The Three Different Cases for the Function: 

	Case 1: a > 1 What is in common to all of these exponential functions? What changes when the value of a increases?
	Case 2: a = 1 What effect does the value of x have on the graph? Is this an exponential function?
	Case 3: 0 < a < 1 What is in common to all of these exponential functions? What changes when the value of a increases? Compare Cases 1 and 3. What is the difference in a, and what difference does this cause in the graph?

Laws of Exponents 

Exponential Law	Explanation
	When you multiply two exponential numbers with the same base, keep the base and add the exponents.
	When you divide two exponential numbers with the same base, keep the base and subtract the exponents.
	From the exponential law immediately above this one we know that , and from arithmetic we know that any number divided by itself (with the exception of zero) is 1. There fore:
	When you take a power of a power, keep the base and multiply the exponents.
	From the third exponential law in this list we know that . If we substitute using this proprerty, our law now reads: . From the second exponential law in this list, we now conclude that:

There are three scenarios shown below. The Real TV show money offerings for the number of consecutive questions answered correctly is shown under the Heading, "Real Show".

In the "Truly Exponential" scenario, each value is 2.511886432 taken to the power of the number of consecutive correct answers. This base of 2.511886432 was calculated so that 1) the number of consecutive correct answers would be the exponent, and 2) to the 15th power, the answer would be one million.

In the "Same Start" scenario, we begin with 100 dollar prize and then truly double each answer (which the real TV show does not always do).

Below the scenario data in the table is the plot of each scenario.

Blue: The Real Show; Yellow: Truly Exponential; Purple: Same Start

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 1.5 This module enables you to graph exponential functions with various bases and their tangent lines in order to estimate more closely the value of a for which the tangent has a slope of 1.

Check Concepts

#1: Which of the following grows the most quickly?	Choose One y = 2x y = 2^x y = x^2
#2: Exponential functions fit:	Choose One Population Models Decay Models Population & Decay Models Neither
#3: For the general function , there are ____ case(s) that need to be considered.	Choose One 1 2 3 4 an infinited number of
#4: If we know how to graph , how would we alter the graph to represent ?	Choose One Move graph down two and right 3 Move graph down two and left 3 Invert graph and move left 3 Invert graph and move right 3
#5 The function  is special for use in calculus because....	Choose One The slope of the tangent to the curve at (0,1) is 1. The slope of the tangent to the curve at any point is always equal to its y-coordinate. e is an irrational number.