = (y₂ - y₁)/(x₂ - x₁) using your two points, (2) Find y-intercept: pick either point, substitute x, y, and m into y = mx + b, solve for b, (3) Write it: y = [your m]x + [your b]. To check, verify both original points work in your function!

This question tests your ability to construct linear functions from given information like points. To construct a linear function from two points, we find the slope using m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept using one of the points: substitute the point and slope into y = mx + b and solve for b. Once you have m and b, you've got your function y = mx + b! Let's find the slope: m = (19 - 7)/(6 - 2) = 12/4 = 3. Now using point (2, 7): 7 = 3(2) + b, so 7 = 6 + b, which gives us b = 1. Choice B correctly constructs y = 3x + 1 by finding slope = 3 and y-intercept = 1, giving the function y = 3x + 1. Perfect! Choice A incorrectly calculates the slope as 2 instead of 3 - remember to carefully compute the rise over run! The two-point linear function recipe: (1) Find slope: m = (y₂ - y₁)/(x₂ - x₁) using your two points, (2) Find y-intercept: pick either point, substitute x, y, and m into y = mx + b, solve for b, (3) Write it: y = [your m]x + [your b]. To check, verify both original points work in your function!

This question tests your ability to construct linear functions from given information like descriptions. From context descriptions, listen for the clues: 'per,' 'each,' or 'constant rate' suggest linear (use that rate as slope), while 'percent growth,' 'doubles,' or 'halves' suggest exponential. 'Starting with' or 'initial' tells you the y-intercept (linear) or initial value (exponential). The gym charges a $25 sign-up fee (this is our y-intercept, the cost when m = 0) plus $10 per month (this is our slope, the rate of change). So C(m) = 10m + 25. Choice B correctly constructs C(m) = 10m + 25 by identifying the monthly rate as the slope (10) and the sign-up fee as the y-intercept (25). Perfect! Choice A incorrectly swaps these values - remember, the 'per month' rate is what multiplies the variable, while the one-time fee is the constant term. Context tip: if the problem says 'starts at [value],' that's your y-intercept. If it says 'increases by [number] each time,' that's slope. The language tells you exactly what you need!

This question tests your ability to construct exponential functions from given information like descriptions. From context descriptions, listen for the clues: 'percent growth' suggests exponential (convert to growth factor). 'Starting with' tells you the initial value. The bacteria start at 200 (initial value a = 200) and grow by 10% each hour. Growing by 10% means multiplying by 1.10 (100% + 10% = 110% = 1.10), so b = 1.1. The function is P(t) = $200(1.1)^t$. Let's verify: P(0) = $200(1.1)^0$ = 200 ✓; P(1) = $200(1.1)^1$ = 220, which is 200 + 10% of 200 ✓. Choice C correctly constructs P(t) = $200(1.1)^t$ by identifying initial value = 200 and growth factor = 1.1, giving the function P(t) = $200(1.1)^t$. Perfect! Choice A uses 0.9 (which would be 10% decay, not growth), B is linear, and D has no initial value coefficient. Context tip: if the problem says 'increases by [percent] each time,' that's exponential with growth factor = 1 + (percent as decimal). For 10% growth, b = 1 + 0.10 = 1.10. The language tells you exactly what you need!

This question tests your ability to construct linear functions from given information like descriptions. From context descriptions, listen for the clues: 'per,' 'each,' or 'constant rate' suggest linear (use that rate as slope), while 'percent growth,' 'doubles,' or 'halves' suggest exponential (convert to growth/decay factor). 'Starting with' or 'initial' tells you the y-intercept (linear) or initial value (exponential). Here, the $15 per month is the constant rate (slope a=15), and the $25 sign-up is the initial fixed cost (intercept b=25), so C(m)=15m+25. Choice B correctly constructs the linear function by using the monthly rate as the slope 15 and the sign-up as the intercept 25, giving the function C(m)=15m+25. Perfect! It's easy to swap the coefficients if you're not careful, but remembering 'per month' multiplies with m helps avoid that. Context tip: if the problem says 'starts at [value],' that's your y-intercept (linear) or initial value (exponential). If it says 'increases by [number] each time,' that's slope (linear). If it says 'increases by [percent] each time' or 'multiplies by [number],' that's exponential with that as your rate or factor. The language tells you exactly what you need!

This question tests your ability to construct exponential functions from given information like descriptions. From context descriptions, listen for the clues: 'per,' 'each,' or 'constant rate' suggest linear (use that rate as slope), while 'percent growth,' 'doubles,' or 'halves' suggest exponential (convert to growth/decay factor). 'Starting with' or 'initial' tells you the y-intercept (linear) or initial value (exponential). Starts with 200 (initial a=200), increases by 10% each hour means growth factor b=1+0.10=1.10, so $P(t)=200(1.10)^t$. Choice A correctly constructs the exponential by using initial 200 and factor 1.10, giving $P(t)=200(1.10)^t$. Perfect! A mistake could be using decay like 0.90, but since it's increase, add to 1 for the factor. Context tip: if the problem says 'starts at [value],' that's your y-intercept (linear) or initial value (exponential). If it says 'increases by [number] each time,' that's slope (linear). If it says 'increases by [percent] each time' or 'multiplies by [number],' that's exponential with that as your rate or factor. The language tells you exactly what you need! For exponential from a table: (1) Find the initial value by looking at x = 0 (if in table) or work backward using the pattern, (2) Find the common ratio by dividing consecutive y-values: y₂/y₁ (should be the same for all consecutive pairs), (3) Write y = $a·b^x$ with your values. Example: if y goes 5, 10, 20 as x goes 0, 1, 2, then a = 5, b = 10/5 = 2, so y = $5·2^x$!

This question tests your ability to construct linear functions from given information like points. To construct a linear function from two points, we find the slope using m = (y₂ - y₁)/(x₂ - x₁), then find the y-intercept using one of the points: substitute the point and slope into y = mx + b and solve for b. Once you have m and b, you've got your function y = mx + b! Here, points are like (0,4) and (3,13): m=(13-4)/(3-0)=9/3=3, and b=4 given at x=0, so f(x)=3x+4. Choice B correctly constructs the function by finding slope 3 and intercept 4, giving f(x)=3x+4. Perfect! Swapping numbers might give 4x+3, but remember slope is change in y over change in x. The two-point linear function recipe: (1) Find slope: m = (y₂ - y₁)/(x₂ - x₁) using your two points, (2) Find y-intercept: pick either point, substitute x, y, and m into y = mx + b, solve for b, (3) Write it: y = [your m]x + [your b]. To check, verify both original points work in your function!