Chapter 6.4, Problem 37E

To determine

Write a rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)=e^x$ : horizontal shrink by a factor of $\frac{1}{2}$ , followed by a translation $5$ units up.

Answer to Problem 37E

The rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)=e^x$ is $g\left(x\right)=e^{2x}+5$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=e^x$ .

Calculation:

Consider the given function

  $f\left(x\right)=e^x$ .

First, perform horizontal shrink by a factor of $\frac{1}{2}$ .

For horizontal shrink, multiply $x$ value by the given unit.

[Hint: For performing shrink the function $f\left(x\right)$ by a unit of $\frac{1}{d}$ then, the function becomes $f\left(d\cdot x\right)$ .]

Therefore, after the shrink the function becomes

  $f\left(x\right)=f\left(2x\right)$

  $f\left(x\right)=e^{2x}$.

After that, translate the graph upwardby5 units.

For the translation upward, add units to the function $f$ .

[Hint: Fortranslating $f\left(x\right)$ by $d$ units, the function becomes, $f\left(x\right)+d$ ]

Therefore after the translation, the function becomes

  $f\left(x\right)=e^{2x}+5$.

Hence, the rule for $g$ that represents the indicated transformation of the graph $f$ is $g\left(x\right)=e^{2x}+5$.