Chapter 1.6, Problem 14E

Solution

To Describe:

The series of transformations that will change graph $y=x^3$ into the graph of given equation.

  $y={\left(2x\right)}^3$ is the graph of $y=x^3$ squeezed by a factor of two along the horizontal direction.

Given:

  $y={\left(2x\right)}^3$ .

Concepts Used:

The graph of the $y=f\left(ax\right);a>1$ is the graph of the equation $y=f\left(x\right)$ squeezed horizontally by a factor $a$ .

Calculations:

Rewrite $y={\left(2x\right)}^3$ in terms of the function $f\left(x\right)=x^3$ .

  $\begin{array}{c}y={\left(2x\right)}^3\\ =f\left(2x\right)\end{array}$

  $y=f\left(2x\right)$ is the graph of $y=f\left(x\right)$ squeezed horizontally by a factor $2$ .

Conclusion:

  $y={\left(2x\right)}^3$ is the graph of $y=x^3$ squeezed by a factor of two along the horizontal direction.