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$y=-\frac{1}{5}{(x-5)}^2+5$

Step-by-step solution

Given information

Given the base is $10ft$ wide and $5ft$ 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 $10$feet wide the point $(10,0)$represents the lower right side.

The bridge is $5$feet high at the highest point. The highest point is the vertex of the parabola so the $y$coordinate of the vertex will be $5$

So the vertex is$(5,5)$

Step 3:Substitute in standard form and form the equation

Calculating, we get

$$\begin{gathered} y=a(x-h{)}^2+k \\ y=a{(x-5)}^2+5 \\ 0=a{(0-5)}^2+5 \\ 25a=-5 \\ a=-\frac{1}{5} \end{gathered}$$

The equation is$y=-\frac{1}{5}{(x-5)}^2+5$