Linear, Quadratic, and Exponential Models: Interpreting Parameters in Linear and Exponential Models (CCSS.F-LE.5)
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Common Core High School Functions › Linear, Quadratic, and Exponential Models: Interpreting Parameters in Linear and Exponential Models (CCSS.F-LE.5)
A babysitter charges according to the function $C(t)=5t+12$, where $t$ is hours worked (with $t\ge 0$) and $C$ is the total cost in dollars. The slope is 5 and the intercept is 12. Hours are billed only for time worked.
What does the intercept represent in this situation?
An upfront fee of 12 dollars before any hours are worked
The babysitter earns 12 dollars per hour
The babysitter travels 12 miles to the job
The total cost when $t=12$ hours
Explanation
In $C(t)=5t+12$, the intercept 12 is $C(0)$, the cost at zero hours, so it represents a 12-dollar upfront fee. Choice B confuses the rate with the initial fee (the hourly rate is 5, not 12).
A car worth 24,000 dollars depreciates by 7% each year. Its value after $t$ years is modeled by $V(t) = 24000(0.93)^t$ for $t \ge 0$.
What does the base represent in this situation?
Each year, the car retains 93% of its value from the previous year (a 7% decrease per year).
The car's initial value is 0.93 dollars.
It takes 0.93 years for the car to lose all its value.
For each dollar of value, time decreases by 0.93 years.
Explanation
In $V(t) = 24000(0.93)^t$, the base 0.93 is the yearly multiplier: the car keeps 93% of its value each year. Choice B confuses the base with the initial value, which is 24000; C is unrelated to what the base means.
A bacteria culture doubles in size each day. Its population after $d$ days is modeled by $B(d) = 300\cdot 2^d$ for $d \ge 0$.
What does the intercept represent in this situation?
For each bacterium present, the time doubles.
The population increases by 300 bacteria each day.
There are 300 days between measurements.
At day 0, the culture has 300 bacteria.
Explanation
In $B(d) = 300\cdot 2^d$, the intercept $B(0)=300$ is the starting population at day 0. Choice B confuses exponential doubling with adding 300 per day; exponential growth multiplies by 2 each day.
Maya opens a savings account with 200 dollars and deposits 25 dollars each week. The balance after $w$ weeks is modeled by $S(w) = 200 + 25w$ for $w \ge 0$.
What does the intercept represent in this situation?
The account grows by 200 dollars each week.
At week 0, the account already has 200 dollars.
You deposit 200 dollars in total over the entire time.
There are 200 weeks in a year.
Explanation
In $S(w) = 200 + 25w$, the intercept $S(0)=200$ is the initial balance at week 0. Choice A confuses the intercept with the weekly rate, which is 25, not 200. Choice D is unrelated to the model.
After a dose, the amount of a medicine in the bloodstream declines by 20% each hour. The amount remaining after $t$ hours is modeled by $C(t) = 160(0.8)^t$ for $t \ge 0$.
What does the base represent in this situation?
The amount of medicine decreases by 0.8 milligrams each hour.
The starting amount is 0.8 milligrams.
Each hour, 80% of the previous amount remains (a 20% decrease per hour).
For each milligram present, time decreases by 0.8 hours.
Explanation
In $C(t) = 160(0.8)^t$, the base 0.8 is the hourly multiplier: each hour the amount is 80% of the previous hour. Choice A treats 0.8 as a fixed subtraction, not a multiplier; B confuses the base with the initial amount 160.
A car's value depreciates according to $V(t)=18000(0.85)^t$, where $t$ is years since purchase (with $t\ge 0$) and $V$ is the value in dollars. The initial value is 18,000 and the base is 0.85. This models a constant percent decrease each year.
What does the base represent in this situation?
The car loses 0.85 dollars each year
The initial price of the car is 0.85 dollars
The car loses all its value in 0.85 years
Each year the value is multiplied by 0.85 (a 15% decrease per year)
Explanation
In $V(t)=18000(0.85)^t$, the base 0.85 means the value is scaled to 85% of the previous year each year (a 15% decrease). Choice A treats 0.85 as an additive dollar amount rather than a multiplicative factor.
A bacteria culture grows as $P(t)=200\cdot 2^{t}$, where $t$ is hours since the start (with $t\ge 0$) and $P$ is the number of cells. The base is 2 and the initial amount is 200. The culture doubles regularly over time.
What does the base represent in this situation?
The culture adds 2 cells every hour
The initial population is 2 cells
The population is multiplied by 2 each hour
Growth begins 2 hours after the start
Explanation
In $P(t)=200\cdot 2^t$, the base 2 means the population is multiplied by 2 for each additional hour (doubling). Choice B is wrong because the initial amount is 200, not 2.
A hiking trail starts at an elevation of 520 feet. The elevation after walking $d$ miles is modeled by $h(d) = 520 + 90d$ for $d \ge 0$.
What does the slope represent in this situation?
The elevation increases by 90 feet for each mile walked.
For each foot of elevation, you travel 90 miles.
You start the hike at 90 feet of elevation.
The trail is 90 miles long.
Explanation
In $h(d) = 520 + 90d$, the slope 90 is the rate of change: 90 feet per mile. That means each additional mile adds 90 feet of elevation. Choice B swaps the variables (feet per mile vs. miles per foot), and C confuses the rate with the initial elevation 520.
Water drains from a tank according to $V(t)=120-3t$, where $t$ is minutes since draining began (with $t\ge 0$ until empty) and $V$ is volume in liters. The slope is $-3$ and the intercept is 120. Draining continues at a steady rate.
What does the slope represent in this situation?
The starting volume is 3 liters
The water decreases by 3 liters each minute
The tank is empty after 3 minutes
The tank holds 120 liters per minute
Explanation
The slope $-3$ indicates the volume changes by $-3$ liters per minute, i.e., it decreases by 3 liters each minute. Choice A confuses the rate with the initial value (the initial volume is 120 liters, not 3).
Donations for a charity run follow $T(k)=30+2k$, where $k$ is kilometers run (with $k\ge 0$) and $T$ is the total pledge in dollars. The slope is 2 and the intercept is 30. Runners pay a sign-up fee plus a per-kilometer pledge.
What does the slope represent in this situation?
The total increases by 2 dollars for each kilometer run
The sign-up fee is 2 dollars
The runner starts at 2 kilometers
The race lasts 2 hours
Explanation
In $T(k)=30+2k$, the slope 2 is the rate of change: each additional kilometer adds 2 dollars. Choice B confuses the slope with the intercept (the sign-up fee is 30 dollars, not 2).