Question

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

$x=-{(y+4)}^2+3$

Short answer

The graph of the equation is

Step-by-step solution

Step 1. Given Information

Given equation of the parabola is$x=-{(y+4)}^2+3$

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$x=-{(y+4)}^2+3$

Here$a=-1,k=-4,h=3$

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:

$y=-4$

Hence the axis of symmetry is$y=-4$

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" $(3,-4)$

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:

$$\begin{gathered} x=-{(y+4)}^2+3 \\ x=-{(0+4)}^2+3 \\ x=-{\left(4\right)}^2+3 \\ x=-16+3 \\ x=-13 \end{gathered}$$

Hence the x-intercept is $(-13,0)$

y-intercept:

Plugging $x=0$in the equation:

role="math" localid="1645430029824" $$\begin{gathered} x=-{(y+4)}^2+3 \\ 0=-{(y+4)}^2+3 \\ -3=-{(y+4)}^2 \\ {(y+4)}^2=3 \\ y+4=\pm \sqrt{3} \\ y=\pm \sqrt{3}-4 \end{gathered}$$

Hence y-intercepts are $(0,\sqrt{3}-4),(0,-\sqrt{3}-4)$

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 $y=1$:

$$\begin{gathered} x=-{(y+4)}^2+3 \\ x=-{(1+4)}^2+3 \\ x=-{\left(5\right)}^2+3 \\ x=-25+3 \\ x=-22 \end{gathered}$$

When $y=-1$:

$$\begin{gathered} x=-{(y+4)}^2+3 \\ x=-{(-1+4)}^2+3 \\ x=-{\left(3\right)}^2+3 \\ x=-9+3 \\ x=-6 \end{gathered}$$

Hence the points are $(-22,1),(-6,-1)$

Step 7. Graphing the parabola

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

Conclusion

The graph of the parabola is