Unit 2: Polynomial Functions

Graphs of Polynomial Functions2.2

JMerrill 2005

Revised 2008

Learning Goal• To find zeros and use transformations to sketch

graphs of polynomial functions

• To use the Leading Coefficient Test to determine end behavior

Significant features• The graphs of polynomial functions are

continuous (no breaks—you draw the entire graph without lifting your pencil). This is opposed to discontinuous functions (remember piecewise functions?).

• This data is continuous as opposed to discrete.

Examples of Polynomials

Degree Name Example

0 Constant 5

1 Linear 3x+2

2 Quadratic X2 – 4

3 Cubic X3 + 3x + 1

4 Quartic -3x4 + 4

5 Quintic X5 + 5x4 - 7

Significant features• The graph of a polynomial function has only

smooth turns. A function of degree n has at most n – 1 turns. − A 2nd degree polynomial has 1 turn− A 3rd degree polynomial has 2 turns− A 5th degree polynomial has…

Cubic Parent Function

Draw the parent functions on the graphs.

f(x) = x3

X Y

-3 -27

-2 -8

-1 -1

0 0

1 1

2 8

3 27

Domain ,

Range - ,

X Y

-3 81

-2 16

-1 1

0 0

1 1

2 16

3 81

Quartic Parent Function

Draw the parent functions on the graphs.

f(x) = x4

Domain ,

Range 0,

Graph and TranslateStart with the graph of y = x3. Stretch it by a factor of 2 in the y direction. Translate it 3 units to the right.

X Y

-3 -27

-2 -8

-1 -1

0 0

1 1

2 8

3 27

X Y

0 -54

1 -16

2 -2

3 0

4 2

5 16

6 54

3Equation 2 3y x

Domain ,

Range ,

X Y

-3 81

-2 16

-1 1

0 0

1 1

2 16

3 81

Graph and Translate

Start with the graph of y = x4. Reflect it across the x-axis. Translate it 2 units down.

X Y

-3 -83

-2 -18

-1 -3

0 -2

1 -3

2 -18

3 -83

4Equation 2y x

Domain ,

Range , 2 X Y

-3 -81

-2 -16

-1 -1

0 0

1 -1

2 -16

3 -81

Max/Min• A parabola has a

maximum or a minimum

• Any other polynomial function has a local max or a local min. (extrema)

Local max

Local min

max

min

Polynomial Quick Graphs• From yesterday’s

activity:

• f(x) = x2 + 2x

f(x) =

f(x) =

Look at the root where the graph of f(x) crossed the x-axis. What was the power of the factor?

A. 3

B. 2

C. 1

Look at each root where the graph of a function“wiggled at” the x-axis. Were the powers even or odd?

A. Even

B. Odd

Look at each root where the graph of a function was tangent to the x-axis. What was the power of the factor?

A. 4

B. 3

C. 2

D. 1

Describe the end behavior of a function if a > 0 and n is even.A. Rise left, rise right

B. Fall left, fall right

C. Rise left, fall right

D. Fall left, rise right

Describe the end behavior of a function if a < 0 and n is even.A. Rise left, rise right

B. Fall left, fall right

C. Rise left, fall right

D. Fall left, rise right

Describe the end behavior of a function if a > 0 and n is odd.A. Rise left, rise right

B. Fall left, fall right

C. Rise left, fall right

D. Fall left, rise right

Describe the end behavior of a function if a < 0 and n is odd.A. Rise left, rise right

B. Fall left, fall right

C. Rise left, fall right

D. Fall left, rise right

Leading Coefficient Test• As x moves without bound to the left or right,

the graph of a polynomial function eventually rises or falls like this:

• In an odd degree polynomial:− If the leading coefficient is positive, the graph

falls to the left and rises on the right− If the leading coefficient is negative, the graph

rises to the left and falls on the right

• In an even degree polynomial:− If the leading coefficient is positive, the graph

rises on the left and right− If the leading coefficient is negative, the graph

falls to the left and right

End Behavior• If the leading coefficient of a polynomial

function is positive, the graph rises to the right.

y = x2 y = x3 + … y = x5 + …

Finding Zeros of a Function• If f is a polynomial function and a is a real

number, the following statements are equivalent:

• x = a is a zero of the function

• x = a is a solution of the polynomial equation f(x)=0

• (x - a) is a factor of f(x)

• (a, 0) is an x-intercept of f

Example• Find all zeros of f(x) = x3 – x2 – 2x

• Set function = 0 0 = x3 – x2 – 2x

• Factor 0 = x(x2 – x – 2)

• Factor completely 0 = x(x – 2)(x + 1)

• Set each factor = 0, solve 0 = x

0 = x – 2; so x = 2

0 = x +1; so x = -1

You Do• Find all zeros of f(x) = - 2x4 + 2x2

• X = 0, 1, -1

Multiplicity (repeated zeros)• How many roots? • How many roots?

3 roots; x = 1, 3, 3.4 roots; x = 1, 3, 3, 4.

3 is a double root. It is tangent to the x-axis

3 is a double root. It is tangent to the x-axis

Roots of Polynomials• How many roots? • How many roots?

5 roots: x = 0, 0, 1, 3, 3. 0 and 3 are double roots

3 roots; x = 2, 2, 2

Double roots

Double roots

(tangent)

Triple root – lies flat

then crosses

axis (wiggles)

Given Roots, Find a Polynomial Function• There are many correct solutions. Our solutions

will be based only on the factors of the given roots:

• Ex: Find a polynomial function with roots 2, 3, 3

• Turn roots into factors: f(x) = (x – 2)(x – 3)(x – 3)

• Multiply factors: f(x) = (x – 2)(x2 – 6x + 9)

• Finish multiplying: f(x) = x3 – 8x2 + 21x -18

You Do• Find a polynomial with roots – ½, 3, 3

• One answer might be: f(x) = 2x3 – 11x2 + 12x +9

Sketch graph

f(x) = (x - 4)(x - 1)(x + 2)

Step 1: Find zeros. zeros: 2,1, and 4Step 2: Mark the zeros on a number line.

Step 3: Determine end behavior

Step 4: Sketch the graph

Fall left, rise right

Sketch graph

f(x)= -(x-4)(x-1)(x+2)

zeros: 2,1, and 4

You Do

f(x) = (x+1)2(x-2)

zeros: 1,2

You Do

f(x) = - (x-4)3

zeros: 4

Sketch graph.

f(x) = (x-2)2(x+3)(x+2)

roots: -3, -2 and 2

Rise left, rise right

Roots: -3, 2 and 6Factors: (x+3), (x-2) and (x-6)

Factored Form:

f(x) = (x+3)(x-2)(x-6)

Write an equation.

Polynomial Form: f(x) = (x+3)(x2 – 8x + 12)

= x3 – 5x2 – 12x + 36

Write equation.

Zeros: -2, -1, 3 and 5

Factors: (x+2), (x+1), (x-3) and (x-5)

Factored Form:

f(x) = (x + 2)(x + 1)(x – 3)(x – 5)

Polynomial Form:

Gateway Problem• Sketch the graph of f(x) = x2(x – 4)(x + 3)3

Double root at x = 0

Root at x = 4

Triple root at x = -3

Roots?

Degree of polynomial? 6

End Behavior? Rise left

Rise right