Section 3.2Properties of a Function’s Graph

Objectives

  1. Determining the Intercepts of a Function
  2. Determining the Domain and Range of a Function from its Graph
  3. Determining Where a Function is Increasing, Decreasing or Constant
  4. Determining Relative Maximum and Relative Minimum Values of a Function
  5. Determining if a Function is Even, Odd or Neither
  6. DeterminingInformation about a Function from a Graph

Objective 1: Determining the Intercepts of a Function

In the previous section, we discussed that every function has a graph in the Cartesian plane. In this section we will take a closer look at a function’s graph by discussing some of its characteristics. A good place to start is to talk about a function’s intercepts.

An intercept of a functionis a pointon the graph of a function where the graph either crosses of touches a coordinate axis. There are two types of intercepts:

1) The y-intercept, which is the y-coordinate of the point where the graph crosses or touches the y-axis.

2) The x-intercepts, which are the x-coordinates of the points where the graph crosses or touches the
x-axis.

The y-intercept:
A function can have at most one y-intercept. The y-intercept exists if is in the domain of the function. The y-intercept can be found by evaluating

The x-intercept(s):
A function may have several (even infinitely many) x-intercepts. The x-intercepts, also calledreal zeros, can be found by finding all real solutions to the equation. Although a function may have several zeros, only the real zeros are x-intercepts.

Objective 2: Determining the Domain and Range of a Function from its Graph

In Section 3.1 we saw that every function has a graph. The figure to the right

illustrates how we can use a function’s graph to determine the

domain and range. The domain is the set of all input values (x-values)

and the range is the set of all output values (y-values). The domain is the

intervalwhile the range is the interval .

The domain and range of y=f(x)

Objective 3: Determining Where a Function is Increasing, Decreasing or Constant

A function f is said to be increasing on an open interval if the value of gets larger as x gets larger on the interval. The graph of f rises from left to right on the interval in which f is increasing.

Likewise, a function f is said to be decreasing on an open interval if the value of gets smaller as x gets larger on the interval. The graph of f falls from left to right on the interval in which f is decreasing.

A graph is constant on an open interval if the values of do not change as x gets larger on the interval. In this case, the graph is a horizontal line on the interval.

The function is increasing on the interval .

The function is decreasing on the interval .

The function is constant on the interval .

Objective 4: DeterminingRelative Maximum and Relative Minimum Values of a Function

When a function changes from increasing to decreasing at a point , then f is said to have a relative maximum at . The relative maximum value is .

Similarly, when a function changes from decreasing to increasing at a point , then f is said to have a relative minimum at . The relative minimum value is .

(a) (b)

Note: The word “relative” indicates that the function obtains a maximum or minimum value relative to some open interval. It is not necessarily the maximum (or minimum) value of the function on the entire domain.

A relative maximum cannot occur at an endpoint and must occur in an open interval. This applies to a relative minimum as well.

Objective 5: Determiningif a Function is Even, Odd or Neither

The graphs of some functions are symmetric about the y-axis. For example, the graphs of the 3 functions sketched in Figure 9 are all symmetric about the y-axis.


Functions whose graphs
are symmetric about the y-axis

are called even functions.

Imagine “folding” each graph along the y-axis. If each graph were folded along the y-axis, the left and right halves of each graph would match. Functions whose graphs are symmetric about the y-axis are called even functions. Notice that for any point on each graph, the point also lies on the graph. Therefore, for any value in the domain, .

DefinitionEven Functions

A function is even if for every in the domain, . Even functions are symmetric about the

y-axis. For each point on the graph, the point is also on the graph.

Functions can also be symmetric about the origin. The functions sketched below illustrate this type of symmetry. If we were to fold each graph along the x-axis then fold again along the y-axis, the parts of each graph located in opposite quadrants would match. Functions whose graphs are symmetric about the origin are called odd functions.

Functions whose graphs

are symmetric about the

origin are called

odd functions.

Notice that for any point on each graph in Figure 10, the point also lies on the graph. Therefore, for any value in the domain, or equivalently .

DefinitionOdd Functions

A function is odd if for every in the domain, . Odd functions are symmetric about the

origin. For each point on the graph, the point is also on the graph.

Objective 6: Determining Information about a Function from a Graph

We are now ready to take all of the concepts learned in this section to answer questions about a function’s graph.