To graph these functions we will note that they all go through the points and where is the base of that exponential.

Notice how a larger base corresponds to a steeper curve.

Part A

Rearranging the equation gives

Using the graph we only have the transformation , which means we shift the graph down by 3 units.

Part B

Using the graph we only have the transformation which shifts the graph to the right by 2 units.

Part C

Rearranging the equation gives:

Using the graph we have two transformations, which shifts the graph to the left by 1 unit and which shifts the graph up by 2 units.

Rearranging the equation gives:

Using the graph this gives us the transforms and . If we track the points (0, 0) and (1, 2):

We can also see in our equation that approaches 0 as x approaches negative inifinity - so f(x) approaches 4 as x approaches negative infinity. This gives us the following graph.

Rearranging the equation gives:

Using the graph this gives us the transforms and . If we track the points (0, 0) and (1, e):

We can also see in our equation that approaches 0 as x approaches negative positive - so f(x) approaches 6 as x approaches positive infinity. This gives us the following graph.

Part A

We just need to replace in our equation with , hence:

Part B and C

Part D Clearly, approaches negative infinity in both directions. Therefore, must approach 0 in both directions.

Part E

Rearranging the equation gives:

Using the graph this gives us the transforms and . If we track the points (0, 0) and (1, e):

We can also see in our equation that approaches 0 as x approaches positive inifinity - so f(x) approaches 4 as x approaches positive infinity. This gives us the following graph.