Skip to ContentGo to accessibility page
OpenStax Logo
Algebra 1

5.9.2 Analyze Underlying Relationship in a Graph of a Function

Algebra 15.9.2 Analyze Underlying Relationship in a Graph of a Function

5.9.2 Analyze Underlying Relationship in a Graph of a Function

5.9.2 • Analyze Underlying Relationship in a Graph of a Function

Activity

The cost, in dollars, to produce 1 watt of solar energy is a function of the number of years since 1977, t t .

From 1977 to 1987, the cost could be modeled by an exponential function f f . Here is the graph of the function.

1. What is the statement f ( 9 ) ≈ 6 f ( 9 ) ≈ 6 saying about this situation?

2. What is f ( 4 ) f ( 4 ) ? What about f ( 3.5 ) f ( 3.5 ) ? What do these values represent in this context?

3. When f ( t ) = 45 f ( t ) = 45 , what is t t ? What does that value of t t  represent in this context?

4. By what factor did the cost of 1 watt of solar energy change each year? (If you get stuck, consider creating a table.)

Video: Understanding Characteristics of Graphs

Watch the following video to learn more about the characteristics of graphs:

Understanding Characteristics of Graphs

An imaginary line that a function value will not touch or cross is called an asymptote.

The value of the range that the function will never touch is called the horizontal asymptote.

In the function graphed, the horizontal asymptote is y = 0 y = 0 .

Self Check

The graph below shows the total smartphones, s , in millions that were sold in the years, t , since 2008. What does the point ( 4 , 125 ) represent in this context?

  1. In 2004, there were 125 million more smartphones than in 2000.
  2. In 2012, there were 125 million smartphones sold.
  3. In 2012, there were 125 million more smartphones than in 2008.
  4. In 2004, there were 125 million smartphones sold.

Additional Resources

Reading Exponential Graphs

The graph below describes the amount of caffeine, c c , in a person’s body t t  hours after an initial measurement of 100 mg.

What does the point c ( 10 ) = 35 c ( 10 ) = 35 mean?

After 10 hours, there is 35 mg of caffeine in a person’s body.

What is the value of t t when c ( t ) = 80 c ( t ) = 80 ?

When the caffeine concentration is 80 mg, time is 2 hours, so t = 2 t = 2 .

Try it

Reading Exponential Graphs

Using the graph above, what is the value of c ( 5 ) c ( 5 ) , and what does it mean in the context of the situation?

PreviousNext
Order a print copy
Citation/Attribution
Reuse and redistribution of this content in digital or print format:
  • 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 prior written permission.
  • This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License, which means that you can reuse and modify the material only for noncommercial purposes, must attribute OpenStax, and must distribute any derivative works under the same license.
  • Any commercial printing of this textbook, including using a local or custom printer, must be approved by OpenStax, and proper citation provided.
  • OpenStax-copyrighted images, activities, assessments, and similar components of this book are subject to the same licensing – CC-BY-NC-SA. They can be used for noncommercial purposes with attribution. Commercial use requires permission.
  • Permission requests: Anyone who intends to incorporate this content (including text, images, and other components) into large language models, use it in AI offerings, use it commercially (including in print), and/or has questions about another use case is welcome to complete our reuse request form.
Attribution information
  • If you are redistributing all or part of this book in a noncommercial 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 noncommercial digital format, then for every page that includes OpenStax content, you must license the derivative work under the same CC-BY-NC-SA license as the original, and 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

The information below includes the information needed to generate citations in most major styles (APA, MLA, etc.); you must reformat and organize the information as needed to fit the requirements of the style. Use the information below to generate a citation. We recommend using a citation tool such as this one.

© Apr 23, 2026 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, and Rice University name, and Rice University logo trademarks, or wordmarks are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.