Algebra Quiz: Interpret Exponential Functions And Growth Rate
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Interpret Exponential Functions And Growth RateQuestion 1 of 20

A medication amount in the body is modeled by M(t)=100(0.97)tM(t)=100(0.97)^t, where tt is in hours. What is the percent decay rate per hour?

97% decay per hour
3% decay per hour
0.97% decay per hour
3% growth per hour
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Algebra Quiz

Algebra Quiz: Interpret Exponential Functions And Growth Rate

Practice Interpret Exponential Functions And Growth Rate in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpret Exponential Functions And Growth Rate, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A medication amount in the body is modeled by M(t)=100(0.97)tM(t)=100(0.97)^t, where tt is in hours. What is the percent decay rate per hour?

  1. 97% decay per hour
  2. 3% decay per hour (correct answer)
  3. 0.97% decay per hour
  4. 3% growth per hour
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.97, then r = 0.97 - 1 = -0.03 = 3% decay (we usually just say '3% decay' and understand it's a decrease). For the function M(t) = 100(0.97)^t, the base is 0.97. To find the percent rate, we calculate r = 0.97 - 1 = -0.03. Converting to percent: -0.03 × 100% = -3%. Since 0.97 is less than 1, this is decay, specifically 3% decay per hour. Choice B correctly identifies the percent rate as 3% decay per hour by showing correct reasoning. Excellent! Choice A confuses the remaining amount with the decay rate: if the medication retains 97% of its amount each hour (multiplying by 0.97), then it loses 3% of its amount, not 97%. The decay rate is what's lost, not what remains! Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form!

Question 2

A machine's value is modeled by V(t)=20000(0.80)tV(t)=20000(0.80)^t, where tt is in years. What is the annual depreciation rate (percent decay rate)?

  1. 80% decay per year
  2. 20% growth per year
  3. 0.20% decay per year
  4. 20% decay per year (correct answer)
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function y = 20000·(0.80)^t, the base is 0.80. To find the percent rate, we calculate r = 0.80 - 1 = -0.20 as decimal. Converting to percent: -0.20 × 100% = -20%. Since 0.80 is less than 1, this is decay, specifically 20% decay per year. Choice D correctly identifies the percent rate as 20% by showing correct reasoning. Excellent! Choice A forgets to subtract 1 from the base before converting to percent. To find the rate, we do r = b - 1: for b = 0.80, that's 0.80 - 1 = -0.20, which as a percent is -20%. If you skip the 'subtract 1' step, you get the wrong rate! To find the percent rate: (1) Identify the base b, (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: base is 1.03, so r = 1.03 - 1 = 0.03 = 3%. For decay like 0.97: r = 0.97 - 1 = -0.03 = -3%, which we call '3% decay.' Easy!

Question 3

The value of a laptop after tt years is modeled by V(t)=25000(0.88)tV(t)=25000(0.88)^t. Is this exponential growth, exponential decay, or neither?​

  1. Neither (constant)
  2. Exponential decay (correct answer)
  3. Exponential growth
  4. Linear decay
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.03), while the rate r is how much it's changing by percent (like 3%). They're related by b = 1 + r, so knowing one gives you the other! Looking at the function V(t) = 25000(0.88)^t, we check the base: 0.88 is less than 1, which means this is exponential decay. Think of it this way: each time t increases by 1, V is multiplied by 0.88, so V is getting smaller—that's decay! Choice C correctly identifies this as exponential decay because the base 0.88 < 1. Excellent! Choice A confuses growth with decay: since the base 0.88 is less than 1, this is decay, not growth. An easy way to remember: bases bigger than 1 mean growing, bases between 0 and 1 mean shrinking! Here's your growth/decay decision tree: (1) Look at the base b, (2) Is b > 1? That's growth. Is 0 < b < 1? That's decay. Is b = 1? No change. That's it! For example, 1.07 > 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 4

A savings account balance is modeled by A(t)=1000(1.05)tA(t)=1000(1.05)^t, where tt is the number of years. What is the annual percent growth rate?​

  1. 0.05% growth
  2. 5% growth (correct answer)
  3. 105% growth
  4. 5% decay
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. For the function A(t) = 1000(1.05)^t, the base is 1.05. To find the percent rate, we calculate r = 1.05 - 1 = 0.05. Converting to percent: 0.05 × 100% = 5%. Since 1.05 is greater than 1, this is growth, specifically 5% growth per year. Choice B correctly identifies the percent rate as 5% growth by showing correct reasoning. Excellent! Choice C makes a common percent mistake: the base 1.05 doesn't mean 105% growth—it means 5% growth! The 1 represents 'what you already have' (100%), and the 0.05 is the additional 5%, for a total of 105% of the previous amount (which is 5% growth). To find the percent rate: (1) Identify the base b, (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: base is 1.03, so r = 1.03 - 1 = 0.03 = 3%. For decay like 0.97: r = 0.97 - 1 = -0.03 = -3%, which we call '3% decay.' Easy!

Question 5

A bacteria culture is modeled by P(t)=800(1.03)tP(t)=800(1.03)^t, where tt is measured in hours. Is this exponential growth, decay, or neither?

  1. Exponential decay
  2. Neither (constant)
  3. Exponential growth (correct answer)
  4. Linear growth
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. Looking at the function P(t) = 800(1.03)^t, we check the base: 1.03 is greater than 1, which means this is exponential growth. Think of it this way: each time t increases by 1, P is multiplied by 1.03, so P is getting bigger—that's growth! Choice C correctly identifies this as exponential growth because the base 1.03 > 1. Excellent! Choice A confuses growth with decay: since the base 1.03 is greater than 1, this is growth, not decay. An easy way to remember: bases bigger than 1 mean growing, bases between 0 and 1 mean shrinking! Here's your growth/decay decision tree: (1) Look at the base b, (2) Is b > 1? That's growth. Is 0 < b < 1? That's decay. Is b = 1? No change. That's it! For example, 1.03 > 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 6

Which function represents exponential decay?​

  1. y=100(1.08)ty=100(1.08)^t
  2. y=100(1.15)ty=100(1.15)^t
  3. y=100(0.90)ty=100(0.90)^t (correct answer)
  4. y=100(1.00)ty=100(1.00)^t
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. Looking at each function, we check the bases: A has base 1.08 > 1 (growth), B has base 0.90 < 1 (decay), C has base 1.00 = 1 (constant), D has base 1.15 > 1 (growth). Only choice B has a base less than 1, making it exponential decay. Choice B correctly identifies y = 100(0.90)^t as exponential decay because the base 0.90 < 1. Excellent! Choices A and D have bases greater than 1, so they represent growth, not decay. Choice C has base exactly 1, which means no change—it stays constant at 100. Here's your growth/decay decision tree: (1) Look at the base b, (2) Is b > 1? That's growth. Is 0 < b < 1? That's decay. Is b = 1? No change. That's it! For example, 1.07 > 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 7

The value of a laptop after tt years is modeled by V(t)=25000(0.88)tV(t)=25000(0.88)^t. Is this exponential growth, exponential decay, or neither?

  1. Neither (constant)
  2. Exponential growth
  3. Linear decay
  4. Exponential decay (correct answer)
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.03), while the rate r is how much it's changing by percent (like 3%). They're related by b = 1 + r, so knowing one gives you the other! Looking at the function V(t) = 25000(0.88)^t, we check the base: 0.88 is less than 1, which means this is exponential decay. Think of it this way: each time t increases by 1, V is multiplied by 0.88, so V is getting smaller—that's decay! Choice C correctly identifies this as exponential decay because the base 0.88 < 1. Excellent! Choice A confuses growth with decay: since the base 0.88 is less than 1, this is decay, not growth. An easy way to remember: bases bigger than 1 mean growing, bases between 0 and 1 mean shrinking! Here's your growth/decay decision tree: (1) Look at the base b, (2) Is b > 1? That's growth. Is 0 < b < 1? That's decay. Is b = 1? No change. That's it! For example, 1.07 > 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 8

An amount of medicine in the bloodstream is modeled by M(t)=1000(0.80)tM(t)=1000(0.80)^t, where tt is in hours. What percent rate of change does this function represent per hour?

  1. 20% growth
  2. 80% decay
  3. 0.20% decay
  4. 20% decay (correct answer)
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function M(t) = 1000(0.80)^t, the base is 0.80. To find the percent rate, we calculate r = 0.80 - 1 = -0.20. Converting to percent: -0.20 × 100% = -20%. Since 0.80 is less than 1, this is decay, specifically 20% decay per hour. Choice D correctly identifies the percent rate as 20% decay by showing correct reasoning. Excellent! Choice B gives the growth factor (b = 0.80) when the question asks for the growth rate (r = 20% decay). Remember: factor is what you multiply by, rate is the percent change. They're related by b = 1 + r! The form y = a(1 + r)^x makes the rate super obvious: if you see y = 500(1 + 0.08)^t, you can read the rate right off—it's 0.08 = 8%. But if you see y = 500(1.08)^t, you have to subtract 1 from the base: 1.08 - 1 = 0.08 = 8%. Same rate, just written differently!

Question 9

In the model y=800⋅(1.03)xy=800\cdot(1.03)^x, what does the base 1.031.03 represent?

  1. A 103% decrease for each 1-unit increase in xx
  2. A constant change of 3 units per 1-unit increase in xx
  3. A 3% increase for each 1-unit increase in xx (correct answer)
  4. An initial value of 1.03
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.03), while the rate r is how much it's changing by percent (like 3%). They're related by b = 1 + r, so knowing one gives you the other! For the function y = 800·(1.03)^x, the base is 1.03. To find the percent rate, we calculate r = 1.03 - 1 = 0.03 as decimal. Converting to percent: 0.03 × 100% = 3%. Since 1.03 is greater than 1, this is growth, specifically 3% growth per unit in x. Choice B correctly identifies the percent rate as 3% by showing correct reasoning. Excellent! Choice C makes a common percent mistake: the base 1.03 doesn't mean 103% growth—it means 3% growth! The 1 represents 'what you already have' (100%), and the 0.03 is the additional 3%, for a total of 103% of the previous amount (which is 3% growth). Don't confuse the factor with the rate: if something grows by 5% per year, the growth RATE is 5% (r = 0.05), but the growth FACTOR is 1.05 (b = 1.05). Each year you have 105% of what you had (100% + 5%), which means multiplying by 1.05.

Question 10

A phone battery's remaining charge is modeled by C(t)=100(0.95)tC(t)=100(0.95)^t, where tt is measured in hours. What percent rate of change does this represent per hour?

  1. 5% decay per hour (correct answer)
  2. 95% decay per hour
  3. 5% growth per hour
  4. 0.05% decay per hour
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. For the function C(t) = 100(0.95)^t, the base is 0.95. To find the percent rate, we calculate r = 0.95 - 1 = -0.05. Converting to percent: -0.05 × 100% = -5%. Since 0.95 is less than 1, this is decay, specifically 5% decay per hour. Choice A correctly identifies the percent rate as 5% decay per hour by recognizing that 0.95 = 1 - 0.05, which means a 5% decrease each hour. Excellent! Choice B confuses the base with the rate: the base 0.95 doesn't mean 95% decay—it means the battery retains 95% of its charge each hour, which is a 5% loss! To find the percent rate: (1) Identify the base b = 0.95, (2) Subtract 1: r = 0.95 - 1 = -0.05, (3) Convert to percent: -0.05 × 100 = -5%. For decay like 0.95: r = 0.95 - 1 = -0.05 = -5%, which we call '5% decay.' Easy!

Question 11

An investment is modeled by A(t)=1500(1.02)12tA(t)=1500(1.02)^{12t}, where tt is in years. The base 1.021.02 is applied each month. What is the monthly interest rate (as a percent)?

  1. 2% per month (correct answer)
  2. 12% per month
  3. 24% per month
  4. 2% per year
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function A(t) = 1500(1.02)^{12t}, the base is 1.02 and it's applied each month (12 times per year). To find the monthly percent rate, we calculate r = 1.02 - 1 = 0.02. Converting to percent: 0.02 × 100% = 2%. Since 1.02 is greater than 1, this is growth, specifically 2% growth per month. Choice A correctly identifies the monthly interest rate as 2% per month by showing correct reasoning. Excellent! Choice C might be thinking about the annual rate (roughly 24% per year with compounding), but the question specifically asks for the monthly rate, which is 2%. The exponent 12t tells us the base is applied 12 times per year, once each month. Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form!

Question 12

A town's population is modeled by the exponential function P(t)=800(1.03)tP(t)=800(1.03)^t, where tt is the number of years since 2020. What is the annual percent growth rate?

  1. 0.03% growth
  2. 103% growth
  3. 1.03% growth
  4. 3% growth (correct answer)
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. For the function P(t) = 800(1.03)^t, the base is 1.03. To find the percent rate, we calculate r = 1.03 - 1 = 0.03. Converting to percent: 0.03 × 100% = 3%. Since 1.03 is greater than 1, this is growth, specifically 3% growth per year. Choice B correctly identifies the percent rate as 3% growth by recognizing that the base 1.03 represents a 3% annual increase. Excellent! Choice C makes a common percent mistake: the base 1.03 doesn't mean 103% growth—it means 3% growth! The 1 represents 'what you already have' (100%), and the 0.03 is the additional 3%, for a total of 103% of the previous amount (which is 3% growth). To find the percent rate: (1) Identify the base b, (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: base is 1.03, so r = 1.03 - 1 = 0.03 = 3%. For decay like 0.97: r = 0.97 - 1 = -0.03 = -3%, which we call '3% decay.' Easy!

Question 13

A town's population starts at 1000 people and increases by 10% each year. Which exponential function models the population after tt years?

  1. P(t)=1000(0.10)tP(t)=1000(0.10)^t
  2. P(t)=1000(1.10)tP(t)=1000(1.10)^t (correct answer)
  3. P(t)=1100(1.10)tP(t)=1100(1.10)^t
  4. P(t)=1000(1.00)tP(t)=1000(1.00)^t
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form! The context tells us initial value is 1000 and rate is 10% increase. Converting the rate to decimal: 10% = 0.10. The growth factor is b = 1 + 0.10 = 1.10. So the exponential function is y = 1000·(1.10)^t. We can also write this as y = 1000(1 + 0.10)^t to show the rate explicitly! Choice B correctly identifies the function as y=1000(1.10)^t by showing correct reasoning. Excellent! Choice A confuses growth with decay (or vice versa): since the base 0.10 is less than 1, this is decay, not growth. An easy way to remember: bases bigger than 1 mean growing, bases between 0 and 1 mean shrinking! The form y = a(1 + r)^x makes the rate super obvious: if you see y = 500(1 + 0.08)^t, you can read the rate right off—it's 0.08 = 8%. But if you see y = 500(1.08)^t, you have to subtract 1 from the base: 1.08 - 1 = 0.08 = 8%. Same rate, just written differently!

Question 14

A laptop depreciates according to V(t)=1200(0.90)tV(t)=1200(0.90)^t, where tt is in years. What is the annual percent depreciation rate?

  1. 0.10% depreciation
  2. 10% depreciation (correct answer)
  3. 10% growth
  4. 90% depreciation
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function V(t) = 1200(0.90)^t, the base is 0.90. To find the percent rate, we calculate r = 0.90 - 1 = -0.10. Converting to percent: -0.10 × 100% = -10%. Since 0.90 is less than 1, this is decay, specifically 10% decay (depreciation) per year. Choice B correctly identifies the percent rate as 10% depreciation by calculating 1 - 0.90 = 0.10 = 10% decrease per year. Excellent! Choice A gives the growth factor (b = 0.90) when the question asks for the depreciation rate. Remember: if the base is 0.90, that means you keep 90% of the value each year, which is a 10% loss, not a 90% loss! Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form!

Question 15

A quantity is modeled by y=400(1−0.12)xy=400(1-0.12)^x. What is the percent rate of change per unit xx?

  1. 12% decay (correct answer)
  2. 12% growth
  3. 0.12% decay
  4. 88% decay
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function y = 400(1 - 0.12)^x, we first simplify: 1 - 0.12 = 0.88, so y = 400(0.88)^x. The base is 0.88. To find the percent rate, we calculate r = 0.88 - 1 = -0.12. Converting to percent: -0.12 × 100% = -12%. Since 0.88 is less than 1, this is decay, specifically 12% decay per unit x. Choice A correctly identifies the percent rate as 12% decay by recognizing that (1 - 0.12) = 0.88 represents keeping 88% of the quantity, which is a 12% decrease. Excellent! Choice D gives the wrong rate: the base 0.88 means we keep 88% each time, which is a 12% loss, not an 88% loss! Remember to subtract from 1 to find the decay rate. The form y = a(1 - r)^x makes the decay rate super obvious: if you see y = 500(1 - 0.12)^t, you can read the rate right off—it's 0.12 = 12% decay. This form explicitly shows what percent is being lost each time period!

Question 16

An investment account is modeled by A(t)=1000(1+r)tA(t)=1000(1+r)^t, where tt is in years. If the account grows by 8% per year, which function correctly models the balance?

  1. A(t)=1000(1.08)tA(t)=1000(1.08)^t (correct answer)
  2. A(t)=1000(0.92)tA(t)=1000(0.92)^t
  3. A(t)=1000(8)tA(t)=1000(8)^t
  4. A(t)=1000(1.8)tA(t)=1000(1.8)^t
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. The context tells us the initial value is $1000 and the rate is 8% growth. Converting the rate to decimal: 8% = 0.08. The growth factor is b = 1 + 0.08 = 1.08. So the exponential function is A(t) = 1000(1.08)^t. We can also write this as A(t) = 1000(1 + 0.08)^t to show the rate explicitly! Choice A correctly identifies the function as A(t) = 1000(1.08)^t by adding the 8% growth rate to 1 to get the growth factor 1.08. Excellent! Choice B confuses growth with decay: for 8% growth, we need b = 1 + 0.08 = 1.08, not b = 1 - 0.08 = 0.92. The base 0.92 would represent 8% decay, not growth! The form y = a(1 + r)^x makes the rate super obvious: if you see y = 500(1 + 0.08)^t, you can read the rate right off—it's 0.08 = 8%. But if you see y = 500(1.08)^t, you have to subtract 1 from the base: 1.08 - 1 = 0.08 = 8%. Same rate, just written differently!

Question 17

An investment is modeled by A(t)=5000(1.02)tA(t)=5000(1.02)^t, where tt is in years. What is the value of the growth factor bb in A=a⋅btA=a\cdot b^t?

  1. b=0.02b=0.02
  2. b=1.02b=1.02 (correct answer)
  3. b=5000b=5000
  4. b=2b=2
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.03), while the rate r is how much it's changing by percent (like 3%). They're related by b = 1 + r, so knowing one gives you the other! Looking at the function y = 5000·(1.02)^t, we check the base: 1.02 is greater than 1, which means this is exponential growth. Think of it this way: each time t increases by 1, y is multiplied by 1.02, so y is getting bigger—that's growth! Choice C correctly identifies the growth factor as b=1.02 by showing correct reasoning. Excellent! Choice B gives the growth rate (r = 0.02) when the question asks for the growth factor (b = 1.02). Remember: factor is what you multiply by, rate is the percent change. They're related by b = 1 + r! Don't confuse the factor with the rate: if something grows by 5% per year, the growth RATE is 5% (r = 0.05), but the growth FACTOR is 1.05 (b = 1.05). Each year you have 105% of what you had (100% + 5%), which means multiplying by 1.05.

Question 18

An amount of medicine in the bloodstream is modeled by M(t)=1000(0.80)tM(t)=1000(0.80)^t, where tt is in hours. What percent rate of change does this function represent per hour?​

  1. 80% decay
  2. 20% decay (correct answer)
  3. 20% growth
  4. 0.20% decay
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function M(t) = 1000(0.80)^t, the base is 0.80. To find the percent rate, we calculate r = 0.80 - 1 = -0.20. Converting to percent: -0.20 × 100% = -20%. Since 0.80 is less than 1, this is decay, specifically 20% decay per hour. Choice D correctly identifies the percent rate as 20% decay by showing correct reasoning. Excellent! Choice B gives the growth factor (b = 0.80) when the question asks for the growth rate (r = 20% decay). Remember: factor is what you multiply by, rate is the percent change. They're related by b = 1 + r! The form y = a(1 + r)^x makes the rate super obvious: if you see y = 500(1 + 0.08)^t, you can read the rate right off—it's 0.08 = 8%. But if you see y = 500(1.08)^t, you have to subtract 1 from the base: 1.08 - 1 = 0.08 = 8%. Same rate, just written differently!

Question 19

A car's value is modeled by V(t)=20000(1+r)tV(t)=20000(1+r)^t and it depreciates 15% each year. Which function correctly models this situation?

  1. V(t)=20000(1.15)tV(t)=20000(1.15)^t
  2. V(t)=20000(0.15)tV(t)=20000(0.15)^t
  3. V(t)=20000(0.85)tV(t)=20000(0.85)^t (correct answer)
  4. V(t)=20000(1−0.85)tV(t)=20000(1-0.85)^t
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). The context tells us the car depreciates 15% each year. Converting the rate to decimal: 15% = 0.15. Since it's depreciation (decay), r = -0.15. The growth factor is b = 1 + (-0.15) = 0.85. So the exponential function is V(t) = 20000(0.85)^t. Choice C correctly identifies this as V(t) = 20000(0.85)^t by showing correct reasoning. Excellent! Choice A confuses growth with decay: since the car depreciates (loses value), we need a base less than 1, not greater than 1. A base of 1.15 would mean 15% growth, not 15% decay! Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form!

Question 20

A medicine amount in the bloodstream is modeled by M(t)=60(0.95)tM(t)=60(0.95)^t, where tt is in hours. What percent rate of change does this function represent per hour?

  1. 0.95% decay per hour
  2. 95% decay per hour
  3. 5% growth per hour
  4. 5% decay per hour (correct answer)
Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function M(t) = 60(0.95)^t, the base is 0.95. To find the percent rate, we calculate r = 0.95 - 1 = -0.05. Converting to percent: -0.05 × 100% = -5%. Since 0.95 is less than 1, this is decay, specifically 5% decay per hour. Choice A correctly identifies the percent rate as 5% decay per hour by recognizing that a base of 0.95 means keeping 95% of the medicine each hour, which is a 5% loss. Excellent! Choice B makes a common percent mistake: the base 0.95 doesn't mean 95% decay—it means 5% decay! The 0.95 represents 'what remains' (95%), and the medicine decreases by 5% each hour. To find the percent rate: (1) Identify the base b, (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: base is 1.03, so r = 1.03 - 1 = 0.03 = 3%. For decay like 0.97: r = 0.97 - 1 = -0.03 = -3%, which we call '3% decay.' Easy!