<< Chapter < Page Chapter >> Page >
  • 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask