Question

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

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

Short answer

The graph of the equation is

Step-by-step solution

Step 1. Given Information

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

The graph of the parabola is to be determined.

Step 2. Equation of the parabola

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

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

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

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

Step 3. Axis of symmetry

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

Plugging the values:

$y=2$

Hence the axis of symmetry is$y=2$

Step 4. Vertex of a parabola

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

The vertex of the given parabola is$\left(3,2\right)$

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:

$$\begin{gathered} x={\left(y-2\right)}^2+3 \\ x={\left(0-2\right)}^2+3 \\ x={\left(-2\right)}^2+3 \\ x=4+3 \\ x=7 \end{gathered}$$

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

y-intercept:

Plugging $x=0$in the equation:

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

The value of a square can never be negative. Hence there is no y-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 $y=1$:

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

When $y=-1$:

$$\begin{gathered} x={\left(y-2\right)}^2+3 \\ x={\left(-1-2\right)}^2+3 \\ x={\left(-3\right)}^2+3 \\ x=9+3 \\ x=12 \end{gathered}$$

Hence the points are$\left(12,-1\right),\left(4,1\right)$

Step 7. Graphing the parabola

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

Conclusion

The graph of the parabola is