1.3: Rates of Change and Behavior of Graphs (2024)

Last updated
Save as PDF

Page ID: 1294

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$

$ \newcommand{\vectorC}[1]{\textbf{#1}}$

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Learning Objectives

Find the average rate of change of a function.
Use a graph to determine where a function is increasing, decreasing, or constant.
Use a graph to locate local maxima and local minima.
Use a graph to locate the absolute maximum and absolute minimum.

Gasoline costs have experienced some wild fluctuations over the last several decades. Table $\PageIndex{1}$ lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.

Table $\PageIndex{1}$
$y$	2005	2006	2007	2008	2009	2010	2011	2012
$C(y)$	2.31	2.62	2.84	3.30	2.41	2.84	3.58	3.68

If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.

Finding the Average Rate of Change of a Function

The price change per year is a rate of change because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in Table $\PageIndex{1}$ did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.

\[\begin{align*} \text{Average rate of change}&=\dfrac{\text{Change in output}}{\text{Change in input}} \\[4pt] &=\dfrac{\Delta y}{\Delta x}\\[4pt] &=\dfrac{y_2-y_1}{x_2-x_1}\\[4pt] &=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\end{align*} \label{1.3.1}\]

The Greek letter $\Delta$ (delta) signifies the change in a quantity; we read the ratio as “delta-$y$ over delta-$x$” or “the change in $y$ divided by the change in $x$.” Occasionally we write $\Delta f$ instead of $\Delta y$, which still represents the change in the function’s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.

In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was

\[\dfrac{\Delta y}{\Delta x}=\dfrac{$1.37}{7 \text{years}}\approx \text{0.196 dollars per year.} \label{1.3.2}\]

On average, the price of gas increased by about 19.6¢ each year. Other examples of rates of change include:

Table \(\PageIndex{2}\)
\(t\) (hours)	0	1	2	3	4	5	6	7
\(D(t)\) (miles)	10	55	90	153	214	240	292	300

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. Figure $\PageIndex{3}$ shows examples of increasing and decreasing intervals on a function.

1.3: Rates of Change and Behavior of Graphs (4)

While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input 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 input 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.

Clearly, 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.

For the function whose graph is shown in Figure $\PageIndex{4}$, the local maximum is 16, and it occurs at $x=−2$. The local minimum is −16 and it occurs at $x=2$.

1.3: Rates of Change and Behavior of Graphs (5)

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. Figure $\PageIndex{5}$ illustrates these ideas for a local maximum.

1.3: Rates of Change and Behavior of Graphs (6)

These observations lead us to a formal definition of local extrema.

Local Minima and Local Maxima

A function $f$ is an increasing function on an open interval if $f(b)>f(a)$ for every $a$, $b$ interval where $b>a$.
A function $f$ is a decreasing function on an open interval if $f(b)<f(a)$ for every $a$, $b$ interval where $b>a$.

A function $f$ has a local maximum at a point $b$ in an open interval $(a,c)$ if $f(b)$ is greater than or equal to $f(x)$ for every point $x$ ($x$ does not equal $b$) in the interval. Likewise, $f$ has a local minimum at a point $b$ in $(a,c)$ if $f(b)$ is less than or equal to $f(x)$ for every $x$ ($x$ does not equal $b$) in the interval.

Example $\PageIndex{7}$ Finding Increasing and Decreasing Intervals on a Graph

Given the function $p(t)$ in Figure $\PageIndex{6}$, identify the intervals on which the function appears to be increasing.

1.3: Rates of Change and Behavior of Graphs (7)

Solution

We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from $t=1$ to $t=3$ and from $t=4$ on.

In interval notation, we would say the function appears to be increasing on the interval $(1,3)$ and the interval $(4,\infty)$.

Analysis

Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at $t=1$, $t=3$, and $t=4$. These points are the local extrema (two minima and a maximum).

Example $\PageIndex{8}$: Finding Local Extrema from a Graph

Graph the function $f(x)=\frac{2}{x}+\frac{x}{3}$. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.

Solution

Using technology, we find that the graph of the function looks like that in Figure $\PageIndex{7}$. It appears there is a low point, or local minimum, between $x=2$ and $x=3$, and a mirror-image high point, or local maximum, somewhere between $x=−3$ and $x=−2$

1.3: Rates of Change and Behavior of Graphs (8)

Analysis

Most graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure $\PageIndex{8}$ provides screen images from two different technologies, showing the estimate for the local maximum and minimum.

1.3: Rates of Change and Behavior of Graphs (9)

Based on these estimates, the function is increasing on the interval $(−\infty,−2.449)$ and $(2.449,\infty)$. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact location of the extrema is at $\pm\sqrt{6}$, but determining this requires calculus.)

Exercise $\PageIndex{8}$

Graph the function $f(x)=x^3−6x^2−15x+20$ to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.

Solution

The local maximum appears to occur at $(−1,28)$, and the local minimum occurs at $(5,−80)$. The function is increasing on $(−\infty,−1)\cup(5,\infty)$ and decreasing on $(−1,5)$.

Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).

Example $\PageIndex{9}$: Finding Local Maxima and Minima from a Graph

For the function f whose graph is shown in Figure $\PageIndex{9}$, find all local maxima and minima.

1.3: Rates of Change and Behavior of Graphs (11)

Solution

Observe the graph of $f$. The graph attains a local maximum at $x=1$ because it is the highest point in an open interval around $x=1$.The local maximum is the y-coordinate at $x=1$, which is 2.

The graph attains a local minimum at $x=−1$ because it is the lowest point in an open interval around $x=−1$. The local minimum is the y-coordinate at $x=−1$, which is −2.

Analyzing the Toolkit Functions for Increasing or Decreasing Intervals

We will now return to our toolkit functions and discuss their graphical behavior in Figure $\PageIndex{10}$, Figure $\PageIndex{11}$, and Figure $\PageIndex{12}$.

1.3: Rates of Change and Behavior of Graphs (12)

1.3: Rates of Change and Behavior of Graphs (13)

Figure $\PageIndex{12}$

Use A Graph to Locate the Absolute Maximum and Absolute Minimum

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The y-coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively. To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function (Figure $\PageIndex{13}$).

1.3: Rates of Change and Behavior of Graphs (15)

Not every function has an absolute maximum or minimum value. The toolkit function $f(x)=x^3$ is one such function.

Absolute Maxima and Minima

The absolute maximum of $f$ at $x=c$ is $f(c)$ where $f(c)≥f(x)$ for all $x$ in the domain of $f$.
The absolute minimum of $f$ at $x=d$ is $f(d)$ where $f(d)≤f(x)$ for all $x$ in the domain of $f$.

Example $\PageIndex{10}$: Finding Absolute Maxima and Minima from a Graph

For the function f shown in Figure $\PageIndex{14}$, find all absolute maxima and minima.

1.3: Rates of Change and Behavior of Graphs (16)

Solution

Observe the graph of $f$. The graph attains an absolute maximum in two locations, $x=−2$ and $x=2$, because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y-coordinate at $x=−2$ and $x=2$, which is 16.

The graph attains an absolute minimum at x=3, because it is the lowest point on the domain of the function’s graph. The absolute minimum is the y-coordinate at x=3,which is−10.

Key Equations

Average rate of change: $\dfrac{\Delta y}{\Delta x}=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$

Key Concepts

A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See Example.
Identifying points that mark the interval on a graph can be used to find the average rate of change. See Example.
Comparing pairs of input and output values in a table can also be used to find the average rate of change. See Example.
An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See Example and Example.
The average rate of change can sometimes be determined as an expression. See Example.
A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See Example.
A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
Minima and maxima are also called extrema.
We can find local extrema from a graph. See Example and Example.
The highest and lowest points on a graph indicate the maxima and minima. See Example.

\(y\)	2005	2006	2007	2008	2009	2010	2011	2012
\(C(y)\)	2.31	2.62	2.84	3.30	2.41	2.84	3.58	3.68