Constructing Linear and Exponential Functions - Algebra 1

A gym charges a $25 sign-up fee plus $10 per month. Let $m$ be the number of months and $C(m)$ be the total cost (in dollars). Write the linear function $C(m)$ .
Answer: $C(m) = 10m + 25$ 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!
The arithmetic sequence $12, 9, 6, 3, \dots$ can be represented by a linear function where $x=1$ corresponds to the first term, $x=2$ to the second term, and so on. What is the function $f(x)$ ?
Answer: $f(x)=-3x+15$ 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). An arithmetic sequence has a constant difference between terms, which becomes the slope of our linear function. The difference is 9-12 = -3, so slope m = -3. When x = 1, y = 12, so: 12 = -3(1) + b, giving us b = 15. Choice B correctly constructs f(x) = -3x + 15 by finding the common difference -3 as the slope and using point (1, 12) to find b = 15. Perfect! Choice A uses 12 as the y-intercept, but that's the value when x = 1, not x = 0 - be careful about which term corresponds to which x-value! 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!
A population of bacteria starts at 200 and grows by 10% each hour. Let $t$ be the number of hours. Which function models the population $P(t)$ ?
Answer: $P(t) = 200(1.1)^t$ 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!
If $f$ is linear, $f(0)=4$ , and $f(3)=13$ , what is $f(x)$ ? Write in the form $f(x)=mx+b$ .
Answer: $f(x)=3x+4$ 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!
A rental car company charges a flat fee plus a per-mile rate. Two customers provide the following data: Customer A drove 120 miles and paid $89, while Customer B drove 200 miles and paid $129. What is the per-mile rate the company charges?
Answer: $0.50 per mile Let the flat fee be $b$ and the per-mile rate be $m$ . We have two equations: $120m + b = 89$ and $200m + b = 129$ . Subtracting the first from the second: $80m = 40$ , so $m = 0.50$ . Choice A results from incorrectly calculating $40 ÷ 100 = 0.40$ . Choice C comes from using $129 - 89 = 40$ divided by $120 - 200 = -80$ and taking the absolute value. Choice D comes from dividing total cost by total miles for one customer: $89 ÷ 120 ≈ 0.74$ .
A bacteria culture starts with 200 bacteria and increases by 10% each hour. Write a function $P(t)$ for the population after $t$ hours.
Answer: $P(t)=200(1.10)^t$ 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· $2^x$ !
An exponential function passes through the points $(0, 7)$ and $(2, 63)$ . Find the function in the form $y=a\cdot b^x$ .
Answer: $y=7\cdot 3^x$ 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 or decay factor b (what we multiply by each time x increases by 1). If we have two points where x differs by 1, the ratio of their y-values gives us b directly: if (1, 6) and (2, 18) are on the function, then b = 18/6 = 3, and checking x = 0 or using a given point gives us a. From point (0, 7), we directly get a = 7. To find b, we use both points: 63 = 7·b². Solving: b² = 63/7 = 9, so b = 3. Choice A correctly constructs y = 7· $3^x$ by identifying a = 7 from the point (0, 7) and finding b = 3 by solving 63 = 7·b². Perfect! Choice B incorrectly uses b = 9 instead of b = 3 - remember that if x goes from 0 to 2, we apply the base twice, so b² = 9 means b = 3! 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· $2^x$ !
A bacteria culture starts with 50 bacteria. Every 3 hours, the population triples. After observing this pattern, a researcher needs to predict the population after 15 hours. Which function correctly models the bacteria population $P(t)$ after $t$ hours?
Answer: $P(t) = 50 \cdot 3^{t/3}$ Since the population triples every 3 hours, we need an exponential function where the base 3 is raised to a power that equals 1 when t = 3. The function $P(t) = 50 \cdot 3^{t/3}$ satisfies this: when t = 3, we get $P(3) = 50 \cdot 3^{3/3} = 50 \cdot 3^1 = 150$ , which is triple the initial population. Choice B has the wrong exponent (would triple every 1/3 hour). Choice C uses addition instead of multiplication for exponential growth. Choice D would triple every hour, not every 3 hours.
If $f$ is linear, $f(0)= -3$ , and $f(4)=9$ , what is $f(x)$ in the form $f(x)=mx+b$ ?
Answer: $f(x)=3x-3$ 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! We're given f(0) = -3, which directly tells us the y-intercept b = -3. Now find the slope: m = (9 - (-3))/(4 - 0) = 12/4 = 3. Choice A correctly constructs f(x) = 3x - 3 by finding slope = 3 and recognizing that f(0) = -3 gives us the y-intercept directly. Perfect! Choice B incorrectly calculates the slope as 2 instead of 3 - remember to carefully compute the change in y over the 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!
The arithmetic sequence $12, 17, 22, 27, \dots$ can be represented by a linear function where $x=1$ corresponds to the first term. What is the function $f(x)$ ? Write in the form $f(x)=mx+b$ .
Answer: $f(x)=5x+7$ 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). The sequence has common difference 5 (slope m=5); for x=1, y=12, so using y=mx+b: 12=5*1 + b, b=7, giving f(x)=5x+7. Choice B correctly constructs the linear function by using the common difference as slope 5 and solving for intercept 7, giving the function f(x)=5x+7. Perfect! If you start x at 0 instead of 1, you might get a different b, but matching the given correspondence is key—nice work! 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!
An exponential function has initial value $a = 6$ and triples each time $x$ increases by 1. Write the function in the form $y = a \cdot b^x$ .
Answer: $y = 6 \cdot 3^x$ This question tests your ability to construct exponential functions from given information like descriptions. For exponential functions, we need the initial value a (the y-value when x = 0) and the growth or decay factor b (what we multiply by each time x increases by 1). If the function triples each time x increases by 1, then b = 3. We're told the initial value a = 6. Therefore, our exponential function is y = 6 · $3^x$ . Choice B correctly constructs y = 6 · $3^x$ by identifying the initial value as 6 and the growth factor as 3. Perfect! Choice A incorrectly swaps the initial value and growth factor - remember, the coefficient out front is always the starting value when x = 0. Context tip: if the problem says 'starts at [value],' that's your initial value (exponential). If it says 'multiplies by [number]' or 'triples/doubles,' that's your base b. The language tells you exactly what you need!
A bacteria culture starts with 200 bacteria and grows by 10% each hour. Write a function $P(t)$ for the population after $t$ hours. Write in the form $P(t)=a(1+r)^t$ .
Answer: $P(t)=200(1.10)^t$ 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!
A gym charges a $25 sign-up fee plus $15 per month. Write a function $C(m)$ for the total cost (in dollars) after $m$ months.
Answer: $C(m)=15m+25$ 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). The sign-up fee is the fixed cost (y-intercept b=25), and per month is the rate (slope m=15), so C(m)=15m+25. Choice B correctly constructs the linear function by using 15 as slope and 25 as intercept, giving C(m)=15m+25. Perfect! It's easy to swap the numbers, like putting 25 as slope, but remember the 'per month' part is the changing rate, so that's m. 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! 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!
A linear function passes through the points $(2, 7)$ and $(6, 19)$ . What is the equation of the function in the form $y = mx + b$ ?
Answer: $y = 3x + 1$ 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!
If $f$ is linear and $f(0) = 7$ and $f(3) = 19$ , what is $f(x)$ in the form $f(x) = mx + b$ ?
Answer: $f(x) = 4x + 7$ 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. We have f(0) = 7 and f(3) = 19. The slope is m = (19 - 7)/(3 - 0) = 12/3 = 4. Since f(0) = 7, this directly gives us the y-intercept b = 7 (when x = 0, y = 7). Therefore, f(x) = 4x + 7. Choice B correctly constructs f(x) = 4x + 7 by finding slope = 4 and recognizing that f(0) = 7 gives the y-intercept directly. Perfect! Choice A with slope 3 would mean the function only increases by 9 over 3 units, but we need it to increase by 12. The two-point linear function recipe: (1) Find slope: m = (y₂ - y₁)/(x₂ - x₁), (2) Find y-intercept: when given f(0), that's your b value directly!, (3) Write f(x) = mx + b. Always verify by checking both points work!
A population starts at 500 and decreases by 10% each year. Let $t$ be the number of years. Write the function $P(t)$ that models the population.
Answer: $P(t)=500(0.90)^t$ 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 population starts at 500 (initial value a = 500) and decreases by 10% each year. A 10% decrease means we keep 90% = 0.90 of the population each year, so our decay factor b = 0.90. Choice B correctly constructs P(t) = 500 $(0.90)^t$ by identifying the initial population 500 and the decay factor 0.90 (keeping 90% when losing 10%). Perfect! Choice A uses 1.10 which would represent 10% growth, not decay - remember that losing 10% means multiplying by 0.90, not 1.10! 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· $2^x$ !
A population of rabbits doubles every 4 months. If there are currently 80 rabbits (at month 0), which function models the population $P(t)$ after $t$ months?
Answer: $P(t) = 80 \cdot 2^{t/4}$ When you encounter exponential growth problems, you need to identify three key components: the initial value, the growth factor, and the time period for that growth. This question tells you the population doubles (growth factor = 2) every 4 months, starting with 80 rabbits. The correct answer is D: $P(t) = 80 \cdot 2^{t/4}$ . Here's why this works: since the population doubles every 4 months, after 4 months you have $80 \cdot 2^{4/4} = 80 \cdot 2^1 = 160$ rabbits. After 8 months: $80 \cdot 2^{8/4} = 80 \cdot 2^2 = 320$ rabbits. The exponent $t/4$ ensures that every 4 months, you get the next whole number power of 2. Choice A uses $2^{4t}$ , which makes the population grow far too quickly—after just 1 month, you'd have $80 \cdot 2^4 = 1,280$ rabbits instead of the gradual growth that should occur. Choice B adds $2^{t/4}$ to 80 instead of multiplying, which represents linear growth plus a small exponential term, not true exponential growth where the entire population doubles. Choice C uses base 4 with exponent $t/2$ . While $4^{t/2}$ does equal $2^t$ algebraically, this doesn't match our doubling period. After 4 months, you'd get $80 \cdot 4^2 = 1,280$ rabbits, which is incorrect. Remember: in exponential growth problems, if something doubles every $n$ time units, your exponent should be $t/n$ so that when $t = n$ , the exponent equals 1 and you get exactly one doubling.
The graph shows a linear relationship. Using the two marked points, determine which function represents this line.
Answer: $f(x) = -\frac{1}{2}x + 4$ From the graph, the line passes through points (2, 3) and (6, 1). The slope is $m = \frac{1-3}{6-2} = \frac{-2}{4} = -\frac{1}{2}$ . Using point-slope form with (2, 3): $y - 3 = -\frac{1}{2}(x - 2)$ , which simplifies to $y = -\frac{1}{2}x + 1 + 3 = -\frac{1}{2}x + 4$ . Choice A has the wrong slope (-2 instead of -1/2). Choice C has a positive slope when the line is decreasing. Choice D also has the wrong slope and sign.
An exponential function has the values shown in the table. Write the function in the form $f(x)=a\cdot b^x$ . $x: 0, 1, 2, 3$ $f(x): 6, 12, 24, 48$
Answer: $f(x)=6\cdot 2^x$ This question tests your ability to construct exponential functions from given information like tables. When data is in a table, first check if it's linear or exponential: calculate differences between consecutive y-values (if constant, it's linear and that's your slope), or calculate ratios of consecutive y-values (if constant, it's exponential and that's your base b). This tells you which type and gives you the key parameter! Looking at ratios: 12/6=2, 24/12=2, 48/24=2. The constant ratio of 2 tells us this is exponential with base b = 2. When x = 0, f(x) = 6, so our initial value a = 6. Choice C correctly constructs f(x) = 6· $2^x$ by finding the initial value a = 6 (when x = 0) and the common ratio b = 2 from consecutive y-values. Perfect! Choice A incorrectly identifies the base as 3 instead of 2 - make sure to divide consecutive y-values, not find their difference! 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· $2^x$ !
The arithmetic sequence $-2,\ 1,\ 4,\ 7,\ldots$ can be represented by a linear function where $x$ is the term number and $f(x)$ is the value of the term. If $f(1)=-2$ , what is $f(x)$ ?
Answer: $f(x)=3x-5$ 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). The sequence has common difference 3, so slope m=3; since f(1)=-2, use point (1,-2): -2=3*1 + b, b=-5, so f(x)=3x-5. Choice B correctly constructs the linear function by finding slope 3 from differences and intercept -5, giving f(x)=3x-5. Perfect! For sequences, people might forget to adjust for x starting at 1, leading to wrong b, but use the given f(1) to solve. 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! 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!

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