Exponential Functions and Their Graphs

Rules for Exponents Exponents give us shortcuts for multiplying and dividing quickly. Each of the key rules for exponents has an important parallel in the world of logarithms which is learned later.

Properties of Exponents Product Rule: Quotient Rule: Power Rule

Multiplying with Exponents To multiply powers of the same base, keep the base and add the exponents. Can’t do anything about the y3 because it’s not the same base. Keep x, add exponents 7 + 5

Dividing with Exponents To divide powers of the same base, keep the base and subtract the exponents. Keep 5, subtract 12-4 Keep 7, subtract 10-6

Powers with Exponents To raise a power to a power, keep the base and multiply the exponents. This means t7·t7·t7 = t7+7+7

These are examples of exponential functions: The exponential function f with base b is defined by f (x) = abx where a 0, b  0 or 1, and x is any real #. a is the initial value and b is the base; x is the power to which the base is raised. These are examples of exponential functions: a ,the # outside the ( ) is the y-intercept or the initial value of the function. Copyright © by Houghton Mifflin Company, Inc. All rights reserved

Exponential functions are often asymptotic to the x-axis, meaning they get closer and closer and closer to it, but can never reach it because y cannot equal zero. y =abx If b > 1, the exponential is increasing, growing as the values of x go up from left to right.

Graph of f(x) = abx, b > 1 y Range: (0, ) (a, 1) x Horizontal Asymptote y = 0 Domain: (–, ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved

If 0 < b <1, the function is a decreasing exponential. Graph of f (x) = abx, 0<b <1 y Range: (0, ) Horizontal Asymptote y = 0 (0, 1) x Domain: (–, ) If 0 < b <1, the function is a decreasing exponential. Copyright © by Houghton Mifflin Company, Inc. All rights reserved

Determine if the graphs below represent increasing or decreasing functions. Y=4(1.5)x f(x)=10(1.2)x Y=6(.42)x F(x)=12(.88)x

What do you notice about the functions: and They are reflections across the y-axis. Copyright © by Houghton Mifflin Company, Inc. All rights reserved

Identify the initial value and the rate of change for each exponential function below. Are they increasing or decreasing?

Transformations of Exponential Functions Compare the following graph of f(x)=3x and y= 3x+1 f(x)=3x Compare the following graph of f(x)=3x and y = 3x-2 y = 3x-2 What happened in each case?

Sketch the graph of g(x) = 2x – 1. State the domain and range. Y=-1 is the horizontal asymptote Copyright © by Houghton Mifflin Company, Inc. All rights reserved

b is the rate of change in the function b is the rate of change in the function. b = 1 ± r where r = the % of change Nowheresville has a population of 3138. If the population is decreasing at a yearly rate of 3.5%, write an equation to represent this function and determine the population in 5 years. Y=abx

Mrs. Layton has $5000 to invest in a starting company Mrs. Layton has $5000 to invest in a starting company. If they promise a 6% rate of return each year, what formula represents this exponential function? In 5 years, how much will her investment be worth? Y=abx

Sometimes, rather than an integer, the base of the exponential function is the irrational number, e. e is approximately equal to 2.781828… but, because it is irrational, its integers do not repeat in a recognizable pattern. would be an increasing exponential since its base, e, is greater than 1.

Graph of f(x) = ex y x X-axis is the horizontal asymptote 6 4 2 –2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved

Graph the following functions: Copyright © by Houghton Mifflin Company, Inc. All rights reserved