Transformations of Exponential Functions

By S Taylor

The basic graph of an exponential function in the form

(where a is positive) looks like

For a review of basic features of an exponential graph, click here. But what would happen if our function was changed slightly? Suppose we have the function . The calculator shows us the following graph for this function.

The graph of our exponential has been moved up three spaces.

Summary: In general, when we have an exponential in the form

then the graph will be moved up or down k units.

What happens if we make a change to the exponent in our exponential function? Look at the graph of .

This graph has been shifted to the left 2 spaces.

Summary: A left or right shift is what happens when we make a change to the exponent. In general, if we have the function

then the graph will be moved left c units if c is positive and right c units if c is negative. If a negative is placed in front of an exponential function, then it will be reflected over the x-axis.

These are the same rules discussed for transforming quadratic graphs, they just look a little different when applied to exponential functions. But the effect is still the same.

To practice with an EXCEL model of these graphs, click here.

Use the rules of moving graphs left, right, up, and down to make a conjecture about what the graph of each function will look like. Then, graph each function.