8.1 Exponential Growth Algebra 2 Mrs. Spitz Spring 2007
Objective • Graph exponential growth functions • Use exponential growth functions to model real-life situations, such as Internet growth. Assignment • Class Activity – Graphing Exponential Functions
Graphing Exponential Growth Functions • An exponential function involves the expression bx where base b is a positive number other than 1. In this lesson, you will study exponential functions for which b > 1. To see the basic shape of the graph of an exponential function such as f(x) = 2x, you can make a table of values and plot points as shown on the next slide.
Notice the end behavior of the graph. As which means that the graph moves up to the right. As which means that the graph has the line y = 0 as an asymptote. An asymptote is a line that a graph approaches as you move away from the origin.
YOU NEED THIS TO DO THE WORK! • The graph passes through the point (0, a). That is, the y-intercept is a. • The x-axis is an asymptote of the graph. • The domain is all real numbers. • The range is y > 0 if a > 0 and y< 0 if a < 0
There is no a ● 4x, so the y intercept is (0, 1) The x-axis is an asymptote of the graph. The domain is all real numbers. Range is y > 0 because a > 0. When x = 1, y = 4 Ex. 1: Graph:
There is no a ● 2x-3 +1, so the y intercept is (0, 1) The x-axis is an asymptote of the graph. The domain is all real numbers. Range is y > 0 because a > 0. When x = 1, y = 4 Ex. 2: Graph:
To graph a general exponential function: Begin by sketching the graph of y = abx . Then translate the graph horizontally by h units and vertically by k units.
Ex. 2: Graphing a general exponential function First graph the first function then move it over 1 place to the right and then four down.
On a TI-83 or 84 press y= insert 1 divided by 2 times 3 power x press graph press 2nd graph to get the table of values.
Ex. 3: All of the properties of rational exponents apply to real exponents as well. Lucky you! Simplify: Recall the product of powers property, am an = am+n
Ex. 4: All of the properties of rational exponents apply to real exponents as well. Lucky you! Simplify: Recall the power of a power property, (am)n= amn
Definition of Exponential Function • An equation of the form y = ax, where a > 0 and a ≠ 1, is called an exponential function.
Several exponential functions have been graphed at the right. Compare the graphs of functions where a > 1 to those where a < 1. Notice that when a > 1, the value of y increases as the value of x increases. When a < 1, the value of y decreases as the value of x increases. What do they look like graphed?
Property of Equality for Exponential Functions • The following property is very useful when solving equations involving exponential functions. • Suppose a is a positive number other than 1. Then: If and only if x1 = x2 Notice that a cannot equal 1. Since 1x = 1 for any real number, for any choice of x1 and x2