8.1: Exponential Growth Function
A population of 10 mice grows at a rate of 300% every month. How many mice are in the population after six months?
Guidance
An exponential function has the variable in the exponent of the expression. All exponential functions have the form:
Example A
Graph
Solution: Let’s start by making a table. Include some positive and negative values for




3 

8 
2 

4 
1 

2 
0 

1 
1 


2 


3 


This is the typical shape of an exponential growth function. The function grows “exponentially fast”. Meaning, in this case, the function grows in powers of 2. For an exponential function to be a growth function,
Notice that the function gets very, very close to the
Example B
Graph
Solution: This is not considered a growth function because
Notice that the point (0, 1) from
The domain of all exponential functions is all real numbers. The range will be everything greater than the asymptote. In this example, the range is
Example C
Graph the function
Solution: In this example, we will outline how to use the graphing calculator to graph an exponential function. First, clear out anything in Y=. Next, input the function into Y1= (1/2)4^X and press GRAPH. Adjust your window accordingly.
This is not an exponential growth function, because it does not grow in a positive direction. By looking at the definition of a growth function,
Intro Problem Revisit
This is an example of exponential growth, so we can use the exponential form
Therefore, the mouse population after six months is 7,290.
Guided Practice
Graph the following exponential functions. Determine if they are growth functions. Then, find the
1.
2.
3.
4. Abigail is in a singles tennis tournament. She finds out that there are eight rounds until the final match. If the tournament is single elimination, how many games will be played? How many competitors are in the tournament?
Answers
1. This is not a growth function because
2. This is not a growth function because
3. This is a growth function. The
4. If there are eight rounds to single’s games, there are will be
Vocabulary
 Exponential Function

A function whose variable is in the exponent. The general form is
y=a⋅bx−h+k .
 Exponential Growth Function

A specific type of exponential function where
h=k=0,a>0, andb>1 . The general form isy=abx .
 Asymptote
 A boundary line that restricts the domain or range. This line is not part of the graph.
Practice
Graph the following exponential functions. Find the

y=4x 
y=(−1)(5)x 
y=3x−2 
y=2x+1 
y=6x+3 
y=−14(2)x+3 
y=7x+3−5 
y=−(3)x−4+2 
y=3(2)x+1−5  What is the yintercept of
y=ax ? Why is that?  What is the range of the function
y=ax−h+k ?  March Madness is a singlegame elimination tournament of 64 college basketball teams. How many games will be played until there is a champion? Include the championship game.
 In 2012, the tournament added 4 teams to make it a field of 68 and there are 4 "playin" games at the beginning of the tournament. How many games are played now?
 An investment grows according the function
A=P(1.05)t whereP represents the initial investment,A represents the value of the investment andt represents the number of years of investment. If $10,000 was the initial investment, how much would the value of the investment be after 10 years, to the nearest dollar?  How much would the value of the investment be after 20 years, to the nearest dollar?
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Asymptotes
An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).Exponential Function
An exponential function is a function whose variable is in the exponent. The general form is .Exponential growth
Exponential growth occurs when a quantity increases by the same proportion in each given time period.Exponential Growth Function
An exponential growth function is a specific type of exponential function that has the form , where and .Model
A model is a mathematical expression or function used to describe a physical item or situation.Image Attributions
Here you'll learn how to analyze an exponential growth function and its graph.