College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.2 Exponential Functions Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Graph Exponential Functions Evaluate the Exponential Function Base e Use Exponential Functions to Compute Compound Interest Use Exponential Functions in Applications

Concept 1 Graph Exponential Functions

Graph Exponential Functions (1 of 2) Previously: Now:

Graph Exponential Functions (2 of 2) Exponential Function: Let b be a constant real number such that b > 0 and b ≠ 1. Then for any real number x, a function of the form is called an exponential function of base b.

Example 1 Graph the function. x f(x)

Example 2 Graph the function. x f(x)

Skill Practice 1 Graph the functions.

Example 3 (1 of 3) Graph the function. x f(x) .

Example 3 (2 of 3) Parent function: Where in, ab: If a<0 reflect across the x-axis. Shrink vertically if 0< |a| < 1. Stretch vertically if |a| > 1. x – h: If h > 0, shift to the right. If h < 0, shift to the left.

Example 3 (3 of 3) k: If k > 0, shift upward. If k < 0, shift downward. Where, x + 1: 1 to left -2: reflect across x-axis 1: 1 up

Skill Practice 2 Graph.

Concept 2 Evaluate the Exponential Function Base e

Evaluate the Exponential Function Base e e is a universal constant (like the number ) and an irrational number. Approaches a constant value that we call e e ≈ 2.718281828

Examples 4 – 7 Evaluate. Round to 4 decimal places.

Example 8 Graph the function. x f(x)

Skill Practice 3

Concept 3 Use Exponential Functions to Compute Compound Interest

Use Exponential Functions to Compute Compound Interest Suppose that P dollars in principle is invested (or borrowed) at an annual interest rate r for t years. Then: I = Prt: Amount of simple interest I (in $) Amount A (in $) in the account after t years under continuous compounding. Amount A (in $) in the account after t years and n compounding periods per year.

Example 9 Suppose that $15,000 is invested with 2.5% interest under the following compounding options. Determine the amount in the account at the end of 7 years for each option. Compounded annually Compounded quarterly Compounded continuously

Skill Practice 4 Suppose that $8000 is invested and pays 4.5% per year under the following compounding options. Compounded annually Compounded quarterly Compounded monthly Compounded daily Compounded continuously Determine the total amount in the account after 5 year with each option.

Concept 4 Use Exponential Functions in Applications

Example 10 Weapon-grade plutonium is composed of approximately 93% plutonium-239 (Pu-239). The half-life of Pu-239 is 24,000 years. In a sample originally containing 0.5 kilograms, the amount left after t years is given by Evaluate the function for the given values of t and interpret the meaning in context. A(24,000) A(72,000)

Example 11 A 2010 estimate of the population of Mexico is 111 million people with a projected growth rate of 0.994% per year. At this rate, the population P(t) (in millions) can be approximated by where t is the time in years since 2010. Evaluate P(4) and interpret its meaning in the context of this problem. Evaluate P(75) and interpret its meaning in the context of this problem.

Skill Practice 5 Cesium-137 is a radioactive metal with a short half-life of 30 yr. In a sample originally having 2 g of cesium-137, the amount A(t) (in grams) of cesium-137 present after t years is given by How much cesium-137 will be present after 30 year? 60 year? 90 year?