Question

Use transformations to graph the function$y={(x-3)}^2+2$.

Short answer

The graph of the function is:

Step-by-step solution

Step 1. Given Information

The given function is $y={(x-3)}^2+2$and we need to use transformations to graph the function.

Step 2. Simplifying the equation

The equation can be written as:

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

Using the transformations we get,

$$\begin{gathered} X\equiv (x-3) \\ Y\equiv (y-2) \\ \therefore Y=X^2 \end{gathered}$$

Step 3. Finding the vertex 

Comparing the equation with the standard formula of parabola $x^2=4ay$we get,

Vertex of the parabola $Y=X^2$is $X=0,Y=0$.

$$\begin{gathered} \therefore x-3=0 \\ \Rightarrow x=3 \\ y-2=0 \\ \Rightarrow y=2 \end{gathered}$$

Vertex of the parabola is$(3,2)$.

Step 4. Finding focus of parabola

Focus of the standard parabola is $(0,a)$.

$$\begin{gathered} \therefore 4a=1 \\ \Rightarrow a=\frac{1}{4} \end{gathered}$$

Transforming we get,

$$\begin{gathered} x-3=0 \\ \Rightarrow x=3 \\ y-2=\frac{1}{4} \\ \Rightarrow y=2+\frac{1}{4} \\ \Rightarrow y=\frac{9}{4} \end{gathered}$$

The focus of the parabola is$\left(3,\frac{9}{4}\right)$.

Step 5. Graphing the function

The graph of the function is: