Chapter 13, Problem 3CR

To determine

To graph: To graph the function $f\left(x\right)=2{\left(x+1\right)}^2-4$ by using transformations.

Explanation of Solution

Given information:

The function $f\left(x\right)=2{\left(x+1\right)}^2-4$.

Graph:

To get the function $f\left(x\right)=2{\left(x+1\right)}^2-4$, begin with the graph of the function $y=2{\left(x+1\right)}^2$.

The parent function is $f\left(x\right)=x^2$.

To get the function $y=f\left(x\right)$ shifted left by $k$ units, replace $x$ by $x+k$ in $y=f\left(x\right)$.

To get the function $f\left(x\right)={\left(x+1\right)}^2$, we shift $f\left(x\right)=x^2$ left by 1 unit.

To get the function $y=f\left(x\right)$ stretched vertically by factor $c>1$, multiply $f\left(x\right)$ by $c$.

To get the function $f\left(x\right)={\left(x+1\right)}^2$ stretched vertically by factor $2>1$, multiply $f\left(x\right)$ by $2$.

The function becomes $f\left(x\right)=2{\left(x+1\right)}^2$.

To get the function $y=f\left(x\right)$ shifted down by $h$ units, subtract $h$ from $f\left(x\right)$.

To get the function $f\left(x\right)=2{\left(x+1\right)}^2$ shifted down by $4$ units, subtract $4$ from $f\left(x\right)=2{\left(x+1\right)}^2$.

Therefore, the graph of the function $f\left(x\right)=2{\left(x+1\right)}^2-4$ is:

Interpretation:

The above graph represents the function $f\left(x\right)=2{\left(x+1\right)}^2-4$ obtained by shifting graph of $f(x)=x^2$ to the left by $1$ unit, stretched vertically by factor $2$, and shifting down by $4$ units.