Use the two points to compute the slope of the line: change in total cost divided by change in months. That slope is the monthly fee.

Once you know the slope (monthly fee), write $C(m)=\text{slope}\cdot m + b$ and plug in one of the $(m,C)$ points to solve for $b$, the one-time enrollment fee.

After you find the slope and intercept, write the equation and compare it to the answer choices. You can also quickly test each choice by plugging in $m=3$ and $m=7$ to see which one gives $135 and $255.

In Desmos, enter each option as a separate line, using $x$ for months and $y$ for cost: y=22.5x+67.5, y=45x+30, y=67.5x+22.5, and y=30x+45.

Add the points (3,135) and (7,255) in Desmos (for example, by typing (3,135) on one line and (7,255) on another). These points represent the given membership situations.

Look at the graph and see which of the four lines passes through both points (3,135) and (7,255). The equation corresponding to that line is the correct model for the total cost.

The total cost $C$ depends linearly on the number of months $m$.

• A member who stayed 3 months paid $135, so this gives the point $(m,C)=(3,135)$.
• A member who stayed 7 months paid $255, so this gives the point $(7,255)$. These are two points on the line representing the cost function.

For a linear relationship, the monthly fee is the slope: the change in cost divided by the change in months.

Compute the slope using the two points $(3,135)$ and $(7,255)$:

So the cost increases by $30 for each additional month. This means the coefficient of $m$ in the function is 30.

Use the slope and one of the points to find the fixed starting cost.

Write the linear model as

where $b$ is the one-time enrollment fee.

Substitute the point $(3,135)$ into this equation:

So the one-time enrollment fee is $45.

We found that the total cost function must be

This matches answer choice D, so the correct answer is $C(m)=30m+45$.