Transformations of Exponential Functions

We want to determine how the graph of the given function compares to the graph of the parent function $y=b^x.$ Notice that in our case $b=2,$ so we're looking for transformation(s) of the function $y=2^x.$ Now, let's consider the given function. $$\begin{array}{c}y=3(2{)}^{x-1}+4\end{array}$$ Looking at the function, we can see that it has been stretched by a factor of $3,$ translated $1$ unit to the right, and translated $4$ units up. Let's show these transformations one at a time.

Stretch

Let's start by considering the function $y=2^x.$ If we multiply $2^x$ by $3,$ we obtain a vertical stretch by a factor of $3.$ The resulting function is $y=3(2{)}^x.$

Horizontal Translation

Now, we need to consider the function $y=3(2{)}^{x-1}.$ This is a horizontal translation of $y=3(2{)}^x$ to the right by $1$ unit.

Vertical Translation

Finally, we will consider the function $y=3(2{)}^{x-1}+4.$ This is a vertical translation of $y=3(2{)}^{x-1}$ up by $4$ units.