- The amount of a medicine in the bloodstream is modeled by M(t)=200(0.9)t, where t is hours after the dose and M(t) is in milligrams. What does it mean that the base is 0.9?
Answer: The medicine amount is multiplied by 0.9 each hour (a 10% decrease per hour).
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·bx, the parameter a is the initial value (what y equals when x = 0, since b0 = 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·bx: 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! - A town's population is modeled by P(t)=15,000(1.02)t, where t is years and P(t) is the population. What is the percent growth rate per year?
Answer: 2% per year
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 P(t) = 15,000(1.02)t, the 15,000 is the initial value (starting population of 15,000), 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. Each year, the population is 2% larger than the year before! Choice A is correct because it properly identifies that a base of 1.02 represents 2% growth per year. Perfect! Choice C has the growth rate wrong: a base of 1.02 means 2% growth, not 102%. The trick is that 1.02 = 1 + 0.02, and that 0.02 is the 2% rate. Subtract 1 from the base to get the decimal rate! For exponential functions y = a·bx: 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 1.02 → 0.02 = 2% growth), or subtract b from 1 if b < 1 (like 0.98 → 1 - 0.98 = 0.02 = 2% decay). Quick check for exponential: if the base b = 1.02, think '1 plus 0.02, so that's 2% growth.' The distance from 1 is the rate! - A streaming service charges a flat monthly fee plus a per-movie rental charge. The total cost T (in dollars) for renting n movies in a month is T=4n+12. What does the 4 represent in this context?
Answer: The cost increases by $4 per movie rented.
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!
- The value of a laptop after t years is modeled by V(t)=1200(0.85)t, where V is in dollars. What does 0.85 represent in this context?
Answer: Each year, the laptop keeps 85% of its value (a 15% decrease per year).
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·bx, the parameter a is the initial value (what y equals when x = 0, since b0 = 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·bx: 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! - A gym charges a monthly membership fee plus a one-time sign-up fee. The total cost C (in dollars) after m months is modeled by C=35m+60. What does the 60 represent in this context?
Answer: The one-time sign-up fee is $60.
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!
- A laptop originally costs 1,200 and loses value each year. Its value is modeled by V(t)=1200(0.90)t, where t is the number of years. What does it mean that the base is 0.90?
Answer: The laptop keeps 90% of its value each year (a 10% decrease per year).
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·bx, the parameter a is the initial value (what y equals when x = 0, since b0 = 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 0.90 means multiply by 0.90 each year, and since 0.90 = 1 - 0.10, this represents a 10% decrease per year—we subtract 0.90 from 1 to find the decay rate: 1 - 0.90 = 0.10 = 10%. Each year, the laptop's value is 10% less than the year before! Choice B is correct because it properly identifies that 0.90 represents the decay factor with the correct interpretation of a 10% decrease per year. Choice C gets the direction wrong, saying gains when actually it loses value—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·bx: 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 b from 1 if b < 1 (like 0.90 → 1 - 0.90 = 0.10 = 10% decay). - A town's population is modeled by P(t)=25,000(1.02)t, where t is the number of years since 2026. What is the initial value of the population in this model?
Answer: 25,000 people
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·bx, the parameter a is the initial value (what y equals when x = 0, since b0 = 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) = 25,000(1.02)t, the 25,000 is the initial value (starting population of 25,000 people in 2026), 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. Each year, the population is 2% larger than the year before! Choice A is correct because it properly identifies that 25,000 represents the initial value of the population. Perfect! Choice B misidentifies which parameter is which: in y = a·bx, 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·bx: 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 1.02 → 0.02 = 2% growth), or subtract b from 1 if b < 1 (like 0.95 → 1 - 0.95 = 0.05 = 5% decay). - A bike rental shop charges C=6h+15, where C is the total cost (in dollars) and h is the number of hours the bike is rented. In the equation, interpret the parameter 6.
Answer: A cost of $6 per hour rented
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 = 6h + 15, the slope 6 represents the cost increase of $6 per hour, and the y-intercept 15 represents a $15 initial fee. So the full story is: you pay $15 upfront (base rental fee) plus $6 for each hour you rent the bike. Choice B is correct because it properly identifies that 6 represents the hourly rate of $6 per hour rented. Perfect! Choice A confuses the slope with the y-intercept: the 6 is actually the slope (hourly rate), not the starting 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).
- A culture of bacteria is modeled by N(t)=300⋅2t, where t is time in hours and N is the number of bacteria. What does the 300 represent in this context?
Answer: The initial number of bacteria at t=0 is 300.
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·bx, the parameter a is the initial value (what y equals when x = 0, since b0 = 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) = 300·2t, 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·bx, 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·bx: 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). - A gym charges a monthly membership fee plus a one-time sign-up fee. The total cost C (in dollars) after m months is modeled by C=25m+40. What does the 40 represent in this context?
Answer: The one-time sign-up fee is $40.
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 = 25m + 40, the slope 25 represents the monthly membership fee ($25 per month), and the y-intercept 40 represents the one-time sign-up fee (the cost when m = 0, before any months have passed). So the full story is: you pay $40 upfront as a sign-up fee, plus $25 for each month of membership. Choice B is correct because it properly identifies that 40 represents the one-time sign-up fee—the initial cost before any monthly fees. Perfect! Choice A confuses the y-intercept with the slope: the 40 is actually the starting value (y-intercept), not the rate of change. 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). 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!
- The value of a laptop depreciates over time. Its value is modeled by V(t)=1200(0.85)t, where t is years and V(t) is in dollars. What does 0.85 represent in this context?
Answer: Each year the laptop keeps 85% of its value (a 15% decrease per year).
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·bx, the parameter a is the initial value (what y equals when x = 0, since b0 = 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·bx: 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! - A bakery's daily profit is modeled by P=15n+120, where P is profit in dollars and n is the number of cakes sold that day. In the equation P=15n+120, interpret the parameter 120.
Answer: When 0 cakes are sold, the profit is $120 (the starting profit/constant term).
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 P = 15n + 120, the slope 15 represents the rate of $15 per cake sold, and the y-intercept 120 represents the starting profit of $120 when no cakes are sold. So the full story is: there's a base profit of $120 plus $15 for each cake sold that day. Choice C is correct because it properly identifies that 120 represents the initial profit when n=0 with context. Choice A confuses the slope with the y-intercept: the 120 is actually the y-intercept, which represents the starting profit, not the per-cake increase—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 $15 per cake), and b is always the starting value (the amount when x = 0—like $120 base profit); 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!
- A car's value depreciates according to V(t)=28000(0.85)t, where t is years after purchase. The owner plans to sell the car when its value drops to exactly half the original price. In this context, what information would you need to determine when this occurs, and what role does 0.85 play in that calculation?
Answer: You need only the given function; 0.85 represents the fraction of value retained annually
To find when the car's value is half the original ($14,000), you only need the given function: solve 14000 = 28000(0.85)t. The parameter 0.85 represents the fraction of value retained each year (the car keeps 85% of its value annually, losing 15%). - A phone plan's total monthly cost is modeled by C=15+0.10t, where C is in dollars and t is the number of text messages sent in a month. What does the 15 represent in this context?
Answer: The base monthly fee is $15 when 0 texts are sent.
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 = 15 + 0.10t, which can be rewritten as C = 0.10t + 15, the slope 0.10 represents the cost per text message ($0.10 per text), and the y-intercept 15 represents the base monthly fee ($15 when t = 0, before any texts are sent). So the full story is: you pay $15 as a base monthly fee plus $0.10 for each text message sent. Choice C is correct because it properly identifies that 15 represents the base monthly fee when 0 texts are sent—the initial cost before any per-text charges. Perfect! Choice A confuses the y-intercept with the slope: the 15 is actually the starting value (y-intercept), not the rate per text. 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). 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!
- A streaming service charges according to C=8n+20, where C is the total cost (in dollars) and n is the number of months. What does it mean that the y-intercept is 20?
Answer: When n=0 months, the cost is $20 (a starting fee).
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! - A phone battery's remaining charge is modeled by B(t)=100(0.85)t, where t is the number of hours since it was unplugged and B(t) is the percent charge remaining. What does 0.85 represent in this context?
Answer: Each hour, the battery charge is multiplied by 0.85 (a 15% decrease per hour).
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·bx, the parameter a is the initial value (what y equals when x = 0, since b0 = 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 hour, and since 0.85 = 1 - 0.15, this represents a 15% decrease per hour—we subtract 0.85 from 1 to find the decay rate: 1 - 0.85 = 0.15 = 15%. Each hour, the battery charge is 15% less than the hour before! Choice B is correct because it properly identifies that 0.85 represents the decay factor with the correct interpretation of a 15% decrease per hour. 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·bx: 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). - A taxi fare is modeled by F=2.50d+4, where F is the fare (in dollars) and d is the distance traveled (in miles). In this function, what is the meaning of the 4?
Answer: The fare starts at $4 when the distance is 0 miles.
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!
- A savings account balance is modeled by A(t)=600(1.05)t, where t is the number of years and A(t) is in dollars. What is the annual interest rate?
Answer: 5% per year
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·bx, the parameter a is the initial value (what y equals when x = 0, since b0 = 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! - A taxi fare is modeled by F=2.50m+4.00, where F is the fare in dollars and m is the number of miles traveled. In this model, interpret the parameter 2.50.
Answer: The fare increases by $2.50 per mile.
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 = 2.50m + 4.00, the slope 2.50 represents the fare per mile ($2.50 per mile traveled), and the y-intercept 4.00 represents the initial fare ($4.00 when m = 0, before any miles are traveled). So the full story is: you pay $4.00 as a base fare plus $2.50 for each mile traveled. Choice B is correct because it properly identifies that 2.50 represents the fare increase per mile—each additional mile costs $2.50. Perfect! Choice A confuses the slope with the y-intercept: the 2.50 is actually the rate per mile (slope), not the initial fare. 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 2.50'—say 'the slope is 2.50 dollars per mile, meaning each additional mile costs $2.50.' This shows you understand the math represents something real!
- A taxi fare is modeled by F=2.50m+4.00, where F is the fare in dollars and m is the number of miles traveled. What does the slope represent in this context?
Answer: The taxi charges $2.50 per mile.
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 = 2.50m + 4.00, the slope 2.50 represents the rate of $2.50 per mile, and the y-intercept 4.00 represents a $4.00 starting fee. So the full story is: you pay $4.00 upfront plus $2.50 for each mile traveled. Choice A is correct because it properly identifies that the slope represents $2.50 per mile. Perfect! Choice C confuses the slope with the y-intercept: the slope is actually 2.50, which represents the rate per mile. 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—$2.50 per mile), and b is always the starting value (the amount when x = 0—$4.00 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!
