Characteristics of Polynomials:

Domain, Range, & Intercepts

1. What is interval notation?

2. What is the domain & range of a function?

3. How do I find the intercepts of a functions graphically and algebraically?

Daily Questions…….

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 (

( , 2) Draw a number line first if needed….

Let’s do another type….

4 9x Draw a number line first if needed….

4,9

Domain

all the x-values Read the graph from left to right

all the y-values Read the graph from bottom to top

Range

(2,4)

(-1,-5)

(4,0)

What is the domain of f(x)?

y = f(x)

Ex. 1

Domain

¿

(2,4)

(-1,-5)

(4,0)

y = f(x)

Ex. 2: What is the range of f(x)?

Range

[−𝟓 ,𝟒]

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,

,

, LB,

,UB

Zeros/x-intercepts/Solutions/Roots

Where the graph

crosses the x-axis

What’s a zero?

x-interceptsWhere the graph crosses

the x-axis. Also called zeros.

1,0 & 5,0

Analyze the Graph of a Function

Zeros: 1, 5

X-Intercepts: (-1,0)(1,0)(2,0)

Zeros, Roots

x = -1, 1, 2

X-Intercepts: (-2, 0) (-2, 0) (3,0)

Zeros, Roots:

x = -2, -2, 3

y-intercepts

Where the graph crosses the y-axis

y-Intercept: (0,-12)

y-Intercept: (0,2)

Find the y-intercepts & number of zeros:

a)

b)

4 2f x 3x 5x 1

2f x 2x 3x 15

Y- int: (0, -1)

Y- int: (0, 15)

# of zeros: 4

# of zeros: 2

Find the following

1.Domain:

2.Range:

3. x-intercepts:

4. y-intercepts:

All reals

All reals

(-2,0)(-2,0)(1,0)

(0, -4)

Find the following

1.Domain:

2.Range:

3. Zeros:

4. y-intercepts:

All reals

[-4, ∞)

-2, 2

(0, -4)

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

More About Polynomials…

1. When is a function increasing, decreasing, & constant?

2. Where are the max & min of a function?

Decreasing

Incr

easin

g

Constant

(1,-2)

(-1,2)

(-, -1) (1, )(-1, 1)

increasing increasingdecreasing

Ex.Increasing and decreasing are

stated in terms of domain

(2, 1)(0, 1)

(-, 0) (0, 2)increasing decreasing

(2, )constant

Ex. Increasing and decreasing are stated in terms of domain

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

decreasing (- ,0) 2,

Increasing (0,2)

Relative (Local) Minimum & Maximum Values

Relative Minimum: all of the lowest points

Relative Maximum: all of the highest points

Absolute (Global) Minimum & Maximum

Absolute Minimum: the lowest point

Absolute Maximum: the highest point

Relative maximum

Relative minimum

Find the following

1.Domain:

2.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, ) ( 2,0)

(-2, 0)max (0, -4)min

Find the following

1.Domain:

2.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)

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

(0, -4) (min)

(0, -4) (min)

(0, ∞)(-∞, 0)

End Behavior, Extrema & Sketching

End Behavior

You look left and right to figure out what’s happening up and down!

( ) _______

( ) _______

x f x

x f x

right

left

up, down, or flat

up, down, or flat

End Behavior: From a Graph

( ) _______

( ) _______

x f x

x f x

( ) _______

( ) _______

x f x

x f x

1.

2.2.

End Behavior: From a Graph3.

4.

3.

( ) _______

( ) _______

x f x

x f x

( ) _______

( ) _______

x f x

x f x

Determine the left and right behavior based on the equation.

6. f(x) = -x5 +3x4 – x

5. f(x) = x4 + 2x2 – 3x

7. f(x) = 2x3 – 3x2 + 5

( ) _______

( ) _______

x f x

x f x

( ) _______

( ) _______

x f x

x f x

( ) _______

( ) _______

x f x

x f x

Tell me what you know about the equation…

Odd exponent

Positive leading coefficient

Tell me what you know about the equation…

Even exponent

Positive leading coefficient

Tell me what you know about the equation…

Odd exponent

Positive leading coefficient

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

3

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

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

4

What if you didn’t have a graph?

11. f(x) = -x5 +3x4 – x

10. f(x) = x4 + 2x2 – 3x

12. f(x) = 2x3 – 3x2 + 5

Number of Extrema: ____

Number of Extrema: ____

Number of Extrema: ____

Sketching:

# of Zeros: _________ Y-Int: ________

( ) _______

( ) _______ _______

x f x

x f x extrema

213. 8 20

: 10, 2

f x x x

given zeros

Sketching:

( ) _______

( ) _______ _______

x f x

x f x extrema

3 214. 2 2

: 2, 1,1

f x x x x

given zeros

# of Zeros: _________ Y-Int: ________