Question

an engineer is designing a parabolic dish to receive radio signals. she graphs her design on of x, the diameter in inches. which equation represents this graph?

Explanation:

Step1: Recall parabola vertex - form

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

From the graph, the vertex of the parabola is $(20,- 20)$. So, $h = 20$ and $k=-20$, and the equation becomes $y=a(x - 20)^2-20$.

Step3: Find the value of $a$

The parabola passes through the point $(0,30)$. Substitute $x = 0$ and $y = 30$ into the equation $y=a(x - 20)^2-20$:
\[

$$\begin{align*} 30&=a(0 - 20)^2-20\\ 30&=400a-20\\ 400a&=50\\ a&=\frac{50}{400}=\frac{1}{8} \end{align*}$$

\]

Step4: Write the final equation

The equation of the parabola is $y=\frac{1}{8}(x - 20)^2-20$.

Answer:

$y=\frac{1}{8}(x - 20)^2-20$