Question

Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.

Short answer

The equation is$y=-\frac{1}{20}{(x-20)}^2+20$

Step-by-step solution

Given information

Given the base is$40ft$wide and$20ft$high

Calculate the vertex 

Let the lower left side of the bridge be the origin of the coordinate grid at the point

Since the base is $40$feet wide the point $(40,0)$represents the lower right side.

The bridge is $20$feet high at the highest point. The highest point is the vertex of the parabola so the $y$-coordinate of the vertex will be $20$

So the vertex is$(20,20)$

.

Substitute in standard form and form the equation

Substituting, we get

$$\begin{gathered} y=a{(x-h)}^2+k \\ y=a{(x-20)}^2+20 \\ 0=a{(0-20)}^2+20 \\ 400a=-20 \\ a=-\frac{1}{20} \end{gathered}$$

The equation is$y=-\frac{1}{20}{(x-20)}^2+10$