Chapter 1.6, Problem 63E

Solution

To determine the transformation that translates $f\left(x\right)$ vertically upward by $2$ units and then reflecting the resultant function across the $y$ -axis.

The transformation that translates $f\left(x\right)$ vertically upward by $2$ units and then reflecting the resultant function across the $y$ -axis:

  $y=f\left(-x\right)+2$ .

Concepts Used:

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

Vertical translation of a function:

The translation $y=f\left(x\right)+k$ translates $y=f\left(x\right)$ vertically upwards by $k$ units if $k>0$ and it translates $y=f\left(x\right)$ vertically downwards by $k$ units if $k<0$ .

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

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

Finding the transformation:

Observe that the transformation $y=f\left(x\right)+2$ translates $y=f\left(x\right)$ vertically upward by $2$ units.

Now, observe that the transformation $y=f\left(-x\right)+2$ reflects $y=f\left(x\right)+2$ about the $y$ -axis.

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

Conclusion:

The transformation that translates $f\left(x\right)$ vertically upward by $2$ units and then reflecting the resultant function across the $y$ -axis:

  $y=f\left(-x\right)+2$ .