Fifth Edition
James Stewart  Lothar Redlin  Saleem Watson

3

3.3
Getting Information from the
Graph of a Function

The Graph of a Function
Many properties of a function are more easily obtained from a graph than from the rule that describes the function.
• We will see in this section how a graph tells us:
– Whether the values of a function are increasing or decreasing
– Where the maximum and minimum values of a function are.

The Values of a Function
A complete graph of a function contains all the information about a function.
• This is because the graph tells us which input values correspond to which output values.
To analyze the graph of a function, we must keep in mind that the height of the graph is the value of the function .
• So we can read off the values of a function from its graph.

E.g. 1 —Find the Values of a Function from a Graph
The function T graphed here gives the temperature between noon and 6 P.M. at a certain weather station.

E.g. 1 —Find the Values of a Function from a Graph
(a) Find T (1), T (3), and T (5).
(b) Which is larger, T (2) or T (4)?
(c) Find the value(s) of x for which T ( x ) = 25.
(d) Find the value(s) of x for which T ( x ) ≥ 25.

E.g. 1 —Find Values of a Function
Example (a)
T (1) is the temperature at 1:00 P.M.
It is represented by the height of the graph above the x -axis at x = 1.
• Thus, T (1) = 25.
• Similarly,
T (3) = 30 and
T (5) = 20.

E.g. 1 —Find Values of a Function
Example (b)
The graph is higher at x = 2 than at x = 4.
• It follows that T (2) is larger than T (4).

E.g. 1 —Find Values of a Function
Example (c)
The height of the graph is 25 when x is 1 and when x is 4.
• In other words, the temperature is 25 at 1:00
P.M.
and 4:00
P.M.

E.g. 1 —Find Values of a Function
Example (d)
The graph is higher than 25 for x between 1 and 4.
• In other words, the temperature was 25 or greater between 1:00
P.M.
and 4:00
P.M.

Finding the Domain and Range from a Graph
The graph of a function helps us to picture the domain and range of the function on the x -axis and y -axis, as shown.

E.g. 2 —Finding the Domain and Range from a Graph
(a) Use a graphing calculator to draw
 
(b) Find the domain and range of f .

E.g. 2 —Finding the Domain and Range from a Graph
(a) The graph is shown.
(b) We see that:
• The domain is [–2, 2].
• The range is [0, 2].

Increasing and Decreasing Functions

Increasing and Decreasing Functions
The graph shown here rises, falls, then rises again as we move from left to right.
• It rises from A to B , falls from B to C , and rises again from C to D .

Increasing and Decreasing Functions
f
• Increasing when its graph rises.
• Decreasing when its graph falls.

Increasing Function —Definition f is increasing on an interval I if f ( x
1
) < f ( x
2
) whenever x
1
< x
2 in I .

Decreasing Function —Definition f is decreasing on an interval I if f ( x
1
) > f ( x
2
) whenever x
1
< x
2 in I .

E.g. 3 —Intervals Where a Func. Increases & Decreases
The graph gives the weight W of a person at age x .
• Determine the intervals on which the function W is increasing and on which it is decreasing.

E.g. 3 —Intervals Where a Func. Increases & Decreases
• Increasing on [0, 25] and [35, 40].
• Decreasing on [40, 50].
• Constant (neither increasing nor decreasing) on [25, 35] and [50, 80].

E.g. 3 —Intervals Where a Func. Increases & Decreases
• The person gained weight until age 25.
• He gained weight again between ages 35 and 40.
• He lost weight between ages 40 and 50.

E.g. 4 —Finding Where a Function Incrs. & Decrs.
(a) Sketch the graph of the function f ( x ) = 12 x 2 + 4 x 3 – 3 x 4
(b) Find the domain and range of f .
(c) Find the intervals on which f increases and decreases.

E.g. 4 —Sketching the Graph
Example (a)
We use a graphing calculator to sketch the graph here.

E.g. 4 —Domain
Example (b)
The domain of f is because f is defined for all real numbers.

E.g. 4 —Range
Example (b)
Using the TRACE feature on the calculator, we find that the highest value is f (2) = 32.
• So, the range of f is ( –∞, 32].

E.g. 4 —Increase & Decrease
We see that f is:
Example (c)
• Increasing on ( – ∞, 1] and [0, 2].
• Decreasing on
[
–1
, 0] and [2, ∞).

E.g. 5 —Finding Where a Function Incrs. & Decrs.
(a) Sketch the graph of the function f ( x ) = x 2/3 .
(b) Find the domain and range of the function.
(c) Find the intervals on which f increases and decreases.

E.g. 5 —Sketching the Graph
Example (a)
We use a graphing calculator to sketch the graph here.

E.g. 5 —Domain & Range
Example (b)
From the graph, we observe that:
• The domain of f is .
• The range is [0, ∞).

E.g. 5 —Increase & Decrease
We see that f is:
• Decreasing on (-∞, 0].
• Increasing on
[0, ∞).
Example (c)

Local Maximums and Minimums of a Function
Finding the largest or smallest values of a function is important in many applications.
• For example, if a function represents revenue or profit, then we are interested in its maximum value.
• For a function that represents cost, we would want to find its minimum value.
• See Focus on Modeling: Modeling with Functions on pages 280-289 for many such examples.

Local Maximums and Minimums of a Function
We can easily find these values from the graph of a function.
We first define what we mean by a local maximum or minimum.

Local Maximums of a Function —Definition
The function value f ( a ) is a local maximum value of f if f ( a ) ≥ f ( x ) when x is near a
• This means that f ( a ) ≥ f ( x ) for all x in some open interval containing a .
• In this case, we say that f has a local maximum at x = a .

Local Minimums of a Function —Definition
The function value f ( a ) is a local minimum value of f if f ( a ) ≤ f ( x ) when x is near a
• This means that f ( a )
≤ f ( x ) for all x in some open interval containing a .
• In this case, we say that f has a local minimum at x = a .

Using Graphing Devices to Find Extreme Values
Suppose there is a viewing rectangle such that the point ( a , f ( a )) is the highest point on the graph of f within the viewing rectangle
(not on the edge).

Local Maximum Value
f
a
f
• Notice that f ( a ) ≥ f ( x ) for all numbers x that are close to a .

Using Graphing Devices to Find Extreme Values
Similarly, suppose there is a viewing rectangle such that the point ( b , f ( b )) is the lowest point on the graph of f within the rectangle.

Local Minimum Value
f
b
f
• In this case, f ( b ) ≤ f ( x ) for all numbers x that are close to b.

E.g. 6 —Finding Local Maxima & Minima from a Graph
Find the local maximum and minimum values of the function f ( x ) = x 3 – 8 x + 1 correct to three decimals.

E.g. 6 —Local Maxima & Minima from a Graph
Looking at the graph of f , there appears to be:
• One local maximum between x = –2 and x = –1.
• One local minimum between x = 1 and x = 2.
Let’s find the coordinates of the local maximum point first.

E.g. 6 —Local Maxima & Minima from a Graph
• Using the TRACE feature on the graphing device, we move the cursor along the curve and observe how the y -coordinates change.

E.g. 6 —Local Maxima & Minima from a Graph
y
• This value occurs when x is –1.633, correct to three decimals.

E.g. 6 —Local Maxima & Minima from a Graph
By zooming in to the viewing rectangle shown, we find that the local minimum value is about –7.709.
• This value occurs when x ≈ 1.633.

Using Graphing Devices to Find Extreme Values
The maximum and minimum commands on a TI-82 or TI-83 calculator provide another method for finding extreme values of functions.
• We use this method in the next example.

E.g. 7 —A Model for the Food Price Index
A model for the food price index (the price of a representative “basket” of foods) between
1990 and 2000 is given by the function
I t
 
0.0113
t 3 
0.0681
t 2 
0.198
t

99.1
where:
• t is measured in years since midyear 1990, so 0 ≤ t ≤ 10.
•
I ( t ) is scaled so that I (3) = 100.

E.g. 7 —A Model for the Food Price Index

E.g. 7 —A Model for the Food Price Index
The graph of I as a function of t is shown.
There appears to be a maximum between t = 4 and t = 7.

E.g. 7 —A Model for the Food Price Index
Using the maximum command, we see the maximum value of I is about 100.38.
• It occurs when t ≈ 5.15, which corresponds to August 1995.