1. Section 1.3 More on Functions and Their Graphs

3. The open intervals describing where functions increase, decrease, or are constant, use x-coordinates and not the y-coordinates.

4. Increasing, Decreasing, Constant is just like riding a roller coaster!
Example 1
Find where the graph is increasing? Where is it decreasing? Where is it constant?
Increasing:
(-5, -2)
Decreasing:
(-∞, -5)
Constant:
(0, 2)
No where
(-2, 0)
(5, ∞)
(2, 5)
Increasing, Decreasing, Constant is just like riding a roller coaster!
Increasing
Decreasing
Increasing
Decreasing
Increasing
Decreasing

5. Example 2
Find where the graph is increasing? Where is it decreasing? Where is it constant?
(2, ∞)
(-∞, 2)
Constant:
Decreasing:
Increasing:
No where
Increasing
Decreasing

6. Example 3
Find where the graph is increasing? Where is it decreasing? Where is it constant?
Increasing:
Decreasing:
Constant:
[-2, 0)
[0, 2)
[2, 4)
No Where
No Where
Constant
Constant
Constant

7. Relative Maxima And Relative Minima

8. Example 4
Where are the relative minimums? Where are the relative maximums?
Why are the maximums and minimums called relative or local?
(-2, 2)
(2, 2)
(-5, -5)
(5, -5)

9. Even and Odd Functions and Symmetry

10. Even and Odd Functions and Symmetry
A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (- x, y) is also on the graph. All even functions have graphs with this kind of symmetry.
A graph is symmetric with respect to the origin if for every point (x, y) on the graph, the point (-x, -y) is also on the graph. All odd functions have graphs with origin symmetry (or 180 degree rotational symmetry).

11. It matches up with mirror symmetry over the y-axis.
Example 5
Is this an even or odd function?
Even!
It matches up with mirror symmetry over the y-axis.

12. It has 𝟏𝟖𝟎 𝟎 rotational symmetry.
Example 6
Is this an even or odd function?
Odd!
It has 𝟏𝟖𝟎 𝟎 rotational symmetry.

13. It has 𝟏𝟖𝟎 𝟎 rotational symmetry.
Example
Is this an even or odd function?
Odd!
It has 𝟏𝟖𝟎 𝟎 rotational symmetry.

14. Even or Odd Algebraically
Even functions are when
Odd functions are when
Examples 1. 2.
Not the Same,
so not EVEN.
Now factor out a -1.
The Same,
so it’s EVEN.
Now it’s the Same,
so it’s ODD.

15. Even and Odd Algebraically
Neither.
Neither.
Odd.
Neither.
Odd.

16. Interpreting Information based on a graph
For the graph pictured, find the following
The domain of f
The range of f
The x-intercepts
The y-intercepts
Intervals where f is increasing
Intervals where f is decreasing
Intervals where f is constant
The point at which f has a relative minimum/maximum
(-∞, ∞)
(-2, 2) and (2, 2)
[-5, ∞)
maximums
(-6.2, 0), (-3, 0), (0, 0), (3, 0), (6.2, 0)
(0, 0)
(-5, -2), (0, 2), (5, ∞)
minimums
(-5, -5) and (5, -5)
(-∞, -5), (-2, 0), (2, 5)
No where.
f(-2) = 2 and f(2) = 2
and f(6.5) = 2 and f(-6.5) = 2
The values for f (-3) = ? and f (x) = 2
Is f even, odd, or neither?
f(-3) = 0
Even.

17. Piecewise Functions
A function that is defined by two or more equations over a specific domain is called piecewise function. Many cell phone plans can be represented with piecewise functions.
Example
Find and interpret each of the following.

18. Example
Graph the following piecewise function.

19. Piecewise Functions – Step Function – Greatest Integer Function
Some piecewise functions are called STEP FUNCTIONS because their graph form DISCONTINUOUS steps. Check out the picture!
One such function is called the GREATEST INTEGER FUNCTION, which is symbolized by int(x) or [x]. This means the greatest integer that is less than or equal to x. 1 1 1 1 2 2 2 2

20. More examples! Please add them in to your paper.
= 3
= 94
= -7
= 29

21. Weight For Letters (In ounces)
Example
The charge would be $0.59.
The charge would be $0.76.
$1.25
$1.00
$0.75
Pricing For Letters
(In US dollars)
$0.50
$0.25 1 2 3 4 5
Weight For Letters (In ounces)

22. Functions and Difference Quotients
See next slide.

23. Continued on the next slide.

25. Example
Find and simplify the expressions if

26. Example
Find and simplify the expressions if

27. Example
Find and simplify the expressions if

28. Time to review
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(b)
(c)
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(c)

29. (a)
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(c)
(d)
(a)

30. (a)
(b)
(c)
(d)
(c)