Q. 3 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

The required graph is 

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 $h and k$.

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

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