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  • Graph exponential functions.
  • Graph exponential functions using transformations.

As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.

Graphing exponential functions

Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form f ( x ) = b x whose base is greater than one. We’ll use the function f ( x ) = 2 x . Observe how the output values in [link] change as the input increases by 1.

x 3 2 1 0 1 2 3
f ( x ) = 2 x 1 8 1 4 1 2 1 2 4 8

Each output value is the product of the previous output and the base, 2. We call the base 2 the constant ratio . In fact, for any exponential function with the form f ( x ) = a b x , b is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of a .

Notice from the table that

  • the output values are positive for all values of x ;
  • as x increases, the output values increase without bound; and
  • as x decreases, the output values grow smaller, approaching zero.

[link] shows the exponential growth function f ( x ) = 2 x .

Graph of the exponential function, 2^(x), with labeled points at (-3, 1/8), (-2, ¼), (-1, ½), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.
Notice that the graph gets close to the x -axis, but never touches it.

The domain of f ( x ) = 2 x is all real numbers, the range is ( 0 , ) , and the horizontal asymptote is y = 0.

To get a sense of the behavior of exponential decay , we can create a table of values for a function of the form f ( x ) = b x whose base is between zero and one. We’ll use the function g ( x ) = ( 1 2 ) x . Observe how the output values in [link] change as the input increases by 1.

x -3 -2 -1 0 1 2 3
g ( x ) = ( 1 2 ) x 8 4 2 1 1 2 1 4 1 8

Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio 1 2 .

Notice from the table that

  • the output values are positive for all values of x ;
  • as x increases, the output values grow smaller, approaching zero; and
  • as x decreases, the output values grow without bound.

[link] shows the exponential decay function, g ( x ) = ( 1 2 ) x .

Graph of decreasing exponential function, (1/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1/2), (2, 1/4), and (3, 1/8). The graph notes that the x-axis is an asymptote.

The domain of g ( x ) = ( 1 2 ) x is all real numbers, the range is ( 0 , ) , and the horizontal asymptote is y = 0.

Characteristics of the graph of the parent function f ( x ) = b x

An exponential function with the form f ( x ) = b x , b > 0 , b 1 , has these characteristics:

  • one-to-one function
  • horizontal asymptote: y = 0
  • domain: ( ,   )
  • range: ( 0 , )
  • x- intercept: none
  • y- intercept: ( 0 , 1 )
  • increasing if b > 1
  • decreasing if b < 1

[link] compares the graphs of exponential growth    and decay functions.

Graph of two functions where the first graph is of a function of f(x) = b^x when b>1 and the second graph is of the same function when b is 0<b<1. Both graphs have the points (0, 1) and (1, b) labeled.

Given an exponential function of the form f ( x ) = b x , graph the function.

  1. Create a table of points.
  2. Plot at least 3 point from the table, including the y -intercept ( 0 , 1 ) .
  3. Draw a smooth curve through the points.
  4. State the domain, ( , ) , the range, ( 0 , ) , and the horizontal asymptote, y = 0.

Questions & Answers

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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