an engineer sketches a design for a flashlight that uses a mirror in the shape of a parabola to maximize the…

Explanation:

Step1: Recall vertex - form of parabola

The vertex - form of a parabola is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola.

Step2: Identify the vertex of the parabola

From the graph, the vertex of the parabola is at the point $(6,2)$. So, $h = 6$ and $k = 2$.

Step3: Determine the value of $a$

Since the parabola opens upwards, $a>0$.

Step4: Check each option

• For $f(x)=\frac{1}{2}(x - 6)^2+2$, $h = 6$, $k = 2$ and $a=\frac{1}{2}>0$.
• For $f(x)=-\frac{1}{2}(x - 0)^2+20$, the vertex is $(0,20)$ and $a=-\frac{1}{2}<0$.
• For $f(x)=3(x - 6)^2+2$, $h = 6$, $k = 2$ and $a = 3>0$, but the parabola seems to be less steep than the one with $a = 3$.
• For $f(x)=-3(x - 0)^2+20$, the vertex is $(0,20)$ and $a=-3<0$.

Answer:

$f(x)=\frac{1}{2}(x - 6)^2+2$