Chapter 5, Problem 9CR

To determine

To graph: The function $f\left(x\right)=3{\left(x+1\right)}^3-2$ using transformations.

Explanation of Solution

Given information:

The function $f\left(x\right)=3{\left(x+1\right)}^3-2$ .

Graph:

The highest degree of $x$ in $f\left(x\right)=3{\left(x+1\right)}^3-2$ is $3$ . Therefore, parent function to graph $f\left(x\right)$ is $y=x^3$ .

Graph of function $y=x^3$ is:

  

The first transformation is $y={\left(x+1\right)}^3$ . This is the horizontal shifting of graph $y=x^3$ to the left by $1$ unit. So, every point on the graph of $y=x^3$ will shift to the left by $1$ unit. Every point $\left(x,y\right)$

  $y=x^3$ changes to $\left(x-1,y\right)$ . So, point $\left(0,0\right)$ shifts to $\left(-1,0\right)$ .

So, graph of $y={\left(x+1\right)}^3$ would be:

  

Next transformation is $y=3{\left(x+1\right)}^3$ . Since $a=3>1$ , it is vertical stretching of graph $y={\left(x+1\right)}^3$ to get graph of $y={\left(x+1\right)}^3$ . So, every point $\left(x,y\right)$ on $y={\left(x+1\right)}^3$ changes to $\left(x,3y\right)$ . Point $\left(0,1\right)$ will changes to $\left(0,3\right)$ , and $\left(-1,0\right)$ stays as it is.

So, graph of $y=3{\left(x+1\right)}^3$ would be:

  

Next transformation is $y=3{\left(x+1\right)}^3-2$ . It is vertical shifting of graph $y=3{\left(x+1\right)}^3$ down by $2$ units. So every point $\left(x,y\right)$ on $y=3{\left(x+1\right)}^3$ changes to $\left(x,y-2\right)$ . Point $\left(-1,0\right)$ will change to $\left(-1,-2\right)$ , and $\left(0,3\right)$ will change to $\left(0,1\right)$ .

Therefore, the graph of $y=3{\left(x+1\right)}^3-2$ is:

  

Interpretation:

The graph represents the function $y=3{\left(x+1\right)}^3-2$ using transformations.