Question

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

$x={(y+2)}^2+1$

Short answer

The graph of the equation is

Step-by-step solution

Step 1. Given Information

Given equation of the parabola is$x={(y+2)}^2+1$

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 $x={(y+2)}^2+1$

Here role="math" localid="1645422325977" $a=1,k=-2,h=1$

Since a is positive, the parabola opens to the right.

Step 3. Axis of symmetry

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

Plugging the values:

role="math" localid="1645422310707" $y=-2$

Hence the axis of symmetry isrole="math" localid="1645422316314" $y=-2$

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="1645422338252" $(1,-2)$

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:

$$\begin{gathered} x={(y+2)}^2+1 \\ x={(0+2)}^2+1 \\ x=4+1 \\ x=5 \end{gathered}$$

Hence the x-intercept is $(5,0)$

y-intercept:

Plugging $x=0$in the equation:

$$\begin{gathered} x={(y+2)}^2+1 \\ 0={(y+2)}^2+1 \\ {(y+2)}^2=-1 \end{gathered}$$

Square can not be negative. Hence y has no intercept.

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 $y=1$:

$$\begin{gathered} x={(y+2)}^2+1 \\ x={(1+2)}^2+1 \\ x={\left(3\right)}^2+1 \\ x=9+1 \\ x=10 \end{gathered}$$

When $y=-1$:

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

Hence the points are$(10,1),(2,-1)$

Step 7. Graphing the parabola

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

Conclusion

The graph of the parabola is