Exponential Graphs 📈

What is an exponential function?

An exponential function is a function used to show exponential growth and decay.

Exponential Form

For exponential growth → y=a^x, where a is a number greater than 0, and x is what is being raised to
For exponential decay → y=a^x, but where a is a number less than 0.

How to Graph Exponential Functions

A factor that differentiates exponentials from other functions is its asymptote.
- An asymptote shows where the function will stop and not go any lower
To find the asymptote, simply look at the y intercept of the function
- Ex.) y=2x+2. Similarly in a y=mx + b equation, the y intercept can be found simply by looking at the equation, or setting x=0. In both cases, the y intercept and the asymptote would be 2
As you can see by the graph, the graph never goes below y=2

For exponential growth functions, the asymptote is always horizontal
- Graphing for exponential growth functions work much the same, but the asymptote is vertical rather than horizontal.

Transformations

Just like other functions and graphs, exponentials may have transformations. However, the rules stay the same.
Shift across horizontal axis → y=axb
- If it is x+b, it is to the right. Is it is x-b, it is a shift to the left
Shift across vertical axis y=2x+ c
- If it is positive c, the graph will go up. If it is negative c, the graph will go down
Flipped on y axis → y=2-x+2
Flipped on x axis→ y= -2x+2

Practice Problems

Given the following exponential equations, sketch the graph.

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