Exercises: Interpret Parameters in Context

In the linear function $f(x) = 5x - 3$, which value is the slope?

In the exponential function $g(x) = 750 \cdot (1.04)^x$, what is the initial value (the value when $x = 0$)?

In $h(x) = 2000 \cdot (0.90)^x$, is this function growing or decaying?

A taxi fare is modeled by $C(m) = 2.80m + 12$, where $m$ is miles driven and $C$ is cost in dollars.

(a) Interpret the slope in context (include direction, magnitude, and units).
(b) Interpret the y-intercept in context.

A cooling object's temperature is $T(t) = -3t + 90$, where $t$ is time in minutes and $T$ is temperature in degrees Celsius.

Which interpretation of the y-intercept is correct?

A town's population is modeled by $P(t) = 12000 \cdot (1.03)^t$, where $t$ is years after 2015.

(a) Interpret the initial value in context.
(b) Interpret the base in context.

A car's value is modeled by $V(t) = 25000 \cdot (0.88)^t$, where $t$ is years after purchase.

Which interpretation of the base is correct?

An exponential function has base $b = 1.065$. What is the annual percent growth rate? Enter as a decimal (e.g., 0.065 for 6.5%).

A hiking trail has an elevation model $E(d) = 4d + 200$, where $d$ is horizontal distance in meters and $E$ is elevation in meters above sea level.

Which correctly interprets the slope?

An account balance is modeled by $A(t) = 1500 \cdot (1.045)^t$, where $t$ is years.

The initial value is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . It means the starting balance was   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   dollars.
The base is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . The annual growth rate is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   %.

An exponential function has $b = 1.08$. Which interpretation is correct?

A population model shows $P(t) = 5000 \cdot (0.97)^t$. Which interpretation of the base is correct?

A physical trainer charges $f(x) = 75x + 100$, where $x$ is the number of sessions and $f$ is total cost in dollars.

A student says "the cost increases by 75% per session." Explain the error and provide the correct interpretation of the slope.

A city's water usage is modeled by $W(t) = 0.5t + 80$, where $t$ is years after 2010 and $W$ is millions of gallons per day.

Write a complete interpretive sentence for each parameter (slope and y-intercept), including value, meaning, and units.

Two models describe a quantity over time:

• Model A: $f(t) = 50t + 200$
• Model B: $g(t) = 200 \cdot (1.05)^t$

Interpret the "initial value" parameter in each model and explain what is different about how each model grows after $t = 0$.

What is wrong with the student's response?

What is the error?

Two phone plans are modeled as follows:

• Plan A: $C_A(x) = 0.10x + 50$, where $x$ = minutes per month
• Plan B: $C_B(x) = 0.05x + 65$, where $x$ = minutes per month

Interpret all four parameters (slope and y-intercept for each plan) in context, then explain at what usage level the two plans cost the same.

A linear model says a city grows by 2,000 people per year. An exponential model says the same city grows by 3% per year. Both models use a starting population of 50,000.

Interpret the growth parameter in each model. Then explain, using the parameter interpretations, why the two models will give very different long-term predictions.