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Characteristics of Graphs of Functions

Author: Sophia
PDF Version

what's covered
In this lesson, you will use the graph of a function to determine how it changes. Specifically, this lesson will cover:

Table of Contents

1. Using a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant

When we take a graph and examine it over an interval of x-values, we can observe many things.

Here are a few questions we could ask:

  • Is the graph rising, falling, or flat?
  • Does the graph have any high points?
  • Does the graph have any low points?
  • How do the high and low points relate to where the graph is rising or falling?
In this tutorial, we will answer all these questions.

There are three key properties of functions in terms of how they change: increasing, decreasing, and constant. Imagine you are traversing a graph from left to right.

big idea
A function is increasing on an interval of x-values if its y-values increase (the graph rises) as the x-values increase.

A function is decreasing on an interval of x-values if its y-values decrease (the graph falls) as the x-values increase.

A function is constant on an interval of x-values if its y-values do not change (flat, horizontal) as the x-values increase.

To see how this works, consider the graph of f open parentheses x close parentheses equals open vertical bar x minus 2 close vertical bar plus open vertical bar x minus 4 close vertical bar shown below.

As we can see, the graph is falling until the point open parentheses 2 comma space 2 close parentheses comma then is flat until open parentheses 4 comma space 2 close parentheses comma then begins to rise.

More formally, we say that the graph of f open parentheses x close parentheses is decreasing on the interval open parentheses short dash infinity comma space 2 close parentheses comma constant on the interval open parentheses 2 comma space 4 close parentheses comma and increasing on the interval open parentheses 4 comma space infinity close parentheses.

Notice that all these intervals are open. To understand why, consider x equals 2, for instance. There is no meaningful way to decide whether 2 should be included in the constant interval or the decreasing interval. Therefore, we leave all intervals open.

EXAMPLE

Use the graph below to determine the open intervals over which p open parentheses t close parentheses is increasing, decreasing, or constant.



Moving from left to right, it appears that p open parentheses t close parentheses is decreasing until the point open parentheses 1 comma space short dash 1 close parentheses comma then increasing until reaching the point open parentheses 3 comma space 0.9 close parentheses comma then decreasing until reaching the point open parentheses 4 comma space 0.75 close parentheses comma then increasing indefinitely after that.

Therefore, p open parentheses t close parentheses is decreasing on the intervals open parentheses short dash infinity comma space 1 close parentheses union open parentheses 3 comma space 4 close parentheses and increasing on the intervals open parentheses 1 comma space 3 close parentheses union open parentheses 4 comma space infinity close parentheses.

Now, one for you to try.

try it
Consider this graph:



Write all open intervals over which the function is increasing, decreasing, or constant.
Increasing on open parentheses short dash infinity comma space short dash 3 close parentheses union open parentheses 3 comma space infinity close parentheses comma decreasing on open parentheses short dash 3 comma space 3 close parentheses.

2. Using a Graph to Locate Local Maxima and Local Minima

A function f open parentheses x close parentheses has a local (or relative) maximum at x equals a if f open parentheses a close parentheses greater or equal than f open parentheses x close parentheses for all x close to x equals a. In other words, f open parentheses a close parentheses is the largest value of a function f open parentheses x close parentheses for values around x equals a. A function f open parentheses x close parentheses has a local (or relative) minimum at x equals a if f open parentheses a close parentheses less or equal than f open parentheses x close parentheses for all x close to x equals a. In other words, f open parentheses a close parentheses is the smallest value of a function f open parentheses x close parentheses for values around x equals a.

hint
To locate local minimum and maximum points on a graph, look for “peaks” and “valleys” on the curve.

EXAMPLE

List all local minimum and maximum points on the following graph.


  • open parentheses 1 comma space short dash 1 close parentheses is a local minimum point. This means that the local minimum value is -1 and occurs when x equals 1.
  • open parentheses 3 comma space 0.9 close parentheses is a local maximum point. This means that the local maximum value is 0.9 and occurs when x equals 3.
  • open parentheses 4 comma space 0.75 close parentheses is a local minimum point. This means that the local minimum value is 0.75 and occurs when x equals 4.

try it
Consider the graph of y equals f open parentheses x close parentheses shown below.



Determine all the local minimum values on the graph.
-2 (occurs at x equals short dash 1).
Determine all the local maximum values on the graph.
2 (occurs at x equals 1).

terms to know
Local (or Relative) Maximum
A function f open parentheses x close parentheses has a local (or relative) maximum at x equals a if f open parentheses a close parentheses greater or equal than f open parentheses x close parentheses for all x close to x equals a. In other words, f open parentheses a close parentheses is the largest value of a function f open parentheses x close parentheses for values around x equals a.
Local (or Relative) Minimum
A function f open parentheses x close parentheses has a local (or relative) minimum at x equals a if f open parentheses a close parentheses less or equal than f open parentheses x close parentheses for all x close to x equals a. In other words, f open parentheses a close parentheses is the smallest value of a function f open parentheses x close parentheses for values around x equals a.

3. Using a Graph to Locate the Absolute Maximum and Absolute Minimum

Absolute minima and maxima are the smallest and largest values of f open parentheses x close parentheses on the entire domain.

A function f open parentheses x close parentheses has a global (or absolute) maximum at x equals a if f open parentheses a close parentheses greater or equal than f open parentheses x close parentheses for all x in the domain. In other words, f open parentheses a close parentheses is the largest value of a function f open parentheses x close parentheses comma and occurs when x equals a. A function f open parentheses x close parentheses has a global (or absolute) minimum at x equals a if f open parentheses a close parentheses less or equal than f open parentheses x close parentheses for all x in the domain. In other words, f open parentheses a close parentheses is the smallest value of a function f open parentheses x close parentheses comma and occurs when x equals a.

EXAMPLE

Find all global maximum and minimum values of f open parentheses x close parentheses comma where its graph is shown below.



The highest points on the graph are open parentheses short dash 2 comma space 16 close parentheses and open parentheses 2 comma space 16 close parentheses. Therefore, f open parentheses x close parentheses has a global maximum value of 16, which occurs at x equals short dash 2 and x equals 2.

The lowest point on the graph is located at open parentheses 3 comma space short dash 10 close parentheses semicolon therefore, the global minimum value is -10 and occurs when x equals 3.

try it
Consider the graph of y equals f open parentheses x close parentheses shown below.



Determine the interval(s) over which f   (x  ) is increasing and decreasing.
Increasing on open parentheses short dash 7 comma space short dash 3 close parentheses union open parentheses 3 comma space 7 close parentheses comma decreasing on open parentheses short dash 3 comma space 3 close parentheses.
Determine all local maximum and minimum values.
Local maximum is 50 at x equals short dash 3 semicolon local minimum is -50 at x equals 3.
Determine all global maximum and minimum values.
Global maximum is about 150 at x equals 7 semicolon global minimum is about -230 at x equals short dash 7.

terms to know
Global (or Absolute) Maximum
A function f open parentheses x close parentheses has a global (or absolute) maximum at x equals a if f open parentheses a close parentheses greater or equal than f open parentheses x close parentheses for all x in the domain. In other words, f open parentheses a close parentheses is the largest value of a function f open parentheses x close parentheses comma and occurs when x equals a.
Global (or Absolute) Minimum
A function f open parentheses x close parentheses has a global (or absolute) minimum at x equals a if f open parentheses a close parentheses less or equal than f open parentheses x close parentheses for all x in the domain. In other words, f open parentheses a close parentheses is the smallest value of a function f open parentheses x close parentheses comma and occurs when x equals a.

summary
In this lesson, you learned how to use a graph to determine where a function is increasing, decreasing, or constant, by evaluating whether the y-values are increasing (graph rises), decreasing (graph falls), or do not change (graph is flat) on an interval as the x-values increase. You also learned how to use a graph to locate local maxima and local minima, noticing that these are the transition points between increasing and decreasing behavior. Lastly, you learned how to use a graph to locate the absolute maximum and absolute minimum, which are the smallest and largest values of f left parenthesis x right parenthesis on the entire domain. When you get to calculus, you will learn how to find minimum and maximum values by using the function itself rather than relying on graphs.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM 1) CHAPTER 0 AND 3 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES. 2) OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Terms to Know
Global (or Absolute) Maximum

A function f open parentheses x close parentheses has a global (or absolute) maximum at x equals a if f open parentheses a close parentheses greater or equal than f open parentheses x close parentheses for all x in the domain. In other words, f open parentheses a close parentheses is the largest value of a function f open parentheses x close parentheses comma and occurs when x equals a.

Global (or Absolute) Minimum

A function f open parentheses x close parentheses has a global (or absolute) minimum at x equals a if f open parentheses a close parentheses less or equal than f open parentheses x close parentheses for all x in the domain. In other words, f open parentheses a close parentheses is the smallest value of a function f open parentheses x close parentheses comma and occurs when x equals a.

Local (or Relative) Maximum

A function f open parentheses x close parentheses has a local (or relative) maximum at x equals a if f open parentheses a close parentheses greater or equal than f open parentheses x close parentheses for all x close to x equals a. In other words, f open parentheses a close parentheses is the largest value of a function f open parentheses x close parentheses for values around x equals a.

Local (or Relative) Minimum

A function f open parentheses x close parentheses has a local (or relative) minimum at x equals a if f open parentheses a close parentheses less or equal than f open parentheses x close parentheses for all x close to x equals a. In other words, f open parentheses a close parentheses is the smallest value of a function f open parentheses x close parentheses for values around x equals a.

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