The cost per guest is the rate of change of total cost with respect to the number of guests. Use the two points to compute the slope $\frac{\Delta C}{\Delta g}$.

Once you know the cost per guest (the slope), plug it and one of the points into the form $C(g) = mg + b$ to solve for the fixed setup fee $b$.

After you find the cost per guest and the setup fee, write the function $C(g)$ and see which answer choice has the same slope and intercept.

Create a table and enter the two points: in the first column (x1), type 40 and 90; in the second column (y1), type 560 and 1010. These represent $(g, C(g))$.

In a new expression line, type (1010-560)/(90-40) to have Desmos compute the slope between the two points. This value is the cost per guest.

Once you see the slope value from Desmos, use a new expression like 560 - [slope]*40 (replacing [slope] with the number you just found) to get the setup fee, which is the y-intercept of the linear function.

Type each answer choice function into Desmos one at a time (for example, y = 7x + 280, y = 9x + 200, etc.) and check which line passes through both data points in your table at $(40,560)$ and $(90,1010)$. The function whose graph goes through both points is the correct model.

The problem describes a fixed setup fee plus a constant amount per guest. That means the total cost is a linear function in the form

• $C(g) = (\text{cost per guest})\cdot g + (\text{setup fee})$,

where $g$ is the number of guests and $C(g)$ is the total cost in dollars.

The two situations give you two points in the form $(g, C(g))$:

• 40 guests cost $560, giving the point $(40, 560)$
• 90 guests cost $1,010, giving the point $(90, 1010)$

The cost per guest is the slope of the line:

So the caterer charges $9 per guest.

Use one of the points and the cost per guest in the linear form $C(g) = 9g + b$, where $b$ is the setup fee.

Using the point $(40, 560)$:

So the fixed setup fee is $200.

Now plug the values into the linear form:

• Cost per guest (slope) is $9.
• Setup fee (intercept) is $200.

So the function is

• $C(g) = 9g + 200$.

Comparing with the answer choices, this matches choice B) $C(g)=9g+200$.