8-3: The Number ‘e’ (Day 1) Objective (Ca. Standard 12): Students know the laws of fractional exponents, understand exponential function, and use these functions in problems in problems involving exponential growth and decay.

Investigating the Natural Base e Turn to page 480 complete the table and answer the question in part 2.

Do the values in the table appear to be approaching a fixed decimal number? Yes, the number

The Natural Base e The natural base e is irrational. It is defined as follows:

Example 1: Simplifying Natural Base Expressions Simplify

Example 2: Evaluating Natural Base Expressions

Natural base exponential function has the form The function is an exponential growth function if a > 0 and r > 0.

(2, 7.29) (1, 2.7) (0, 1)

The function is an exponential decay function if a > 0 and r < 0. (-1, 2.7) (0, 1) (-2, 7.29)

Example 3: Graphing Natural Base Functions Graph the function. State the domain and range. Solution: Because a = 2 is positive and r = 0.75, the function is an exponential growth function.

Plot the points and sketch the graph. Domain: all real #’s Range: y > 0

Solution: Because a= 1 is positive and r = is negative, the function is an exponential decay function.

Translate the graph to the right by 2 units and up 1.

Domain = all real #’s Range = y > 1

Homework: P.483 #17-31 Odd, #49-59 Odd