POLYNOMIALS REVIEWThe 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 COEFFICIENT is the numerical value of the term with the HIGHEST DEGREE.

xxxxx 5152811 3957

1697 2468 xxxx

72543 342 xxxx

511364 245 xxxx

Polynomials Review Practice For each polynomial1)Write the polynomial in descending order2)Identify the DEGREE and LEADING COEFFICIENT of the polynomial

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

511364)( 245 xxxxxf

1855)2(11)2(3)2(6)2(4)2( 245 f

Find each value for 1. 2.

3. 4.

6125)( 23 xxxxf)3(f

)2

1(f

)1(f

)3( 2af

__________)2( f

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?

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

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

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

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

LEADING COEFFICIENTNumerical 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

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

532)( 24 xxxf 1473)( 35 xxxxg

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:Description #1:

Degree = 4Leading Coefficient = 2

Description #2:Degree = 6

Leading Coefficient = -3

Description #3:Degree = 3

Leading Coefficient = 1

Description #4:

Degree = 8Leading

Coefficient = -2

Description #5:

Degree = 5Leading

Coefficient = -4

Graphs # 1 – 6 Identify RANGE: Interval or Inequality Notation

(1, 4)

(-5, -9) (-6, -9)

(4, -15)

(-2, 8) (0, 11) (13, 9)

(7, -2)

(-17, -10)

(-3, 3)

(-5, -4) (1, -9)

(6, 11)(-3,12)

(1, -3)

(2, 2)

(4, -5)

(1, 12)

(-5,17)

(-2, 6)

(3, 2)

(4, 8)

Graph #1 Graph #2 Graph #3

Graph #4 Graph #5 Graph #6

Range, y: (-∞, ∞ ) Range, y: (-15, ∞ ) Range, y,: (-∞, ∞ )

Range, y: (-5, ∞ ) Range, y: (-∞, 12 ) Range, y,: (-∞, 17 )

The END BEHAVIOR of a polynomial describes 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

x

Decreasing to the Right

x

xRight:

Left:

x)(xf

)(xf

)(xf

)(xfRight:

Left:

Decreasing to the Right

Decreasing to the

Left

Increasing to the Left

Use Graphs #1 – 6 from the previous Slide

GRAPH #1

• 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

Degree: ODDLC: NEG

GRAPH #2 GRAPH #3

GRAPH #4 GRAPH #5 GRAPH #6

,x

,x )(xf

)(xf

Degree: EVENLC: POS

,x

,x )(xf

)(xf

Degree: EVENLC: POS

,x

,x )(xf

)(xf

Degree: ODDLC: NEG

,x

,x )(xf

)(xf

Degree: EVENLC: NEG

,x

,x )(xf

)(xf

Degree: EVENLC: NEG

,x

,x )(xf

)(xf

Describing Polynomial Graphs Based on the EquationBased on the given polynomial function:•Identify the Leading Coefficient and Degree.•Sketch possible graph (Hint: How many direction changes possible?)•Identify the END BEHAVIOR

xxxxf 362)( 35 12)( 24 xxxxh

432)( 246 xxxxg 332)( 23 xxxxxp

Degree: 5 Odd LC: 2 Pos

Start Down, End Up)(xf

)(xf,x,x

Degree: 4 EvenLC: -1 Neg

Start Down, End Down)(xh

)(xh,x

,x

Degree: 6 EvenLC: 1 Pos

Start Up, End Up

)(xg

)(xg,x

,x )(xp

)(xp,x

,x

Degree: 3 Odd LC: -2 Neg

Start Down, End Up

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

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

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

• There is no Absolute Minimum on this

graph because the end behavior is:

(there is no bottom point) )(, xfx )(, xfx

A

B

C

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

Identify ALL Maximum or Minimum PointsDistinguish if each is RELATIVE or ABSOLUTE

(-6, -9)(4, -15)

(-2, 8) (0, 11) (13, 9)

(7, -2)

(-17, -10)

(-3, 3)

(-5, -4) (1, -9)

(6, 11)(-3,12)

(1, -3)

(2, 2)

(4, -5)

(-2, 22)

(6, 3)

(1, 4)

(-5, -9)

Graph #1 Graph #2 Graph #3

Graph #4 Graph #5 Graph #6

R: Min

R: Max

R: Max

R: MinA: Min R: Min

R: Min

R: Max R: Max

R: Max

R: Min

R: MinR: Min R: Min A: Min

R: MaxR: Max

R: MaxA: Max

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.

CALCULATOR COMMANDS for POLYNOMIAL FUNCTIONS

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

158 234 xxxxy xxxxy 8232

1 235

[1]

1183)( 34 xxxxf 1462)( 23 xxxxg[2]

[3]

[4]