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Algebra Help: Interpret Exponential Functions And Growth Rate

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

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Question 1

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

97% decay per hour
3% decay per hour (correct answer)
0.97% decay per hour
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 $V(t)=20000(0.80)^t$ , where $t$ is in years. What is the annual depreciation rate (percent decay rate)?

80% decay per year
20% growth per year
0.20% decay per year
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 $t$ years is modeled by $V(t)=25000(0.88)^t$ . Is this exponential growth, exponential decay, or neither?

Neither (constant)
Exponential decay (correct answer)
Exponential growth
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 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 4

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

0.05% growth
5% growth (correct answer)
105% growth
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 $P(t)=800(1.03)^t$ , where $t$ is measured in hours. Is this exponential growth, decay, or neither?

Exponential decay
Neither (constant)
Exponential growth (correct answer)
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 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 6

Which function represents exponential decay?

$y=100(1.08)^t$
$y=100(1.15)^t$
$y=100(0.90)^t$ (correct answer)
$y=100(1.00)^t$

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 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 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 7

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

Neither (constant)
Exponential growth
Linear decay
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 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 $M(t)=1000(0.80)^t$ , where $t$ is in hours. What percent rate of change does this function represent per hour?

20% growth
80% decay
0.20% decay
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 $y=800\cdot(1.03)^x$ , what does the base $1.03$ represent?

A 103% decrease for each 1-unit increase in $x$
A constant change of 3 units per 1-unit increase in $x$
A 3% increase for each 1-unit increase in $x$ (correct answer)
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 $C(t)=100(0.95)^t$ , where $t$ is measured in hours. What percent rate of change does this represent per hour?

5% decay per hour (correct answer)
95% decay per hour
5% growth per hour
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!