Chapter 1.6, Problem 53E

Solution

The graph of $y=f\left(2x\right)$ using the graph of $y=f\left(x\right)$ .

The graph of $y=f\left(2x\right)$ obtain by transformation of a horizontal shrink by factor $1/2$ from original graph of $y=f\left(x\right)$ .

Given:

The graph of $y=f\left(x\right)$ is:

1

Concept Used:

The graph of $y=f\left(x\right)$ is translated or shifted horizontally left or right according as:

1) If $y=f\left(x-c\right)$ , then graph shifted by c units right from $y=f\left(x\right)$ .

2) If $y=f\left(x+c\right)$ , then graph shifted by c units left from $y=f\left(x\right)$ .

And, the graph of $y=f\left(x\right)$ is translated or shifted vertically up or down according as:

1) If $y=f\left(x\right)+c$ , then graph shifted by c units up from $y=f\left(x\right)$ .

2) If $y=f\left(x\right)-c$ , then graph shifted by c units down from $y=f\left(x\right)$ .

The transformation of horizontal stretches or shrinks from $y=f\left(x\right)$ is as follows:

The graph of $y=f\left(\frac{x}{c}\right)$ transform to horizontal stretches by factor c , when $c>1$ and horizontal shrinks by factor c when $c<1$ .

The transformation of vertical stretches or shrinks from $y=f\left(x\right)$ is as follows:

The graph of $y=c\cdot f\left(x\right)$ transform to vertical stretches by factor c , when $c>1$ and shrinks by factor c when $c<1$ .

The graph of $y=-f\left(x\right)$ is the reflection of graph of $y=f\left(x\right)$ across x -axis.

Calculation:

The graph of $y=f\left(2x\right)$ can be obtained from $y=f\left(x\right)$ by horizontal shrinks by factor $1/2$ , since, $y=f\left(2x\right)\text{ or }f\left(\frac{x}{1/2}\right)$ shrinks the graph of $y=f\left(x\right)$ horizontally by factor $1/2$ .

The graph is formed as follows:

  

Hence the graph of $y=f\left(2x\right)$ is obtained by transformation of $y=f\left(x\right)$ .

Conclusion The graph of $y=f\left(2x\right)$ obtain by transformation of a horizontal shrink by factor $1/2$ from original graph of $y=f\left(x\right)$ .