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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

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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