Statistical Reasoning Name___________________________Unit 6 Lesson 2 Date_____________ Period____

Characteristics of Functions

Remember, a function is defined as a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

During this lesson, we will be looking at the graphs of several different functions and identifying all of the following characteristics:

Domain Range Intercepts Even, Odd, or Neither Symmetry Increasing, Decreasing, and Constant Intervals Local and Absolute Maximum and Minimum Rate of Change

First, let’s review what you should already know:

Domain, Range, and Intercepts

The domain of a function is the set of all possible __________ which will make the function “work”, and will output real _________.

The range of a function is the resulting __________ we get after substituting all the possible _________.

An x – intercept is where the graph touches or crosses the _________ or a point in which y = _____. A y – intercept is where the graph touches or crosses the _________ or a point in which x = _____.

Complete the examples below:

1. 2. 3.

Domain: _____________Range: _____________x – intercept(s): _____________y – intercept(s): _____________

Domain: _____________Range: _____________x – intercept(s): _____________y – intercept(s): _____________

Domain: _____________Range: _____________x – intercept(s): _____________y – intercept(s): _____________

6.2.3

Even, Odd, Neither, and Symmetry

We can classify the graphs of functions as either even, odd, or neither.

Even OddA function is an even function if _____________for all x in the domain of f.

*The right side of the equation of an even function does NOT change if x is replaced with –x.

Even functions are symmetric with respect to the ____________________. This means we could fold the graph on the axis, and it would line up perfectly on both sides!

A function is an odd function if ______________for all x in the domain of f.

*Every term on the right side of the equation changes signs if x is replaced with –x.

Odd functions are symmetric with respect to the ____________________. This means we can flip the image upside down and it will appear exactly the same!

If we cannot classify a function as even or odd, then we call it neither!

***Please Note: There is a such thing as symmetry with respect to the x-axis, but if this type of symmetry exists, then it is NOT a function!***

Directions: Determine graphically using possible symmetry, whether the following functions are even, odd, or neither.

1. 2. 3.

4. 5. 6.

Increasing, Decreasing, and Constant Intervals

A function is increasing on an interval if the y-values are ____________ as the x values ____________.

A function is decreasing on an interval if the y-values are ____________ as the x-values ____________.

A function is constant on an interval if the y-values are ________________as the x-values ____________.

From A to B the graph is _________________________ so the rate of change is ___________.

From B to C the graph is _________________________ so the rate of change is ___________.

From C to D the graph is _________________________ so the rate of change is ___________.

From D to E the graph is _________________________ so the rate of change is ___________.

So the increasing intervals would be written as: _______________________.

The decreasing interval would be written as: _____________________.

The constant interval would be written as: ____________________.

Complete the examples below:Example 1:

Increasing Interval(s):

Decreasing Interval(s):

Constant Interval(s):

Example 2:

Increasing Interval(s):

Decreasing Interval(s):

Constant Interval(s):

Example 3:

Increasing Interval(s):

Decreasing Interval(s):

Constant Interval(s):

Local Maximum and Minimum, Absolute Maximum and Minimum

A B C D E

The absolute maximum is the highest maximum of a graph, while the local maximum is the highest maximum over an interval but not the highest of the entire graph.

Example

The absolute maxima would be (-3,3) and (3,3) since both points have the same y – value.

The local maximum would be (0,2) since it is the highest in an interval, but not the highest maximum.

Example

Identify the absolute and local maximum of the graph to the left:

The absolute minimum is the lowest minimum of a graph, while the local minimum is the lowest minimum over an interval but not the lowest of the entire graph.

Example

The absolute minima would be (-1,0) and (1,0), since the points have the same y – value.

There would be no local minima because all of the minima are the lowest points on the entire graph making them the absolute minima.

Example

Identify the absolute and local minimum of the graph the left:

Average Rate of Change

The average rate of change of the function y=f (x ) between x=a and x=b can be found using the formula below:

avergae rate of change= change∈ychange∈x=f (b )−f (a)b−a

Consider the example below:

a. Find the average rate of change between x=−2 and x=0. You should get a negative rate of change, why?

b. Find the average rate of change between x=1 and x=2. Explain your answer.

c. Find the average rate of change between x=2 and x=4. You

should get a positive rate of change, why?

Now try these:

Example 1:

Find the average rate of change between x=5 and x=6.

Example 2:

Find the average rate of change between x=0 and x=1.

Example 3:

Find the average rate of change between x=−1 and x=1.

Complete the practice below.1. Domain

Rangex – intercept(s)y – intercept(s)Even, Odd, Neither

SymmetryIncreasing Int.Decreasing Int.Constant Int.MaximumMinimumRate of Changeb/w x=−2.1 and x=−1.4

2. DomainRangex – intercept(s)y – intercept(s)Even, Odd, NeitherSymmetryIncreasing Int.Decreasing Int.Constant Int.MaximumMinimumRate of Changeb/w x=1 and x=2.3

3. DomainRangex – intercept(s)y – intercept(s)Even, Odd, Neither

SymmetryIncreasing Int.Decreasing Int.Constant Int.MaximumMinimumRate of Changeb/w x=−4 and x=−2

(-1.3, 9) (2.3, 9)

(.5, -1.6)

(0, 0) (1, 0)(3, 0)

(-2, 0)