3.3 Rates of Change and Behavior of Graphs

* 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.

1. 3.3 Rates of Change
Chapter 3 Functions

2. Concepts & Objectives
⚫Objectives for this section are:
⚫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.

3. Rate of Change
⚫The table below shows the average cost, in dollars, of a
gallon of gasoline for the years 2005-2012.
⚫The price change per year is a rate of changebecause it
describes how an output quantity (cost) changes relative
to the change in the input quantity (year).
⚫We can see that the rate of change was not the same
each year, but if we use only the beginning and ending
data, we would be finding the average rate of change
over the specified period of time.
Year20052006200720082009201020112012
Cost2.312.622.843.302.412.843.583.68

4. Average Rate of Change
⚫To find the average rate of change, we divide the change
in the output value by the change in the input value.Change in output
Average rate of change
Change in input
= ()()
21
21
21
21
y
x
yy
xx
fxfx
xx

=

−
=
−
−
=
−
The Greek letter 
(delta) signifies the
change in quantity.

5. Average Rate of Change (cont.)
⚫Example:Findtheaverage rate of change in the price of
gasoline from 2005-2012.
or about 19.6¢ each year
Year20052006200720082009201020112012
Cost2.312.622.843.302.412.843.583.683.682.31
20122005
1.37
0.196
7
y
x
−
=
−
=

6. Average Rate of Change (cont.)
Fromagraph:
⚫Example:Giventhefunctiong(t), find the average rate of
change on the interval [‒1, 2].

7. Average Rate of Change (cont.)
Fromagraph:
⚫Example:Giventhefunctiong(t), find the average rate of
change on the interval [‒1, 2].
At t= ‒1, g(t)= 4
At t= 2, g(t)= 1

8. Average Rate of Change (cont.)
Fromagraph:
⚫Example:Giventhefunctiong(t), find the average rate of
change on the interval [‒1, 2].
At t= ‒1, g(t)= 4
At t= 2, g(t)= 1()
143
1
213
y
x
−−
===−
−−

9. Average Rate of Change (cont.)
From a function:
⚫Example: Compute the average rate of change of the
function on the interval [2, 4].()
21
fxx
x
=−

10. Average Rate of Change (cont.)
From a function:
⚫Example: Compute the average rate of change of the
function on the interval [2, 4].()
21
fxx
x
=− ()
21
22
2
1
4
2
7
2
f=−
=−
= ()
21
44
4
1
16
4
63
4
f=−
=−
=

11. Average Rate of Change (cont.)
From a function:
⚫Example: Compute the average rate of change of the
function on the interval [2, 4].()
21
fxx
x
=− ()
21
22
2
1
4
2
7
2
f=−
=−
= ()
21
44
4
1
16
4
63
4
f=−
=−
= ()()42
42
6314
44
2
49
8
ffy
x
−
=
−
−
=
=

12. Increasing, Decreasing, or Constant
⚫We saythatafunctionisincreasingonanintervalifthe
functionvaluesincreaseastheinputvaluesincrease
within that interval.
⚫Theaveragerateofchangeofanincreasingfunction
ispositive.
⚫Similarly,afunctionisdecreasingonanintervalifthe
functionvalues decrease as the input values increase
over that interval.
⚫Theaveragerateofchangeofadecreasing function is
negative.

13. Increasing, Decreasing, or Constant
⚫This is a graph of ()
3
12fxxx=−
Increasing
Increasing
Decreasing

14. Increasing, Decreasing, or Constant
⚫This is a graph of
⚫Itisincreasingon
⚫It is decreasing on (‒2, 2)()
3
12fxxx=−
Increasing
Increasing
Decreasing( )(),22,−−

15. Local Maxima and Minima
⚫A value of the input where a function changes from
increasing to decreasing (as the input variable
increases) is the location of a local maximum.
⚫Ifafunctionhasmorethanone,wesayithaslocal
maxima.
⚫Similarly, a value of the input where a function changes
from decreasing to increasing as the input variable
increases is the location of a local minimum(plural
minima).
⚫Together, local maxima and minima are called local
extrema.

16. Local Maxima and Minima (cont.)
⚫The local maximum is 16,
which occurs at x= ‒2.
⚫Thelocalminimumis‒16,
which occurs at x= 2.
⚫The extrema give us the
intervals over which the
function is increasing or
decreasing.
Increasing
Increasing
Decreasing

17. Local Maxima and Minima (cont.)
Finding local extrema from a graph using Desmos:
⚫Example:Graphthefunctionand use the graph to
estimate the local extrema for the function.()
2
3
x
fx
x
=+

18. Local Maxima and Minima (cont.)
Finding local extrema from a graph using Desmos:
⚫Example:Graphthefunctionand use the graph to
estimate the local extrema for the function.
⚫()
2
3
x
fx
x
=+ When you enter the
function, Desmos will
automatically plot the
extrema (the gray dots).

19. Local Maxima and Minima (cont.)
Finding local extrema from a graph using Desmos:
⚫Example:Graphthefunctionand use the graph to
estimate the local extrema for the function.
⚫()
2
3
x
fx
x
=+ To find the coordi-
nates, click on the
dots. You will have to
determine whether it
is a maximum or a
minimum.
Minimum
Maximum

20. Absolute Maxima and Minima
⚫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.
⚫They-coordinates(output)atthehighest and lowest
points are called the absolute maximumand absolute
minimum,respectively.
⚫Not every graph has anabsolutemaximum or minimum
value.

21. Classwork
⚫College Algebra 2e
⚫3.3:6-14 (even); 3.2: 28-36 (even); 3.1: 60-86 (even)
⚫3.3ClassworkCheck
⚫Quiz 3.2