Chapter 10.1 Exponential Functions Standard & Honors Algebra II Mr. Gilbert Chapter 10.1 Exponential Functions Standard & Honors 9/21/2018

Agenda Warm up Decay project Lesson Homework 9/21/2018

Homework Review 9/21/2018

Communicate Effectively Asymptote: a line on a graph approached but never crossed. Radical number: Exponent Exponential Function: y=2x Growth: y=abx| a 0, b>1 Decay: y=abx | a 0, 0<b<1 Examples of Exponential Functions: Population growth, Interest Income growth, radioactive decay: m(t)=m0ekt where m0 is the original amount and t is time, k varied by element. 9/21/2018

World Population Growth 9/21/2018

Example 1 Graph an Exponential Function (3) Example 2 Identify Exponential Growth and Decay (3) Example 3 Write an Exponential Function (5) Example 4 Simplify Expressions with Irrational Exponents (3) Example 5 Solve Exponential Equations (4) Example 6 Solve Exponential Inequalities (3) 9/21/2018 Lesson 1 Contents

x Sketch the graph of . Then state the function’s domain and range. Make a table of values. Connect the points to sketch a smooth curve. 16 2 4 1 –1 –2 x 9/21/2018 Example 1-1a

Answer: The domain is all real numbers, while the range is all positive numbers. 9/21/2018 Example 1-1b

Sketch the graph of Then state the function’s domain and range. Answer: The domain is all real numbers; the range is all positive numbers. 9/21/2018 Example 1-1c

Determine whether represents exponential growth or decay. Answer: The function represents exponential decay, since the base, 0.7, is between 0 and 1. 9/21/2018 Example 1-2a

Determine whether represents exponential growth or decay. Answer: The function represents exponential growth, since the base, 3, is greater than 1. Determine whether represents exponential growth or decay. Answer: The function represents exponential growth, since the base, is greater than 1. 9/21/2018 Example 1-2b

Answer: The function represents exponential decay, since the Determine whether each function represents exponential growth or decay. a. b. c. Answer: The function represents exponential decay, since the base, 0.5, is between 0 and 1. Answer: The function represents exponential growth, since the base, 2, is greater than 1. Answer: The function represents exponential decay, since the base, is between 0 and 1. 9/21/2018 Example 1-2d

Cellular Phones In December of 1990, there were 5,283,000 cellular telephone subscribers in the United States. By December of 2000, this number had risen to 109,478,000. Write an exponential function of the form that could be used to model the number of cellular telephone subscribers y in the U.S. Write the function in terms of x, the number of years since 1990. For 1990, the time x equals 0, and the initial number of cellular telephone subscribers y is 5,283,000. Thus the y-intercept, and the value of a, is 5,283,000. For 2000, the time x equals 2000 – 1990 or 10, and the number of cellular telephone subscribers is 109,478,000. 9/21/2018 Example 1-3a

Replace x with 10, y with 109,478,000 and a with 5,283,000. Substitute these values and the value of a into an exponential function to approximate the value of b. Exponential function Replace x with 10, y with 109,478,000 and a with 5,283,000. Divide each side by 5,283,000. Take the 10th root of each side. 9/21/2018 Example 1-3b

To find the 10th root of 20. 72, use selection To find the 10th root of 20.72, use selection under the MATH menu on the TI-83 Plus. ENTER MATH Keystrokes: 10 5 20.72 1.354063324 Answer: An equation that models the number of cellular telephone subscribers in the U.S. from 1990 to 2000 is 9/21/2018 Example 1-3c

For 2010, the time x equals 2010 – 1990 or 20. Suppose the number of telephone subscribers continues to increase at the same rate. Estimate the number of US subscribers in 2010. For 2010, the time x equals 2010 – 1990 or 20. Modeling equation Replace x with 20. Use a calculator. Answer: The number of cell phone subscribers will be about 2,136,000,000 in 2010. 9/21/2018 Example 1-3d

Health In 1991, 4. 9% of Americans had diabetes Health In 1991, 4.9% of Americans had diabetes. By 2000, this percent had risen to 7.3%. a. Write an exponential function of the form could be used to model the percentage of Americans with diabetes. Write the function in terms of x, the number of years since 1991. b. Suppose the percent of Americans with diabetes continues to increase at the same rate. Estimate the percent of Americans with diabetes in 2010. Answer: Answer: 11.4% 9/21/2018 Example 1-3e

Simplify . Quotient of Powers Answer: 9/21/2018 Example 1-4a

Simplify . Power of a Power Product of Radicals Answer: 9/21/2018 Example 1-4b

Simplify each expression. a. b. Answer: Answer: 9/21/2018 Example 1-4c

Rewrite 256 as 44 so each side has the same base. Solve . Original equation Rewrite 256 as 44 so each side has the same base. Property of Equality for Exponential Functions Add 2 to each side. Divide each side by 9. Answer: The solution is 9/21/2018 Example 1-5a

Check Original equation Substitute for n. Simplify. Simplify. 9/21/2018 Example 1-5b

Rewrite 9 as 32 so each side has the same base. Solve . Original equation Rewrite 9 as 32 so each side has the same base. Property of Equality for Exponential Functions Distributive Property Subtract 4x from each side. Answer: The solution is 9/21/2018 Example 1-5c

Solve each equation. a. b. Answer: Answer: 1 9/21/2018 Example 1-5d

Property of Inequality for Exponential Functions Solve Original inequality Rewrite as Property of Inequality for Exponential Functions Subtract 3 from each side. Divide each side by –2. Answer: The solution is 9/21/2018 Example 1-6a

Check: Test a value of k less than for example, Original inequality Replace k with 0. Simplify. 9/21/2018 Example 1-6b

Solve Answer: 9/21/2018 Example 1-6c

Homework - Honors See Syllabus 10.1 pp. 528-529: 21-54 (multiples of 3), 57-67 9/21/2018

Homework See Syllabus 10.1 pp. 528-529: 21-54 (multiples of 3), 57-61 9/21/2018