Chapter 3 Exponential and Logarithmic Functions Pre-Calculus Chapter 3 Exponential and Logarithmic Functions

3.1 Exponential Functions Objectives: Recognize and evaluate exponential functions with base a. Graph exponential functions. Recognize, evaluate, and graph exponential functions with base e. Use exponential functions to model and solve real-life problems.

Definition of Exponential Function The exponential function with base a is denoted by where a > 0, a ≠ 1, and x is any real number.

Properties of Exponents

Exponential Function The exponential function is different from the other functions we have studied so far. The variable x is an exponent. A distinguishing characteristic of an exponential function is its rapid increase as x increases (for a > 1).

Characteristics of f (x) = ax

Characteristics of f (x) = a -x

Transformations of f (x) = ax The graph of g(x) = a(x ± h) is a ______ shift of f. The graph h(x) = ax ± k is a ________ shift of f. (Note new horizontal asymptote.) The graph k(x) = –ax is a reflection of f ______. The graph j(x) = a(–x) is a reflection of f ______.

The Natural Base, e The natural base (or Euler’s Number) is a constant that occurs frequently in nature and in science. e ≈ 2.718281828459 Like π, e is irrational. It continues forever in a non-repeating manner. The natural base function is f (x) = ex. The properties and characteristics of exponential functions hold for ex.

Graph of ex Complete the table and graph the function. x ex -2 -1 1 2  

Applications Exponential functions are used to model rapid growth or decay. Examples include: Compound interest Population growth Radioactive decay

Compound Interest

Example 1 A total of $12,000 is invested at an annual rate of 3%. Find the balance after 4 years if the interest is compounded Quarterly Continuously

Example 2 Let y represent a mass of radioactive strontium (90Sr), in grams, whose half-life is 28 years. The quantity of strontium present after t years is What is the initial mass (when t = 0)? How much of the initial mass is present after 80 years?

Example 3 The approximate number of fruit flies in an experimental population after t hours is given by where t ≥ 0. Find the initial number of fruit flies in the population. How large is the population of fruit flies after 72 hours?

Homework 3.1 Worksheet 3.1