For a quadratic profit function, the maximum profit occurs at the vertex of the parabola. What formula gives you the $n$-coordinate of the vertex for $an^2 + bn + c$?

Once you recall that the vertex is at $n = -\dfrac{b}{2a}$, identify $a$ and $b$ from $-0.2n^2 + 3.2n - 7$ and be careful with the negative signs when you substitute.

In Desmos, type y = -0.2x^2 + 3.2x - 7 to graph the profit function, where $x$ represents the subscription fee.

Tap or click on the highest point of the parabola; Desmos will label the vertex. The $x$-coordinate of this vertex is the subscription fee that maximizes the profit. Compare that $x$-value to the answer choices.

Define the function as f(x) = -0.2x^2 + 3.2x - 7, then enter f(4), f(8), f(12), and f(16) in separate lines. The largest output among these corresponds to the correct subscription fee.

The profit is modeled by a quadratic function of $n$:

Because the coefficient of $n^2$ is negative ($-0.2$), the graph is a parabola that opens downward, so it has a maximum point at its vertex. The question is asking for the $n$-value (the subscription fee) at that maximum point.

Compare the function to the standard form $an^2 + bn + c$:

• $a = -0.2$
• $b = 3.2$
• $c = -7$

We will use $a$ and $b$ to find the $n$-coordinate of the vertex.

For a quadratic function $an^2 + bn + c$, the $x$-coordinate (here, the $n$-coordinate) of the vertex is given by

Substitute $a = -0.2$ and $b = 3.2$:

This expression gives the subscription fee at which the profit is maximized; we just need to simplify it.

Now simplify the fraction:

So the subscription fee that maximizes the magazine's monthly profit is $8$ dollars, which corresponds to answer choice B) 8.