Chapter 1.6, Problem 43E

Solution

To Explain: The given basic graph and a sequence of transformation which used to produce given graph function.

Given:

  $y=2{\left(x-3\right)}^2-4$

Graph:

  

Interpretation:

The parent function is the simplest form of the type of function given.

  $y=x^2$

Simplify the term $2{\left(x-3\right)}^2-4$ :

  $\begin{array}{l}y=2x^2-12x+18-4\\ y=2x^2-12x+14\end{array}$

Assume that $y=x^2$ is $f\left(x\right)=x^2$ and $y=2{\left(x-3\right)}^2-4$ is $g\left(x\right)=2x^2-12x+14$ :

  $f\left(x\right)=x^2$

  $g\left(x\right)=2x^2-12x+14$

The transformation being described is from $f\left(x\right)=x^2$ to $g\left(x\right)=2x^2-12x+14$

  $f\left(x\right)=x^2\to g\left(x\right)=2x^2-12x+14$

Find the vertex form of $g\left(x\right)=2x^2-12x+14$ :

  $y=2{(x-3)}^2-4$

Compare and list the transformations:

Parent Function: $y=x^2$

Horizontal Shift: Right $3$ Units

Vertical Shift: Down $4$ Units

Reflection about the $x-$ axis: None

Reflection about the $y-$ axis: None

Vertical Compression or Stretch: Stretched by factor of $2$