Question

Write an equation for each parabola.

Short answer

The equation of the parabola is$2x^2-4x-y+1=0$.

Step-by-step solution

Step 1.  Given information .

Consider the given graph and the points located on it .

Step 2.  Formula used .

The formula used to find the equation of parabola is${\left(x-h\right)}^2=4a\left(y-k\right)$

where h , k are the points of vertex $\left(h,k\right)=\left(1,-1\right)$and focus is $\left(x,y\right)=\left(0,1\right)$.

Step 3.  Find the equation of parabola .

Substitute the value of $\left(h,k\right)=\left(1,-1\right)$and $\left(x,y\right)=\left(0,1\right)$in the equation of parabola .

$$\begin{gathered} {\left(x-h\right)}^2=4a\left(y-k\right) \\ {\left(0-1\right)}^2=4a\left(1+1\right) \\ a=\frac{1}{8} \end{gathered}$$

Further simplify.

Substitute the value of a in the equation of parabola .

localid="1646929258883" $$\begin{gathered} {\left(x-h\right)}^2=4a\left(y-k\right) \\ {\left(x-1\right)}^2=4\times \frac{1}{8}\left(y+1\right) \\ 2x^2-4x-y+1=0 \end{gathered}$$