Question

In the following exercises, graph each equation by using properties.

$x=-{(y+3)}^2+2$

Short answer

The graph of the equation is

Step-by-step solution

Step 1. Given Information

Given equation of the parabola is $x=-{(y+3)}^2+2$

The graph of the parabola is to be determined

Step 2. Equation of the parabola

The equation of a parabola in standard form $x=a{(y-k)}^2+h$

The given equation is $x=-{(y+3)}^2+2$

Here $a=-1,k=-3,h=2$

Since a is negative, the parabola opens to the left.

Step 3. Axis of symmetry

The axis of symmetry of the parabola $x=a{(y-k)}^2+h$is$y=k$

Plugging the values:

$y=-3$

Hence the axis of symmetry is localid="1645428098569" $y=-3$

Step 4. Vertex of a parabola

The vertex of the parabola $x=a{(y-k)}^2+h$is$(h,k)$

The vertex of the given parabola isrole="math" localid="1645428420463" $(2,-3)$

Step 5. Intercepts

x-intercept is the point where the graph crosses the x-axis and y is 0.

y-intercept is the point where the graph crosses the y-axis and x is 0.

x-intercept:

Plugging $y=0$in the equation:

localid="1645428593736" $$\begin{gathered} x=-{(y+3)}^2+2 \\ x=-{(0+3)}^2+2 \\ x=-{\left(3\right)}^2+2 \\ x=-9+2 \\ x=-7 \end{gathered}$$

Hence the x-intercept is $(-7,0)$

y-intercept:

Plugging $x=0$in the equation:

localid="1645428745672" $$\begin{gathered} x=-{(y+3)}^2+2 \\ 0=-{(y+3)}^2+2 \\ -{(y+3)}^2=-2 \\ {(y+3)}^2=2 \\ y+3=\pm \sqrt{2} \\ y=\pm \sqrt{2}-3 \end{gathered}$$

Hence y-intercepts are $(0,\sqrt{2}-3),(0,-\sqrt{2}-3)$

Step 6. Plotting points

Since, the vertex and intercepts are only two points, more points are needed to plot the parabola.

Hence assuming few y values and finding the corresponding x values to points can be obtained.

When $y=1$:

$$\begin{gathered} x=-{(y+3)}^2+2 \\ x=-{(1+3)}^2+2 \\ x=-{\left(4\right)}^2+2 \\ x=-16+2 \\ x=-14 \end{gathered}$$

When $y=-1$:

$$\begin{gathered} x=-{(y+3)}^2+2 \\ x=-{(-1+3)}^2+2 \\ x=-{\left(2\right)}^2+2 \\ x=-4+2 \\ x=-2 \end{gathered}$$

Hence the points are$(-14,1),(-2,-1)$

Step 7. Graphing the parabola

Using the properties, plotting the points and graphing the parabola:

Conclusion

The graph of the parabola is