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

Evaluating a Polynomial: Substitute values of x into polynomial and simplify:

511364)( 245 xxxxxf

Find each value for 1. 2.

3. 4.

6125)( 23 xxxxf)3(f

)21(f

)1(f

)6.2(f

__________)2( f

Graphs of Polynomial Functions:

Constant Linear Quadratic (degree = 0) (degree = 1) (degree = 2)

Cubic Quartic Quintic (deg. = 3) (deg. = 4) (deg. = 5)

OBSERVATIONS of Polynomial Graphs: 1) DEGREE and ROOT (x-intercept or Zero) Observations:• How does the degree of a polynomial function relate the

number of roots (zeroes) of the graph?

2) DEGREE and SHAPE OBSERVATIONS•How EVEN versus ODD degree graphs start and end?•How are the number of direction changes (up, down) related to the degree?

LEADING COEFFICIENT AFFECTS SHAPENumerical Value of Degree

NEGATIVE Leading Coefficient

POSITIVE Leading Coefficient:

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

How does the graph start and end?How many changes in direction?

532)( 24 xxxf 1473)( 35 xxxxg

Degree Practice with Polynomial Functions• Identify the degree as odd or even and state possible degree value.• Identify leading coefficient as positive or negative.

Degree: Odd or Even

Possible Value: ________

LC: Pos or Neg

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 = 8

Leading Coefficient = -2

Description #5:Degree = 5

Leading Coefficient = -4

RANGE of POLYNOMIAL FUNCTIONSDescribes the possible y-values of the function.

Is there a highest or lowest value?

ODD DEGREE

EVEN DEGREE

(3, -9)

(-2, 5)

(-6, 15)

(8, 11)

(1, -8)

Graphs # 1 – 6 Identify RANGE: 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

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.

Another way of saying it starts and ends going UP or DOWN

Determine the end behavior for each of the given graphs

Decreasing to the Right

Negative: “Down”

Decreasing to the

Left

Right: “Ends”

Negative: “Down”Left: “Starts”

END BEHAVIOR of a polynomial: Continued

Decreasing to the Right

Increasing to the Left Negative: “Down”

Right: “Ends” Positive: “Up”

Left: “Starts”

Use Range Graphs #1 – 2• 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 #1

Degree: ODD or EVEN LC: POS or NEG

GRAPH #2

,x,x ________)( xf

________)( xf

Use Range Graphs #3 – 6• 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 #4

GRAPH #5 GRAPH #6

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

432)( 246 xxxxg

Degree: Odd or Even Leading Coefficient: Pos or NegEND BEHAVIOR

Degree: Odd or Even Leading Coefficient: Pos or NegEND BEHAVIOR

12)( 24 xxxxh

332)( 23 xxxxxpDegree: Odd or Even Leading Coefficient: Pos or NegEND BEHAVIOR

Degree: Odd or Even Leading Coefficient: Pos or NegEND BEHAVIOR

xxxxf 583)( 36

11537)( 724 xxxxg

Degree: Odd or Even Leading Coefficient: Pos or NegEND BEHAVIOR

Degree: Odd or Even Leading Coefficient: Pos or NegEND BEHAVIOR

• 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, -3)

(2, 2)

(4, -5)

(1, 4)

(-5, -9)

Graph #1 Graph #2

Graph #3 Graph #4

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

(1, -9)

(6, 11)(-3,12)(-2, 22)

(6, 3)

Graph #5 Graph #6

(1, -27)

(-4,19)

(-7, 1.3)

(8, -2.5)

Graph #7 Graph #8

(-7.5, 6)

(10, -4.5)

(-17, -1.1)

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