*free 30 points* The function f(x) = −x2 + 24x − 80 models the hourly profit, in dollars, a shop makes for selling coffee, where x is the number of cups of coffee sold, and f(x) is the amount of profit. Part A: Determine the vertex. What does this calculation mean in the context of the problem? Part B: Determine the x-intercepts. What do these values mean in the context of the problem?

Part A: To determine the vertex of the quadratic function \( f(x) = -x^2 + 24x - 80 \), we can use the vertex formula for a quadratic function in standard form, \( f(x) = ax^2 + bx + c \), which is given by the point \((h, k)\) where \( h = -\frac{b}{2a} \) and \( k = f(h) \).

For our function, \( a = -1 \), \( b = 24 \), and \( c = -80 \). Therefore, we can compute \( h \) as follows:

\[ h = -\frac{b}{2a} = -\frac{24}{2(-1)} = -\frac{24}{-2} = 12 \]

Now we can compute \( k \) by evaluating \( f(h) \):

\[ k = f(12) = -(12)^2 + 24(12) - 80 = -144 + 288 - 80 = 64 \]

So, the vertex of the function is at the point (12, 64).

In the context of the problem, the vertex represents the maximum hourly profit the shop makes and the number of cups of coffee sold to achieve that profit. The number of cups that maximizes the profit is 12, and the maximum profit is $64.

Part B: To determine the x-intercepts (also known as the zeros of the function), we find the values of \( x \) for which \( f(x) = 0 \). So we set the quadratic equation to zero and solve for \( x \):

\[ 0 = -x^2 + 24x - 80 \]

This is a quadratic equation that we can solve by factoring, using the quadratic formula, or by completing the square. Factoring this particular equation may be challenging because the factors are not immediately obvious, so using the quadratic formula is a more straightforward approach. The quadratic formula states that for the quadratic equation \( ax^2 + bx + c = 0 \), the solutions for \( x \) are:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Applying the quadratic formula, we have:

\[ x = \frac{-24 \pm \sqrt{(24)^2 - 4(-1)(-80)}}{2(-1)} = \frac{-24 \pm \sqrt{576 - 320}}{-2} = \frac{-24 \pm \sqrt{256}}{-2} = \frac{-24 \pm 16}{-2} \]

This gives us two solutions:

\[ x_1 = \frac{-24 + 16}{-2} = \frac{-8}{-2} = 4 \] \[ x_2 = \frac{-24 - 16}{-2} = \frac{-40}{-2} = 20 \]

So, the x-intercepts are \( x = 4 \) and \( x = 20 \).

In the context of the problem, the x-intercepts represent the number of cups of coffee sold for which the shop will make zero profit. This means at 4 cups and at 20 cups, the shop isn't making any profit. Between 4 and 20 cups, the shop makes a profit, with the maximum profit occurring at the sale of 12 cups.