Question

In the following exercise graph each equation by using its properties:$x=-y^2+2y+4$

Short answer

The graph can be given as

Step-by-step solution

Given information

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

Check whether the parabola opens to right or left

As the coefficient of the square term is negative the parabola opens to the Left

Find the axis of symmetry

To find the axis of symmetry calculate

$$\begin{gathered} y=-\frac{b}{2a} \\ y=-\frac{2}{-2} \\ y=1 \end{gathered}$$

Find the vertex of parabola 

To find the vertex put $y=1$in the equation

we get,

$$\begin{gathered} x=-1+2+4 \\ x=5 \end{gathered}$$

The vertex of the parabola is $(5,1)$

Find y-intercept

To find x-intercept put$y=0$

we get

$$\begin{gathered} x=0+0+4 \\ x=4 \end{gathered}$$

The coordinate is$(4,0)$

Find y-intercept

Put$y=0$

we get,

$$\begin{gathered} x=0+0+4 \\ x=4 \end{gathered}$$

On solving using the quadratic formula we get $x=3.236orx=-1.236$

Hence the coordinates are

$(0,3.236),(0,-1.236)$

Plot the graph

The graph of parabola can be given as

Conclusion

The graph of the parabola is