To easily understand how to graph exponential functions using transformations, let’s break it down. We first need to know the basic form of these functions and how changing them changes the graph.

The basic exponential function looks like this:

f(x) = a * b^x

Here, a is the starting value and b is the base of the exponent.

Let’s look at three ways we can change the graph:

1. Vertical Stretch/Compression:
   If you change a, the graph can stretch or compress.

   • If |a| > 1, the graph stretches.
   • If 0 < |a| < 1, it compresses.

   For example:

   • In the equation f(x) = 2 * 3^x, the graph stretches by 2 times compared to f(x) = 3^x.

2. Horizontal Shifts:
   If you change the exponent part like this: f(x) = a * b^(x-h), it shifts the graph to the right. The h tells us how much it moves.

   For example:

   • In the equation f(x) = 3^(x-2), this shifts the graph of 3^x two units to the right.

3. Vertical Shifts:
   Adding a number k to the equation moves the graph up or down. The new form looks like this: f(x) = a * b^x + k.

   For example:

   • In the equation f(x) = 3^x + 1, this raises the graph of 3^x by 1 unit.

Visualizing these changes can help you understand better, so try sketching out your graphs step-by-step!