Rates of Change and Behavior of Graphs: Learn It 2

Behaviors of Functions

Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant

As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.

We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. The graph below shows examples of increasing and decreasing intervals on a function.

Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.
The function [latex]f\left(x\right)={x}^{3}-12x[/latex] is increasing on [latex]\left(-\infty \text{,}-\text{2}\right){{\cup }^{\text{ }}}^{\text{ }}\left(2,\infty \right)[/latex] and is decreasing on [latex]\left(-2\text{,}2\right)[/latex].

While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the output where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the output where a function changes from decreasing to increasing as the input variable increases is called a local minimum. The plural form is “local minima.” Together, local maxima and minima are called local extrema, or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local is replaced by the term relative. In this text, we will use the term local.

A function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.

local minima and local maxima

  • A function [latex]f[/latex] is an increasing function on an open interval if [latex]f\left(b\right) > f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b > a[/latex].
  • A function [latex]f[/latex] is a decreasing function on an open interval if [latex]f\left(b\right) < f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b > a[/latex].
  • A function [latex]f[/latex] has a local maximum at [latex]x=b[/latex] if there exists an interval [latex]\left(a,c\right)[/latex] with [latex]a < b < c[/latex] such that, for any [latex]x[/latex] in the interval [latex]\left(a,c\right)[/latex], [latex]f\left(x\right)\le f\left(b\right)[/latex].
  • Likewise, [latex]f[/latex] has a local minimum at [latex]x=b[/latex] if there exists an interval [latex]\left(a,c\right)[/latex] with [latex]a < b < c[/latex] such that, for any [latex]x[/latex] in the interval [latex]\left(a,c\right)[/latex], [latex]f\left(x\right)\ge f\left(b\right)[/latex].

 

For the function below, the local maximum is [latex]16[/latex], and it occurs at [latex]x=-2[/latex]. The local minimum is [latex]-16[/latex] and it occurs at [latex]x=2[/latex].Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum. The local maximum is 16 and occurs at x = negative 2. This is the point negative 2, 16. The local minimum is negative 16 and occurs at x = 2. This is the point 2, negative 16.

To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. The graph below illustrates these ideas for a local maximum.

Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.
Definition of a local maximum.
Given the function [latex]p\left(t\right)[/latex] in the graph below, identify the intervals on which the function appears to be increasing.Graph of a polynomial. As x gets large in the negative direction, the outputs of the function get large in the positive direction. As inputs approach 1, then the function value approaches a minimum of negative one. As x approaches 3, the values increase again and between 3 and 4 decrease one last time. As x gets large in the positive direction, the function values increase without bound.

The behavior of the function values of the graph of a function is read over the x-axis, from left to right. That is,

  • a function is said to be increasing if its function values increase as x increases;
  • a function is said to be decreasing if its function values decrease as x increases.
Graph the function [latex]f\left(x\right)=\dfrac{2}{x}+\dfrac{x}{3}[/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.

Recall that points on the graph of a function are ordered pairs in the form of

[latex]\left(\text{input, output}\right) \quad = \quad \left(x, f(x)\right)[/latex].

If a function’s graph has a local minimum or maximum at some point [latex]\left(x, f(x)\right)[/latex], we say

“the extrema occurs at [latex]x[/latex], and that the minimum or maximum is [latex]f(x)[/latex].”

Graph the function [latex]f\left(x\right)={x}^{3}-6{x}^{2}-15x+20[/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.