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Interpret parameters in linear and exponential functions in context.

Interpret slope as rate of change with units

Problem
For \(C(h)=25h+40\), interpret the slope as a repair-cost rate with units, state its \(\text{one}\text{-}\text{hour}\) meaning and the change over \(3~\text{hours}\), and distinguish it from the intercept.
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Hint

Look at the coefficient of \(h\) and connect it to how the output changes when the input goes up by \(1\).

In a linear function, the slope tells how much the output changes for each \(1\) unit increase in the input.

Solution walkthrough

01

Decode variables and slope

\[C(h)=25h+40;~\text{input}~h=\text{hours};~\text{output}~C=\text{cost}~\text{dollars}\]

In a linear rule, the coefficient of the input is the slope. Therefore 25 compares dollar output change with hour input change.

02

Interpret one hour and three hours

\[\text{slope}=+\$25/\text{hour};~1-\text{hour}~\text{change}=+\$25;~3-\text{hour}~\text{change}=3(+\$25)=+\$75\]

The positive sign indicates increasing cost. Each additional repair hour adds 25 dollars, so three additional hours add 75 dollars.

03

Distinguish the intercept

\[C(0)=40~\text{dollars}\]

The constant term is the cost at zero repair hours: a 40-dollar initial service fee. It is not measured per hour and is not the slope.

04

State the complete slope interpretation

\[\text{slope}~=~+25~\text{dollars}/\text{hour};~\text{one}-\text{hour}~\text{meaning}~=~\text{each}~\text{additional}~\text{repair}~\text{hour}~\text{adds}~\$25;~3-\text{hour}~\text{change}~=~+\$75;~\text{intercept}~\text{distinction}~=~\$40~\text{is}~\text{the}~\text{initial}~\text{service}~\text{fee},~\text{not}~\text{the}~\text{hourly}~\text{rate}\]

The result includes slope units, sign, one-unit meaning, scaled change, and the required contrast with the initial fee.

+

Another way

  1. Compare C(h+1)-C(h)=25 to read the one-hour change directly.

!

Common mistake

Do not call 40 dollars per hour the slope. Forty is a one-time initial fee; 25 is the hourly rate.

Solution walkthrough video