Topic: Exponential and Logarithmic Functions

Subtopic: Exponential Functions & Graphs

**Overview**

We have studied linear and quadratic functions both of which are simple polynomial functions. But not all functions are polynomial. In this lesson we investigate __exponential functions__. These functions have tons of real-world applications! For instance, you've heard of diseases like AIDS spreading "exponentially"? Spread of diseases, population growth, compounding interest, and carbon dating are all examples of applications that can be modeled using exponential functions. Keep your grapher handy as you explore exponential functions such as y=2^{x} (note the variable is in the power not the base like the parabola y=x^{2}). Watch what effect the following things have on the graph: a negative in front, different bases, adding or subtracting a number from the basic function, adding or subtracting a number to the x in the power. Watch for the effects on the shape, axis intercept points, asymptote line(s), domain, and range.

Although your text may not do so until later, I think it is useful to introduce the __scientific constant "e"__ at this time. "e" is an irrational number like π except that e is equal to approximately 2.718. Try graphing y=e^{x} and variations (reflections, shifts, translations) of this function. How does it compare with the graph of y=2^{x} and y=3^{x}? Why?

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 13.1.1 Evaluate exponential functions algebraically and electronically including those containing "e"
- 13.1.2 Graph exponential functions manually and electronically
- 13.1.3 For the basic exponential function f(x)=a
^{x}, know the basic shape of its graph, intercept points, asymptote lines, domain, and range - 13.1.4 Recognize and be able to perform reflections (flips across the x-axis), stretches, vertical shifts (up or down), and horizontal shifts (left or right) in regards to exponential functions
- 13.1.5 Know what values in an exponential function's equation controls the above features (shifts, etc.) of the graph

**Terminology**

Define: exponential function, horizontal asymptote line, the number "e"

**Text Notes**

Everything is important here! New terminology, definitions, graphs, connections between equations and graphs, and more. This section builds the foundation for the rest of the chapter. Learn it well! Be sure you are able to work problems both manually and electronically.

**Supplementary Resources**

A basic introduction to "e" (this lesson) and "ln" (next lesson) is at Algebra Lab: e and ln. Much more info about e is at Math Forum's Ask Dr. Math: FAQs about e including a good collection of Q&A links near the bottom of the article.