Question

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

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

Short answer

The graph of the equation is

Step-by-step solution

Step 1. Given Information

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

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

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

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:

$y=1$

Hence the axis of symmetry is$y=1$

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 is$(2,1)$

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="1645360123002" $$\begin{gathered} x=-{(y-1)}^2+2 \\ x=-{(0-1)}^2+2 \\ x=-{(-1)}^2+2 \\ x=-1+2 \\ x=1 \end{gathered}$$

Hence the x-intercept is $(1,0)$

y-intercept:

Plugging $x=0$in the equation:

localid="1645420321689" $$\begin{gathered} x=-{(y-1)}^2+2 \\ 0=-{(y-1)}^2+2 \\ {(y-1)}^2=2 \\ y-1=\pm \sqrt{2} \\ y=\pm \sqrt{2}+1 \end{gathered}$$

Hence the y-intercepts are$(0,\sqrt{2}+1),(0,-\sqrt{2}+1)$

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 $y=1$:

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

When $y=-1$:

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

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

Step 7. Graphing the parabola

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

Conclusion

The graph of the parabola is