Chapter 1.6, Problem 64E

Solution

To determine the transformation that reflects $f\left(x\right)$ about the $x$ -axis and then shrinking the resultant function by a factor of $\frac{1}{2}$ , horizontally.

The transformation that reflects $f\left(x\right)$ about the $x$ -axis and then shrinking the resultant function horizontally by a factor of $\frac{1}{2}$ :

  $y=-f\left(2x\right)$

Concepts Used:

Given a function $y=f\left(x\right)$ .

Horizontal shrink of a function:

The transformation $y=f\left(ax\right)$ shrinks $y=f\left(x\right)$ horizontally by a factor of $\frac{1}{a}$ when $a>1$ .

Reflection of a function across the $x$-axis:

The transformation $y=-f\left(x\right)$ reflects $y=f\left(x\right)$ about the $x$ -axis.

Finding the transformation:

Observe that the transformation $y=-f\left(x\right)$ reflects $y=f\left(x\right)$ about the $x$ -axis.

Now, observe that the transformation $y=-f\left(2x\right)$ shrinks $y=-f\left(x\right)$ horizontally by a factor of $\frac{1}{2}$ .

Thus, the resultant transformation is $y=-f\left(2x\right)$ .

Conclusion:

The transformation that reflects $f\left(x\right)$ about the $x$ -axis and then shrinking the resultant function horizontally by a factor of $\frac{1}{2}$ :

  $y=-f\left(2x\right)$