Question

In the following exercise a) write the equation in standard form b) use properties of standard form to plot the graph:$x=4y^2+8y$

Short answer

The equation can be given as$x=4{(y+1)}^2-4$

The graph can be given as

Step-by-step solution

Given information

We are given a equation$x=4y^2+8y$

Write the equation in standard form 

We get

$$\begin{gathered} x=4y^2+8y \\ x=4(y^2+2y) \\ x=4(y^2+2y+1)-4 \\ x=4{(y+1)}^2-4 \end{gathered}$$

on comparing with standard equation we getrole="math" localid="1645507741742" $a=4,h=-4,k=1$

 find the axis of symmetry and the vertex of parabola

The axis of symmetry can be given as $y=k$

we get,role="math" localid="1645507944970" $y=-1$

And the vertex can be given as role="math" localid="1645507953505" $(h,k)=(-4,-1)$

Find the x-intercept 

Put $y=0$we get,

role="math" localid="1645507351999" $$\begin{gathered} x=4{(0-1)}^2-4 \\ x=0 \end{gathered}$$

Find the point symmetric to (0,0) across the axis of symmetry

The point can be given as $(0,-2)$

Find y-intercept

Put $x=0$

$$\begin{gathered} x=4y^2+8y \\ 4y^2+8y=0 \\ {y}^2+2y=0 \\ y(y+2)=0 \\ y=0andy=-2 \end{gathered}$$

The coordinates are$(0,0),(0,-2)$

plot the graph

The graph can be given as

Conclusion

The equation can be given as$x={(y+1)}^2-4$

The graph can be given as