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The DEGREE of a polynomial is the largest degree of any single term in the polynomial (Polynomials are often written in descending order of the degree of its terms) COEFFICIENTS are the numerical value of each term in the polynomial

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Polynomials review

The DEGREE of a polynomial is the largest degree of any single term in the polynomial

(Polynomials are often written in descending order of the degree of its terms)

COEFFICIENTS are the numerical value of each term in the polynomial

The LEADING COEFFICIENTis the numerical value of the term with the HIGHEST DEGREE.

POLYNOMIALS REVIEW


Polynomials Review Practice

For each polynomial

Write the polynomial in descending order

Identify the DEGREE and LEADING COEFFICIENT of the polynomial


Finding values of a polynomial substitute values of x into polynomial and simplify
Finding values of a polynomial: Substitute values of x into polynomial and simplify:

Find each value for

1. 2.

3. 4.


Graphs of Polynomial Functions:

Constant Function Linear Function Quadratic Function

(degree = 0) (degree = 1) (degree = 2)

Cubic Function Quartic Function Quintic Function

(deg. = 3) (deg. = 4) (deg. = 5)


OBSERVATIONS of Polynomial Graphs:

1) How does the degree of a polynomial function relate the number of roots of the graph?

The degree is the maximum number of zeros or roots that a graph can have.

2) Is there any relationship between the degree of the polynomial function and the shape of the graph?

Number of Changes in DIRECTION OF THE GRAPH = DEGREE

EVEN DEGREES: Start and End both going UP or DOWN

ODD DEGREES: Start and End as opposites  UP and DOWM


Describe possible shape of the following based on the degree and leading coefficient

OBSERVATIONS of Polynomial Graphs:

Describe possible shape of the following based on the degree and leading coefficient:

3) What additional information (value) related the degree of the polynomial may affect the shape of its graph?

LEADING COEFFICIENT

Numerical Value of Degree

ODD DEGREE:

POSITIVE Leading Coefficient = START Down and END Up

NEGATIVE Leading Coefficient = START Up and END Down

EVEN DEGREE:

POSITIVE Leading Coefficient

= UP

NEGATIVE Leading Coefficient

= DOWN


Degree practice with polynomial functions
Degree Practice with Polynomial Functions

  • Identify the degree as odd or even and state the assumed degree.

  • Identify leading coefficient as positive or negative.


Draw a graph for each descriptions
Draw a graph for each descriptions:

Description #3:

Degree = 3

Leading Coefficient = 1

Description #1:

Degree = 4

Leading Coefficient = 2

Description #2:

Degree = 6

Leading Coefficient = -3

Description #5:

Degree = 5

Leading Coefficient = -4

Description #4:

Degree = 8

Leading Coefficient = -2


Graphs 1 6 identify range interval or inequality notation
Graphs # 1 – 6 Identify RANGE:Interval or Inequality Notation

Graph #3

Graph #2

Graph #1

(-2, 8)

(0, 11)

(13, 9)

(1, 4)

(7, -2)

(-17, -10)

(-6, -9)

(-5, -9)

(4, -15)

Range, y: (-∞, ∞ )

Range, y,: (-∞, ∞ )

Range, y: (-15, ∞ )

Graph #6

Graph #5

Graph #4

(-5,17)

(-3,12)

(6, 11)

(1, 12)

(-3, 3)

(4, 8)

(2, 2)

(-2, 6)

(3, 2)

(1, -3)

(-5, -4)

(1, -9)

(4, -5)

Range, y,: (-∞, 17 )

Range, y: (-∞, 12 )

Range, y: (-5, ∞ )


The END BEHAVIOR of a polynomialdescribes the RANGE, f(x), as the DOMAIN, x, moves LEFT(as x approaches negative infinity: x → - ∞) and RIGHT(as x approaches positive infinity : x → ∞) on the graph.Determine the end behavior for each of the given graphs

Increasing to the Left

Decreasing to the

Left

Decreasing to the Right

Decreasing to the Right

Right:

Left:

Right:

Left:


Use graphs 1 6 from the previous slide
Use Graphs #1 – 6 from the previous Slide

  • Describe the END BEHAVIOR of each graph

  • Identify if the degree is EVEN or ODD for the graph

  • Identify if the leading coefficient is POSITIVE or NEGATIVE

GRAPH #3

GRAPH #2

GRAPH #1

Degree: ODD

LC:NEG

Degree: EVEN

LC:POS

Degree: ODD

LC:NEG

GRAPH #6

GRAPH #5

GRAPH #4

Degree: EVEN

LC:NEG

Degree: EVEN

LC:POS

Degree: EVEN

LC:NEG


Describing polynomial graphs based on the equation
Describing Polynomial Graphs Based on the Equation

  • Based on the given polynomial function:

  • Identify the Leading Coefficient and Degree.

  • Sketch possible graph (Hint: How many direction changes possible?)

  • Identify the END BEHAVIOR

Degree: 4  Even

LC: -1  Neg

Start Down, End Down

Degree: 5  Odd

LC: 2  Pos

Start Down, End Up

Degree: 3  Odd

LC: -2  Neg

Start Down, End Up

Degree: 6  Even

LC: 1  Pos

Start Up, End Up


EXTREMA: MAXIMUM and MINIMUM points are the highest and lowest points on the graph.

C

  • Point A is a Relative Maximumbecause it is the highest point in the immediate area (not the highest point on the entire graph).  

  • Point B is a Relative Minimumbecause it is the lowest point in the immediate area (not the lowest point on the entire graph).

  • Point C is the Absolute Maximumbecause it is the highest point on the entire graph.

  • There is no Absolute Minimumon this graph because the end behavior is:

    (there is no bottom point)

A

B


Identify all maximum or minimum points distinguish if each is relative or absolute
Identify ALL Maximum or Minimum PointsDistinguish if each is RELATIVE or ABSOLUTE

Graph #1

Graph #3

Graph #2

(1, 4)

R: Max

R: Max

R: Max

(-2, 8)

(0, 11)

(13, 9)

R: Max

(-5, -9)

(7, -2)

(-6, -9)

R: Min

(-17, -10)

(4, -15)

R: Min

R: Min

R: Min

A: Min

Graph #6

Graph #5

Graph #4

R: Max

R: Max

(-3,12)

(6, 11)

(-2, 22)

R: Max

(-3, 3)

A: Max

(2, 2)

R: Max

(6, 3)

R: Min

(1, -3)

(1, -9)

(-5, -4)

(4, -5)

R: Min

R: Min

R: Min

A: Min


CALCULATOR COMMANDS for POLYNOMIAL FUNCTIONS

The WINDOW needs to be large enough to see graph!

  • The ZEROES/ ROOTS of a polynomial function are the x-intercepts of the graph.

    Input [ Y=] as Y1 = function and Y2 = 0

    [2nd ] [Calc]  [Intersect]

  • To find EXTEREMA(maximums and minimums):

    Input [ Y=] as Y1 = function

    [2nd ][Calc]  [3: Min] or [4: Max]

    • LEFT and RIGHT bound tells the calculator where on the domain to search for the min or max.

    • y-value of the point is the min/max value.


Using your calculator: GRAPH the each polynomial function and IDENTIFY the ZEROES, EXTREMA, and END BEHAVIOR.

[1]

[2]

[3]

[4]


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