Algebra Quiz: Interpreting Parameters In Linear Exponential Models

What this quiz covers

This quiz focuses on Interpreting Parameters In Linear Exponential Models, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

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.

Question 1

1. The number of bacteria decreases by 10% each hour.
2. The number of bacteria increases by 1.10 bacteria each hour.
3. The culture starts with 1.10 bacteria at $t=0$ hours.
4. The number of bacteria increases by 10% each hour (multiplied by 1.10 each hour). (correct answer)

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1)—if b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! In the function N(t) = 300(1.10)^t, the 300 is the initial value (starting number of bacteria of 300 when t=0), and the base 1.10 means the number is multiplied by 1.10 each hour—since 1.10 = 1 + 0.10, this represents 10% growth per hour; each hour, the number of bacteria is 10% larger than the hour before! Choice B is correct because it properly identifies that 1.10 represents the growth factor with the correct 10% increase interpretation. Choice D gets the direction wrong, saying decreases when actually it increases—with exponential functions, if b > 1 it's growth (getting bigger), if 0 < b < 1 it's decay (getting smaller); check whether your base is above or below 1! Quick check for exponential: if the base b = 1.10, think '1 plus 0.10, so that's 10% growth'; if b = 0.90, think '1 minus 0.10, so that's 10% decay'—the distance from 1 is the rate, and whether it's above or below 1 tells you growth or decay!

Question 2

1. The laptop loses $0.85 each year.
2. Each year, the laptop keeps 85% of its value (a 15% decrease per year). (correct answer)
3. The laptop’s value increases by 85% each year.
4. The initial value of the laptop is $0.85.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1). If b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! The base 0.85 means multiply by 0.85 each year, and since 0.85 = 1 - 0.15, this represents a 15% decrease per year. We subtract 0.85 from 1 to find the decay rate: 1 - 0.85 = 0.15 = 15%. Choice B is correct because it properly identifies that 0.85 represents keeping 85% of the value each year (a 15% decrease per year). Perfect! Choice C gets the direction wrong, saying increases when actually it decreases. With exponential functions, if b > 1 it's growth (getting bigger), if 0 < b < 1 it's decay (getting smaller). Check whether your base is above or below 1! For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period. To find the percent rate: subtract 1 from b if b > 1 (like 1.05 → 0.05 = 5% growth), or subtract b from 1 if b < 1 (like 0.85 → 1 - 0.85 = 0.15 = 15% decay). Quick check for exponential: if the base b = 1.03, think '1 plus 0.03, so that's 3% growth.' If b = 0.97, think '1 minus 0.03, so that's 3% decay.' The distance from 1 is the rate, and whether it's above or below 1 tells you growth or decay!

Question 3

1. The cost increases by $12 per movie rented.
2. The cost increases by $4 per movie rented. (correct answer)
3. The total cost after 4 movies is $12.
4. The monthly fee is $4.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function T = 4n + 12, the slope 4 represents the rate of $4 per movie rented, and the y-intercept 12 represents a$12 flat monthly fee. So the full story is: you pay $12 per month plus$4 for each movie rented. Choice B is correct because it properly identifies that 4 represents the cost increase of $4 per movie rented. Perfect! Choice A confuses the slope with the y-intercept: the 4 is actually the slope, which represents the rate per movie. It's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! For linear functions y = mx + b in context: m is always the rate (the 'per' something amount—$4 per movie), and b is always the starting value (the amount when x = 0—$12 monthly fee). If you can identify what's changing at a constant rate (that's m) vs what's there from the beginning (that's b), you've got it!

Question 4

1. 60 is the starting calories and 20 is calories per mile.
2. 60 is calories burned per mile, and 20 is the calories burned when 0 miles are walked. (correct answer)
3. 60 is the total calories for a 20-mile walk.
4. 20 is miles per calorie, and 60 is a one-time calorie fee.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function C = 60w + 20, the slope 60 represents calories burned per mile (60 calories per mile walked), and the y-intercept 20 represents calories burned when 0 miles are walked (20 calories burned just from the activity of preparing to walk or baseline metabolism). So the full story is: you burn 20 calories as a baseline plus 60 calories for each mile you walk. Choice B is correct because it properly identifies that 60 is calories burned per mile (the rate), and 20 is the calories burned when 0 miles are walked (the starting value). Perfect! Choice A confuses the slope with the y-intercept (has them swapped): the 60 is actually the rate per mile (slope), and 20 is the starting value (y-intercept). It's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! For linear functions y = mx + b in context: m is always the rate (the 'per' something amount—$5 per item, 60 miles per hour), and b is always the starting value (the amount when x = 0—$20 initial fee, 50 degrees starting temperature). In context, always state the full interpretation with units: don't just say 'the slope is 60'—say 'the slope is 60 calories per mile, meaning each additional mile burns 60 calories.' This shows you understand the math represents something real!

Question 5

1. 4% growth per year (correct answer)
2. 1.04% growth per year
3. 104% growth per year
4. $1500 growth per year

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. To find the percent growth or decay rate from an exponential function, look at the base: if it's written as (1 + r), then r is your rate. For example, (1.03)^t means 3% growth because 1.03 = 1 + 0.03. If the base is less than 1, like 0.97 = 1 - 0.03, that's a 3% decay. In the function A(t) = 1500(1.04)^t, the 1500 is the initial balance ($1500 when t = 0), and the base 1.04 means the balance is multiplied by 1.04 each year. Since 1.04 = 1 + 0.04, this represents 4% growth per year. Each year, the balance is 4% larger than the year before! Choice A is correct because it properly identifies that a base of 1.04 represents 4% growth per year. Perfect! Choice B has the growth rate wrong: a base of 1.04 means 4% growth, not 1.04%. The trick is that 1.04 = 1 + 0.04, and that 0.04 is the 4% rate. Subtract 1 from the base to get the decimal rate! For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period. To find the percent rate: subtract 1 from b if b > 1 (like 1.05 → 0.05 = 5% growth), or subtract b from 1 if b < 1 (like 0.95 → 1 - 0.95 = 0.05 = 5% decay).

Question 6

1. The one-time sign-up fee is $60. (correct answer)
2. The cost increases by $60 per month.
3. The gym charges $35 for the first month only.
4. The total cost after 60 months is $35.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function C = 35m + 60, the slope 35 represents the rate of $35 per month, and the y-intercept 60 represents a$60 one-time sign-up fee. So the full story is: you pay $60 upfront plus$35 for each month. Choice B is correct because it properly identifies that 60 represents the one-time sign-up fee. Perfect! Choice A confuses the slope with the y-intercept: the 60 is actually the y-intercept, which represents the starting value. It's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! For linear functions y = mx + b in context: m is always the rate (the 'per' something amount—$35 per month), and b is always the starting value (the amount when x = 0—$60 initial fee). If you can identify what's changing at a constant rate (that's m) vs what's there from the beginning (that's b), you've got it!

Question 7

1. The cost increases by $20 per month.
2. When $n=0$ months, the cost is $20 (a starting fee). (correct answer)
3. The service costs $20 for each month.
4. After 20 months, the cost is $8.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function C = 8n + 20, the slope 8 represents the cost increase of $8 per month, and the y-intercept 20 represents a$20 initial fee. So the full story is: you pay $20 upfront (starting fee) plus$8 for each month of service. Choice B is correct because it properly identifies that the y-intercept 20 represents the starting fee of $20 when n = 0 months. Perfect! Choice A confuses the y-intercept with the slope: the 20 is actually the y-intercept (starting fee), not the monthly rate. It's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! In context, always state the full interpretation with units: don't just say 'the y-intercept is 20'—say 'the y-intercept is$20, meaning there's a $20 starting fee before any months of service.' This shows you understand the math represents something real!

Question 8

1. The booking fee is $2.25.
2. The cost increases by $2.25 per mile. (correct answer)
3. The cost increases by $4.50 per mile.
4. The ride is 2.25 miles when the cost is $0.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function C = 2.25m + 4.50, the slope 2.25 represents the rate of $2.25 per mile, and the y-intercept 4.50 represents the initial booking fee of$4.50 when no miles are traveled. So the full story is: you pay $4.50 upfront plus$2.25 for each mile of the ride. Choice B is correct because it properly identifies that 2.25 represents the per-mile rate increase with units and context. Choice A confuses the slope with the y-intercept: the 2.25 is actually the slope, which represents the per-mile rate, not the initial fee—it's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! For linear functions y = mx + b in context: m is always the rate (the 'per' something amount—like $2.25 per mile), and b is always the starting value (the amount when x = 0—like$4.50 booking fee); if you can identify what's changing at a constant rate (that's m) vs what's there from the beginning (that's b), you've got it!

Question 9

1. The cost increases by $3 for each additional movie rented. (correct answer)
2. The service charges a $3 one-time membership fee.
3. The total cost is $3 when 12 movies are rented.
4. The cost increases by $12 for each additional movie rented.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function C = 3n + 12, the slope 3 represents the cost per movie ($3 per movie rented), and the y-intercept 12 represents the base fee ($12 when n = 0, before any movies are rented). So the full story is: you pay $12 as a base fee plus$3 for each movie you rent. Choice A is correct because it properly identifies that 3 represents the cost increase per movie—each additional movie costs $3. Perfect! Choice B confuses the slope with the y-intercept: the 3 is actually the rate per movie (slope), not a one-time fee. It's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! For linear functions y = mx + b in context: m is always the rate (the 'per' something amount—$5 per item, 60 miles per hour), and b is always the starting value (the amount when x = 0—$20 initial fee, 50 degrees starting temperature). In context, always state the full interpretation with units: don't just say 'the slope is 3'—say 'the slope is 3 dollars per movie, meaning each additional movie costs$3.' This shows you understand the math represents something real!

Question 10

1. 90% decrease per hour
2. 0.9% decrease per hour
3. 9% increase per hour
4. 10% decrease per hour (correct answer)

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. To find the percent growth or decay rate from an exponential function, look at the base: if it's written as (1 + r), then r is your rate. For example, (1.03)^t means 3% growth because 1.03 = 1 + 0.03. If the base is less than 1, like 0.97 = 1 - 0.03, that's a 3% decay. The base 0.9 means multiply by 0.9 each hour, and since 0.9 = 1 - 0.1, this represents a 10% decrease per hour. We subtract 0.9 from 1 to find the decay rate: 1 - 0.9 = 0.1 = 10%. Choice C is correct because it properly identifies that the percent decay rate is 10% per hour. Perfect! Choice A has the growth rate wrong: a base of 0.9 means 10% decay, not 0.9% or 9%. The trick is that 0.9 = 1 - 0.1, and that 0.1 is the 10% rate. Subtract the base from 1 to get the decimal rate! For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period. To find the percent rate: subtract 1 from b if b > 1 (like 1.05 → 0.05 = 5% growth), or subtract b from 1 if b < 1 (like 0.9 → 1 - 0.9 = 0.1 = 10% decay). Quick check for exponential: if the base b = 1.03, think '1 plus 0.03, so that's 3% growth.' If b = 0.97, think '1 minus 0.03, so that's 3% decay.' The distance from 1 is the rate, and whether it's above or below 1 tells you growth or decay!

Question 11

1. The medicine amount increases by 10% each hour.
2. The medicine amount is multiplied by 0.9 each hour (a 10% decrease per hour). (correct answer)
3. The initial dose is 0.9 mg.
4. The medicine decreases by 0.9 mg each hour.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1). If b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! In the function M(t) = 200(0.9)^t, the 200 is the initial value (starting medicine amount of 200 mg), and the base 0.9 means the medicine amount is multiplied by 0.9 each hour. Since 0.9 = 1 - 0.1, this represents a 10% decrease per hour. We subtract 0.9 from 1 to find the decay rate: 1 - 0.9 = 0.1 = 10%. Choice B is correct because it properly identifies that 0.9 means the medicine amount is multiplied by 0.9 each hour, which represents a 10% decrease per hour. Perfect! Choice A gets the direction wrong, saying the medicine increases when actually it decreases. With exponential functions, if b > 1 it's growth (getting bigger), if 0 < b < 1 it's decay (getting smaller). Check whether your base is above or below 1! For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period. To find the percent rate: subtract 1 from b if b > 1 (like 1.05 → 0.05 = 5% growth), or subtract b from 1 if b < 1 (like 0.9 → 1 - 0.9 = 0.1 = 10% decay). Quick check for exponential: if the base b = 0.9, think '1 minus 0.1, so that's 10% decay.' The distance from 1 is the rate, and whether it's above or below 1 tells you growth or decay!

Question 12

1. The laptop loses $0.85 each year.
2. Each year the laptop keeps 85% of its value (a 15% decrease per year). (correct answer)
3. The laptop’s value increases by 15% each year.
4. The initial value of the laptop is $0.85.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1). If b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! In the function V(t) = 1200(0.85)^t, the 1200 is the initial value (starting laptop value of $1200), and the base 0.85 means the value is multiplied by 0.85 each year. Since 0.85 = 1 - 0.15, this represents a 15% decrease per year. We subtract 0.85 from 1 to find the decay rate: 1 - 0.85 = 0.15 = 15%. Choice B is correct because it properly identifies that 0.85 means the laptop keeps 85% of its value each year, which is equivalent to a 15% decrease per year. Perfect! Choice C gets the direction wrong, saying the value increases when actually it decreases. With exponential functions, if b > 1 it's growth (getting bigger), if 0 < b < 1 it's decay (getting smaller). Check whether your base is above or below 1! For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period. To find the percent rate: subtract 1 from b if b > 1 (like 1.05 → 0.05 = 5% growth), or subtract b from 1 if b < 1 (like 0.95 → 1 - 0.95 = 0.05 = 5% decay). Quick check for exponential: if the base b = 0.85, think '1 minus 0.15, so that's 15% decay.' The distance from 1 is the rate, and whether it's above or below 1 tells you growth or decay!

Question 13

1. The fare starts at $4 when the distance is 0 miles. (correct answer)
2. The fare starts at 4 miles when the cost is $0.
3. The fare increases by $4 per mile.
4. The fare is multiplied by 4 for each additional mile.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function F = 2.50d + 4, the slope 2.50 represents the fare increase of $2.50 per mile, and the y-intercept 4 represents a$4 initial fee. So the full story is: you pay $4 upfront (base fare) plus$2.50 for each mile traveled. Choice A is correct because it properly identifies that 4 represents the starting fare of $4 when the distance is 0 miles. Perfect! Choice B confuses the y-intercept with the slope: the 4 is actually the y-intercept (starting fare), not the per-mile rate. It's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! If you can identify what's changing at a constant rate (that's m) vs what's there from the beginning (that's b), you've got it!

Question 14

1. The initial number of bacteria at $t=0$ is 300. (correct answer)
2. The growth factor is 300 each hour.
3. The population increases by 300 bacteria each hour.
4. The bacteria population doubles every 300 hours.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1). If b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! In the function N(t) = 300·2^t, the 300 is the initial value (starting number of bacteria at t=0), and the base 2 means the population is multiplied by 2 each hour. Since 2 = 1 + 1, this represents 100% growth per hour (doubling). Each hour, the population is twice as large as the hour before! Choice B is correct because it properly identifies that 300 represents the initial number of bacteria at t=0. Perfect! Choice A misidentifies which parameter is which: in y = a·b^x, the a is the initial value and b is the growth/decay factor. This choice has them swapped! Think: 'a' comes first alphabetically and represents the first/initial value. For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period. To find the percent rate: subtract 1 from b if b > 1 (like 2 → 1 = 100% growth), or subtract b from 1 if b < 1 (like 0.95 → 1 - 0.95 = 0.05 = 5% decay).

Question 15

1. The initial number of bacteria is 60 at $t=0$ hours. (correct answer)
2. The culture doubles by 60% each hour.
3. The number of bacteria is multiplied by 60 each hour.
4. The number of bacteria increases by 60 each hour.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1). If b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! In the function B(t) = 60·2^t, the 60 is the initial value (60 bacteria at t = 0 hours), and the base 2 means the population doubles each hour. Since 2 = 1 + 1, this represents 100% growth per hour. Each hour, the bacteria population is twice as large as the hour before! Choice B is correct because it properly identifies that 60 represents the initial number of bacteria at t = 0 hours. Perfect! Choice D misidentifies which parameter is which: in y = a·b^x, the 60 is the initial value and 2 is the growth factor. This choice confuses the coefficient with the base! Think: 'a' comes first alphabetically and represents the first/initial value. For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period.

Question 16

1. $600 per year
2. 5% per year (correct answer)
3. 105% per year
4. 1.05% per year

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1)—if b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! To find the percent growth or decay rate from an exponential function, look at the base: if it's written as (1 + r), then r is your rate—for example, (1.03)^t means 3% growth because 1.03 = 1 + 0.03; if the base is less than 1, like 0.97 = 1 - 0.03, that's a 3% decay. In the function A(t) = 600(1.05)^t, the base 1.05 means the balance is multiplied by 1.05 each year, and since 1.05 = 1 + 0.05, this represents 5% growth per year—each year, the balance is 5% larger than the year before! Choice A is correct because it properly identifies that the annual interest rate is 5% per year. Choice C has the growth rate wrong: a base of 1.05 means 5% growth, not 105% or 0.05%—the trick is that 1.05 = 1 + 0.05, and that 0.05 is the 5% rate; subtract 1 from the base to get the decimal rate! Quick check for exponential: if the base b = 1.03, think '1 plus 0.03, so that's 3% growth'; if b = 0.97, think '1 minus 0.03, so that's 3% decay'—the distance from 1 is the rate, and whether it's above or below 1 tells you growth or decay!

Question 17

1. The bill increases by $25 per gigabyte.
2. The monthly base fee is $25 when 0 gigabytes are used. (correct answer)
3. The bill is $25 after 10 gigabytes are used.
4. The cost is $0.25 per gigabyte.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function B = 0.10g + 25, the slope 0.10 represents the rate of $0.10 per gigabyte, and the y-intercept 25 represents a$25 monthly base fee. So the full story is: you pay $25 per month plus$0.10 for each gigabyte used. Choice B is correct because it properly identifies that 25 represents the monthly base fee when 0 gigabytes are used. Perfect! Choice A confuses the slope with the y-intercept: the 25 is actually the y-intercept, which represents the starting value. It's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! For linear functions y = mx + b in context: m is always the rate (the 'per' something amount—$0.10 per gigabyte), and b is always the starting value (the amount when x = 0—$25 base fee). If you can identify what's changing at a constant rate (that's m) vs what's there from the beginning (that's b), you've got it!

Question 18

1. The laptop loses $0.85 each year.
2. The laptop keeps 85% of its value each year (a 15% decrease per year). (correct answer)
3. The laptop gains 85% value each year.
4. The initial value of the laptop is $0.85.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1). If b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! In the function V(t) = 900(0.85)^t, the 900 is the initial value (the laptop's value of $900 when new), and the base 0.85 means the laptop retains 85$0.85, it means multiplying by 0.85 (keeping 85% of the value). This choice confuses exponential change with linear change! Quick check for exponential: if the base b = 1.03, think '1 plus 0.03, so that's 3% growth.' If b = 0.97, think '1 minus 0.03, so that's 3% decay.' The distance from 1 is the rate, and whether it's above or below 1 tells you growth or decay!

Question 19

1. The initial data used at $w=0$ weeks is 2 gigabytes. (correct answer)
2. The weekly data usage rate is 2 weeks per gigabyte.
3. The data usage increases by 2 gigabytes per week.
4. The data usage is multiplied by 2 each week.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In a linear function like y = mx + b, the slope m represents the rate of change—how much y increases (or decreases if negative) for each one-unit increase in x—while the y-intercept b represents the starting value or initial amount when x = 0. In the function D(w) = 3.5w + 2, the slope 3.5 represents the data usage increase of 3.5 gigabytes per week, and the y-intercept 2 represents 2 gigabytes of initial data used. So the full story is: you start with 2 gigabytes of data already used (at week 0) and then use an additional 3.5 gigabytes each week. Choice B is correct because it properly identifies that 2 represents the initial data usage of 2 gigabytes when w = 0 weeks. Perfect! Choice A confuses the y-intercept with the slope: the 2 is actually the y-intercept (initial data used), not the weekly rate. It's easy to mix these up when you're learning, but remember: in y = mx + b, m is the rate and b is the starting value! If you can identify what's changing at a constant rate (that's m) vs what's there from the beginning (that's b), you've got it!

Question 20

1. The population increases by 24,000 people each year.
2. The population in the year 2026 (when $t=0$) is 24,000 people. (correct answer)
3. The population decreases by 24,000 people each year.
4. The population growth rate is 24,000% per year.

Explanation: This question tests your ability to interpret the numbers in linear and exponential functions and understand what they mean in real-world contexts. In an exponential function like y = a·b^x, the parameter a is the initial value (what y equals when x = 0, since b^0 = 1), and the base b is the growth factor (if b > 1) or decay factor (if 0 < b < 1). If b = 1.05, that means multiplying by 1.05 each time, which is a 5% increase! In the function P(t) = 24000(1.02)^t, the 24000 is the initial value (the population of 24,000 people when t = 0), and the base 1.02 means the population is multiplied by 1.02 each year. Since 1.02 = 1 + 0.02, this represents 2% growth per year. The population in 2026 (when t = 0) is 24,000 people! Choice B is correct because it properly identifies that 24000 represents the initial population of 24,000 people in the year 2026 (when t = 0). Perfect! Choice A confuses exponential growth with linear growth: the 24,000 is the starting population, not an annual increase. In exponential functions, the population is multiplied by the base each year, not increased by a fixed amount! For exponential functions y = a·b^x: a is what you have at time zero (plug in x = 0 and you get a), and b tells you the multiplication factor each time period. Always check what happens when you plug in x = 0 to identify the initial value!