1. 4.3B Analyzing Functions

2. Analyzing any Function
Graphs of functions help you visualize relationships between two variables.
Finding key information to help describe the graph is analyzing the function.

3. Analyzing Graphs of Functions
Remember that a graph of a function is the collection of ordered pairs (x, y) placed on a coordinate plane.
Sometimes the number of ordered pairs are infinite (continuous), sometimes they are not (discrete).
The set of all values for “x” is the domain and it read left to right.
The set of all values for “y” is the range and it is read bottom to top.
Determining the domain and range is one element in analyzing graphs.

4. Analyzing Graphs of Functions
Symmetry:
There are two types of symmetry associated with graphs. Functions containing symmetry have special names:
Even – An even function has line (or reflection) symmetry around the y-axis.
Odd – An odd function has point (or rotation) symmetry about the origin.
Describing the shape using symmetry: even, odd or neither is another element in analyzing graphs.

5. Examples of Even and Odd Functions; Symmetry
Even: Odd:

6. Analyzing Graphs of Functions
Graphs may have key points of interest.
The y-intercept, x-intercept, any high or low points may be of interest in analyzing the graph.
The y-intercept is where the graph crosses the y – axis.
The x-intercepts, also known as the zeros of the function, are where the graph crosses the x – axis.
High and low points can be written in terms of relative maximums and relative minimums.
Identifying key points is another element in analyzing graphs.

7. Relative Maximum and Relative Minimum

8. Analyzing Graphs of Functions
As you review a graph from left to right on the number line, the changes in the graph should be noted. The intervals of the independent variable “x” can be used to describe the graph as increasing, decreasing or constant. Graphs can change direction at its relative maximums and relative minimums Identifying increasing, decreasing and constant intervals is one more element in analyzing graphs.

9. Increasing and Decreasing Functions
We right increasing and decreasing in interval notation using x-values.

10. Example: Analyze the function below
y = f (x)
(0, 5)
(-1, 0)
(6, 1)
(1, 0)
(5, 0)
(-2, -5)
(4, -5)
(3, -7)
Answers on next page.

11. Example: Analyzing a function
Domain:
Range:
Relative Minimum:
Relative Maximum:
Y – intercept:
Zeros (or x-intercepts):
Increasing:
Decreasing:
Symmetry:
f(x) > 0:
f(x) < 0:
Neither