Question

question 6(multiple choice worth 1 points) (08.02 mc) 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. determine the vertex, and explain what it means in the context of the problem. (12, 16); the vertex represents the maximum profit. (12, 16); the vertex represents the minimum profit. (14, 4); the vertex represents the maximum profit. (14, 4); the vertex represents the minimum profit.

Explanation:

Step1: Identify coefficients

For the quadratic function $f(x)=-x^{2}+28x - 192$, $a=-1$, $b = 28$, $c=-192$.

Step2: Find x - coordinate of vertex

Use the formula $x=-\frac{b}{2a}$. Substitute $a=-1$ and $b = 28$: $x=-\frac{28}{2\times(-1)}=\frac{-28}{-2}=14$.

Step3: Find y - coordinate of vertex

Substitute $x = 14$ into the function $f(x)=-x^{2}+28x - 192$. So $f(14)=-(14)^{2}+28\times14 - 192=-196 + 392-192 = 4$.

Step4: Determine maximum or minimum

Since $a=-1<0$, the parabola opens downwards, so the vertex represents the maximum profit.

Answer:

C. (14, 4); The vertex represents the maximum profit.