Question

Graph $f\left(x\right)={\left(x-3\right)}^2-2$using transformations.

Short answer

The required graph is

Step-by-step solution

Step 1. Given information.

Graph $f\left(x\right)={\left(x-3\right)}^2-2$using transformations.

Step 2. Determine the horizontal and vertical shift. 

Compare the function with $f\left(x\right)=a{\left(x-h\right)}^2+k$and determine $handk$.

$$\begin{gathered} f\left(x\right)={\left(x-3\right)}^2-2 \\ f\left(x\right)=a{\left(x-h\right)}^2+k \\ So,h=3 \\ k=-2 \\ a=1 \end{gathered}$$

Step 3. plot the graph of the parent function. 

Plot the graph of $f\left(x\right)=x^2$.

Step 4. Horizontal shift of the graph.

Shift the graph of $f\left(x\right)=x^2$to 3 units rights to plot $f\left(x\right)={\left(x-3\right)}^2$.

Step 5. Vertical shift of the graph. 

Shift the graph of $f\left(x\right)={\left(x-3\right)}^2$to 2 units down to plot $f\left(x\right)={\left(x-3\right)}^2-2$.