What are Exponential Functions? Exponential functions – functions that include the expression bx where b is a positive # other than 1. b is called the base.
What’s the Shape? Let’s make a table to find the general shape. If we use f(x) = 2x as an example: x f(x) = 2x -3 -2 -1 0 1 2 3
Asymptotes An asymptote is a line that a graph approaches (but does not touch) as you move away from the origin. For example: Our graph has a horizontal asymptote at y = 0.
Graphing y = abx If a > 0 and b > 1, y = abx is an exponential growth function. For all y = abx , b > 1: Graphs pass through (0, a) (a is the y-int) x-axis is an asymptote Domain: all real #s Range: y > 0 if a > 0 y < 0 if a < 0
To graph: Plot 2 points: (0, a) and (1, __) Plug in 1 for x to fill the blank Connect with a smooth curve that: Starts left of the origin, close to the x-axis Moves up or down quickly to the right
General Exponential Functions General form: As usual: h is horizontal shift k is vertical shift To graph: Sketch the “parent graph” y = abx Shift using h and k
Examples Graph and state the domain and range:
Your Turn! Graph and state the domain and range:
Exponential Growth Models When a real-life quantity increases by a fixed % each year, the amount of the quantity after t years can be modeled by: y = a(1 + r)t where a is the initial amount and r is the % increase (as a decimal). (1 + r) is the growth factor.
Example: In January, 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year. Write a model giving the number h (in millions) of hosts t years after 1993. How many hosts were there in 1996?
Compound Interest Compound interest is interest paid on the original principal and on previously earned interest. Modeled by an exponential function. If interest is compounded n times per year, the amount A in the account after t years is: where P is the initial principal and r is the annual interest rate.
Example: You deposit $1000 in an account that pays 8% annual interest. Find the balance after 1 year if interest is compounded: A. annually B. quarterly C. daily Which is the best investment?