Question

In the following exercise a) write the equation in standard form b) use properties of standard form to plot the graph:$x=y^2+4y+5$

Short answer

The equation in standard form can be given as$x={(y+2)}^2+1$

The graph of the equation can be given as

Step-by-step solution

Given information

We are given an equation$x=y^2+4y+5$

Step 2:Write parabola in standard form

We have$$\begin{gathered} x=y^2+4y+5 \\ x=y^2+4y+4+1 \\ x={(y+2)}^2+1 \end{gathered}$$

On comparing with standard form we get$a=1,k=-2,h=1$

Find the axis of symmetry and vertex

To find the axis of symmetry we have $y=k$

we get, $y=-2$

The vertex can be given asrole="math" localid="1645509461688" $(h,k)=(1,-2)$

Find x-intercept

To find put $y=0$

We get,

$x=5$

The coordinate is$(5,0)$

Find y-intercept

To find y-intercept put $x=0$

$y^2+4y+5=0$

The roots of the above equation are imaginary hence the equation does not have a y-intercept

Plot the graph

The graph can be given as

Conclusion

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

The graph can be given as