Transformations of Exponential Functions

We want to graph the given function $f(x)=2^{x+2}-2$ as a transformation of the function $y=2^x.$ $$\begin{array}{c}f(x)=2^{x+2}-2\iff f(x)=2^{x-(\text{-}2)}+(\text{-}2)\end{array}$$ Looking at the equation, we can see that the function has been translated $2$ units to the left and $2$ units down. Let's show the transformations one at a time.

Horizontal Translation

Firstly, let's consider the function $y=2^{x+2}.$ This is a horizontal translation of $y=2^x$ to the left by $2$ units.

Vertical Translation

Finally, we will consider the function $f(x)=2^{x+2}-2.$ This is a vertical translation of $y=2^{x+2}$ down by $2$ units.

Therefore, we can notice that the obtained graph is shown in choice C.