Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Algebra 1

5.12.2 Equations and Their Graphs

Algebra 15.12.2 Equations and Their Graphs

Search for key terms or text.

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

1. Each of the following functions, f f , g g , h h , and j j , represents the amount of money in a bank account, in dollars, as a function of time x x , in years. They are each written in the form m ( x ) = a · b x m ( x ) = a · b x .

  • f ( x ) = 50 · 2 x f ( x ) = 50 · 2 x
  • g ( x ) = 50 · 3 x g ( x ) = 50 · 3 x
  • h ( x ) = 50 · ( 3 2 ) x h ( x ) = 50 · ( 3 2 ) x
  • j ( x ) = 50 · ( 0.5 ) x j ( x ) = 50 · ( 0.5 ) x

a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.

b. Explain how changing the value of b b  changes the graph.

2. Here are equations defining functions p p , q q , and r r . They are also written in the form m ( x ) = a · b x m ( x ) = a · b x .

  • p ( x ) = 10 · 4 x p ( x ) = 10 · 4 x
  • q ( x ) = 40 · 4 x q ( x ) = 40 · 4 x
  • r ( x ) = 100 · 4 x r ( x ) = 100 · 4 x

a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.

b. Explain how changing the value of a a changes the graph.

3. Here are equations defining functions f f , g g , and h h  written in the form m ( x ) = a · b x + d m ( x ) = a · b x + d .

  • f ( x ) = 4 x + 2 f ( x ) = 4 x + 2
  • g ( x ) = 4 x g ( x ) = 4 x
  • h ( x ) = 4 x 2 h ( x ) = 4 x 2

a. Use graphing technology to graph each function on the same coordinate plane. It may be helpful to use a different color for each function.

a. Explain how changing the value of a a changes the graph.

b. Discuss with a partner how changing the value of d d  changes the asymptote of the graph.

Are you ready for more?

Extending Your Thinking

As before, consider bank accounts whose balances are given by the following functions:

f ( x ) = 10 · 3 x f ( x ) = 10 · 3 x

g ( x ) = 3 x + 2 g ( x ) = 3 x + 2

h ( x ) = 1 2 · 3 x + 3 h ( x ) = 1 2 · 3 x + 3

Which function would you choose? Does your choice depend on x x ?

Video: Exponential Equations and How They Affect the Graphs

Watch the following video to learn more about exponential functions and how they affect the graphs:

Self Check

What is the horizontal asymptote for the function f ( x ) = 2 3 x 5 ?
  1. y = 3
  2. y = 5
  3. y = 2
  4. y = 5

Additional Resources

Vertical Shifts with Exponential Functions

The first transformation occurs when we add a constant d d  to the parent function f ( x ) = b x f ( x ) = b x , giving us a vertical shift d d  units in the same direction as the sign. For example, if we begin by graphing a parent function, f ( x ) = 2 x f ( x ) = 2 x , we can then graph two vertical shifts alongside it, using  d = 3 d = 3 : the upward shift, g ( x ) = 2 x + 3 g ( x ) = 2 x + 3 , and the downward shift, h ( x ) = 2 x 3 h ( x ) = 2 x 3 . Both vertical shifts are shown in the graph below.

Graph of three exponential growth functions. f of x has a y-intercept of 1 and a horizontal asymptote of y equals 0. g of x has a y-intercept of 4 and a horizontal asymptote of y equals 3. h of x has a y-intercept of negative 2 and a horizontal asymptote of y equals negative 3.

Observe the results of shifting f ( x ) = 2 x f ( x ) = 2 x vertically:

  • The domain,  ( , ) ( , ) , remains unchanged.
  • When the function is shifted up 3 units to g ( x ) = 2 x + 3 g ( x ) = 2 x + 3 :
    • The y y -intercept shifts up 3 units to ( 0 , 4 ) ( 0 , 4 ) .
    • The asymptote shifts up 3 units to y = 3 y = 3 .
    • The range becomes ( 3 , ) ( 3 , ) .
  • When the function is shifted down 3 units to h ( x ) = 2 x 3 h ( x ) = 2 x 3 :
    • The y y -intercept shifts down 3 units to ( 0 , 2 ) ( 0 , 2 ) .
    • The asymptote also shifts down 3 units to y = 3 y = 3 .
    • The range becomes ( 3 , ) ( 3 , ) .

Try it

Try It: Vertical Shifts with Exponential Functions

Where would the horizontal asymptote be located for the graph of f ( x ) = 4 · 2 x 6 f ( x ) = 4 · 2 x 6 ?

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

© May 21, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.