Question

The function f(x) = –x2 + 28x – 192 models the hourly profit, in dollars, a shop makes for selling sodas, where x is the number of sodas 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? (5 points) Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)

Answer 1

a)


`\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{llccll} y = &{{ -1}}x^2&{{ +28}}x&{{ -192}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)`

check the picture below.

b)


`\bf f(x)=-x^2+28x-192\implies 0=-x^2+28x-192 \\\\\\ x^2-28x+192=0\implies (x-16)(x-12)=0\implies x= \begin{cases} 16\\ 12 \end{cases}`

what does this mean? check the picture below, notice the x-intercepts points.