Transformations of Exponential Functions

We want to graph the given function $g(x)=\text{-}{2}^{x-3}$ as a transformation of the function $f(x)=2^x.$ $$\begin{array}{c}g(x)=\text{-}{2}^{x-3}\end{array}$$ Looking at the function, we can see that it has been reflected in the $x\text{-}$axis and translated $3$ units to the right. Let's show the transformations one at a time.

Reflection

Let's start by considering the parent function $f(x)=2^x.$ If we perform a reflection across the $x\text{-}$axis, the resulting function is $y=\text{-}{2}^x.$

Horizontal Translation

Now, we need to consider the function $g(x)=\text{-}{2}^{x-3}.$ This is a horizontal translation of $y=\text{-}{2}^x$ to the right by $3$ units.