1. Characteristics of Polynomials: Domain, Range, & Intercepts
Daily Questions…….
What is interval notation?
What is the domain & range of a function?
How do I find the intercepts of a functions graphically and algebraically?

2. How do we write in interval notation?
x < 2….
when you want to include use a bracket [
when you want to exclude use a parenthesis (
Draw a number line first if needed….

3. Draw a number line first if needed….
Let’s do another type….
Draw a number line first if needed….

4. Domain Range all the x-values Read the graph from left to right
all the y-values
Read the graph from bottom to top

5. What is the domain of f(x)?
Ex. 1
[−𝟏,𝟒)
(2,4)
(-1,-5)
(4,0)
y = f(x)
Domain

6. Ex. 2: What is the range of f(x)?
[−𝟓,𝟒]
(2,4)
(-1,-5)
(4,0)
y = f(x)
Range

7. With polynomials…. The DOMAIN is always All Reals,
The RANGE will be one of the following:
All Reals,
Lower Boundary to infinity,
Negative infinity to Upper Boundary,

8. Zeros/x-intercepts/Solutions/Roots Where the graph crosses the x-axis
What’s a zero?
Zeros/x-intercepts/Solutions/Roots
Where the graph crosses the x-axis

9. Where the graph crosses the x-axis. Also called zeros.
Analyze the Graph of a Function
x-intercepts
Where the graph crosses the x-axis. Also called zeros.
Zeros: 1, 5

10. X-Intercepts: (-1,0)(1,0)(2,0)
Zeros, Roots
x = -1, 1, 2

11. X-Intercepts: (-2, 0) (-2, 0) (3,0)
Zeros, Roots:
x = -2, -2, 3

12. y-intercepts
Where the graph crosses the y-axis

13. y-Intercept: (0,-12)

14. y-Intercept: (0,2)

15. Y- int: (0, -1) # of zeros: 4 Y- int: (0, 15) # of zeros: 2
Find the y-intercepts & number of zeros: a) b)
Y- int: (0, -1)
# of zeros: 4
Y- int: (0, 15)
# of zeros: 2

16. All reals All reals (-2,0)(-2,0)(1,0) (0, -4) Find the following
Domain:
Range:
3. x-intercepts:
4. y-intercepts:
All reals
All reals
(-2,0)(-2,0)(1,0)
(0, -4)

17. All reals [-4, ∞) -2, 2 (0, -4) Find the following Domain: Range:
3. Zeros:
4. y-intercepts:
All reals
[-4, ∞)
-2, 2
(0, -4)

18. More About Polynomials…
1. When is a function increasing, decreasing, & constant?
2. Where are the max & min of a function?

19. Increasing
Decreasing
Constant
This is a piecewise function

20. Increasing and decreasing are stated in terms of domain
Ex.
(-, -1)
(-1, 1)
(1, )
increasing
decreasing
increasing
(-1,2)
(1,-2)

21. Increasing and decreasing are stated in terms of domain
Ex.
Increasing and decreasing are stated in terms of domain
(-, 0)
(0, 2)
(2, )
constant
increasing
decreasing
(0, 1)
(2, 1)

22. Determine the intervals over which the function is increasing and decreasing…

23. Relative (Local) Minimum & Maximum Values
Relative Minimum: all of the lowest points
Relative Maximum: all of the highest points

24. Absolute (Global) Minimum & Maximum
Absolute Minimum: the lowest point
Absolute Maximum: the highest point

25. Relative maximum
Relative minimum

26. All reals All reals -2, -2, 1 (0, -4) none (-2, 0)max (0, -4)min
Find the following
Domain:
Range:
3. Zeros:
4. y-intercepts:
5. Absolute Max/Min:
6. Relative Max/Min:
7. Increasing:
8. Decreasing:
All reals
All reals
-2, -2, 1
(0, -4)
none
(-2, 0)max (0, -4)min

27. All reals [-4, ∞) -2, 2 (0, -4) (0, -4) (min) (0, -4) (min) (0, ∞)
Find the following
Domain:
Range:
3. Zeros:
4. y-intercepts:
5. Absolute Max/Min:
6. Relative Max/Min:
7. Increasing:
8. Decreasing:
All reals
[-4, ∞)
-2, 2
(0, -4)
(0, -4) (min)
(0, -4) (min)
(0, ∞)
(-∞, 0)

28. End Behavior, Extrema & Sketching

29. End Behavior up, down, or flat right up, down, or flat left
You look left and right to figure out what’s happening up and down!
up, down, or flat
right
up, down, or flat
left

30. End Behavior: From a Graph1. 2. 2.

31. End Behavior: From a Graph3. 3. 4.

32. 5. f(x) = x4 + 2x2 – 3x 6. f(x) = -x5 +3x4 – x 7. f(x) = 2x3 – 3x2 + 5
Determine the left and right behavior based on the equation.
5. f(x) = x4 + 2x2 – 3x
6. f(x) = -x5 +3x4 – x
7. f(x) = 2x3 – 3x2 + 5

33. Tell me what you know about the equation…
Odd exponent
Positive leading coefficient

34. Tell me what you know about the equation…
Page 261 #53
Even exponent
Positive leading coefficient

35. Tell me what you know about the equation…
Page 261 #54
Odd exponent
Positive leading coefficient

36. Extrema are turns in the graph.
If you are given a graph take the turns and add 1 to get the least possible degree of the polynomial.
If you are given the function, take the degree and subtract 1 to get the max possible number of extrema.
f(x) = 2x3 – 3x2 + 5

37. 8. What is the least possible degree of this function?

38. 9. What is the least possible degree of this function?

39. What if you didn’t have a graph?
10. f(x) = x4 + 2x2 – 3x
Number of Extrema: ____
11. f(x) = -x5 +3x4 – x
Number of Extrema: ____
12. f(x) = 2x3 – 3x2 + 5
Number of Extrema: ____

40. Sketching:
# of Zeros: _________ Y-Int: ________

41. Sketching:
# of Zeros: _________ Y-Int: ________