Question

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

$x=-y^2-4y+5$

Short answer

The graph of the equation is

Step-by-step solution

Step 1. Given Information

Given equation of the parabola is $x=-y^2-4y+5$

The graph of the parabola is to be determined.

Step 2. Equation of the parabola

The equation of a parabola in general form is $x=a{y}^2+by+c$
The given equation is $x=-y^2-4y+5$

Here $a=-1,b=-4,c=5$

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

Step 3. Axis of symmetry

The axis of symmetry of the parabola $x=a{y}^2+by+c$is$y=-\frac{b}{2a}$

Plugging the values:

$$\begin{gathered} y=-\frac{b}{2a} \\ y=-\frac{(-4)}{2(-1)} \\ y=-\left(\frac{-4}{-2}\right) \\ y=-\left(2\right) \\ y=-2 \end{gathered}$$

Hence the axis of symmetry is$y=-2$

Step 4. Vertex of a parabola

The vertex of the parabola $x=a{y}^2+by+c$can be determined by plugging and solving for x.

Plugging $y=-2$in the given equation:

localid="1645290306471" $$\begin{gathered} x=-y^2-4y+5 \\ x=-{(-2)}^2-4(-2)+5 \\ x=-\left(4\right)+8+5 \\ x=9 \end{gathered}$$

The vertex of parabola is localid="1645290322220" $(9,-2)$

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:

$$\begin{gathered} x=-y^2-4y+5 \\ x=-{\left(0\right)}^2-4\left(0\right)+5 \\ x=-0-0+5 \\ x=5 \end{gathered}$$

Hence the x-intercept is localid="1645290357954" $(5,0)$

y-intercept:

Plugging $x=0$in the equation:

$$\begin{gathered} x=-y^2-4y+5 \\ 0=-y^2-4y+5 \\ {y}^2+4y-5=0 \end{gathered}$$

The quadratic equation has to be solved to get y values:

localid="1645259649922" $$\begin{gathered} y^2+4y-5=0 \\ {y}^2-1y+5y-5=0 \\ y(y-1)+5(y-1)=0 \\ (y-1)(y+5)=0 \\ (y-1)=0,(y+5)=0 \\ y=1,y=-5 \end{gathered}$$

Hence the y-intercepts are $(0,1),(0,-5)$

Step 6. Graphing the parabola

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

Conclusion

The graph of the parabola is