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Polynomial Functions Advanced Math Chapter 3

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Page 1: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Polynomial Functions

Advanced MathChapter 3

Page 2: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Quadratic Functions and Models

Advanced MathSection 3.1

Page 3: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 3

Quadratic function

• Polynomial function of degree 2

2f x ax bx c

Page 4: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 4

Parabola

• “u”-shaped graph of a quadratic function

• May open up or down

Page 5: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 5

Axis of symmetry

• Vertical line through the center of a parabola

• Vertex: where the axis intersects the parabola

Math Composer 1. 1. 5http: / /www. mathcomposer. com

vertex

-8 -7 -6 -5 -4 -3 -2 -1 1 2

-3

-2

-1

1

2

3

4

5

6

7

8

x

y

Page 6: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 6

Standard Form

• Convenient for sketching a parabola because it identifies the vertex as (h, k).

• If a > 0, the parabola opens up• If a < 0, the parabola opens down

2

0

f x a x h k

a

Page 7: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 7

Graphing a parabola in standard form

• Write the quadratic function in standard form by completing the square.

• Use standard form to find the vertex and whether it opens up or down

22 1

example

f x x x

Page 8: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 8

Example

• Sketch the graph of the quadratic function

2 4 1f x x x

Page 9: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 9

Writing the equation of a parabola

• Substitute for h and k in standard form

• Use a given point for x and f(x) to find a

example

vertex: 4,-1

point 2,3

Page 10: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 10

Example

• Write the standard form of the equation of the parabola that has a vertex at (2,3) and goes through the point (0,2)

Page 11: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 11

Finding a maximum or a minimum

• Locate the vertex

• If a > 0, vertex is a minimum (opens up)

• If a < 0, vertex is a maximum (opens down)

2:

: ,2 2

function f x ax bx c

b bvertex f

a a

Page 12: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 12

Example

• The profit P (in dollars) for a company that produces antivirus and system utilities software is given below, where x is the number of units sold. What sales level will yield a maximum profit?

20.0002 140 250000P x x

Page 13: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Polynomial Functions of Higher Degree

Advanced MathSection 3.2

Page 14: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 14

Polynomial functions

• Are continuous– No breaks– Not piecewise

• Have only smooth rounded curves– No sharp points

• This section will help you make reasonably accurate sketches of polynomial functions by hand.

Page 15: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 15

Power functions

• n is an integer greater than 0• If n is even, the graph is similar to

f(x)=x2

• If n is odd, the graph is similar to f(x)=x3

• The greater n is, the skinner the graph is and the flatter the it is near the origin.

nf x x

Page 16: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 16

Compare

2

4

6

f x

x

x

f

x

x

f x

Math Composer 1. 1. 5http: / /www. mathcomposer. com

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

-1

1

2

3

4

5

6

7

8

9

10

x

y

Page 17: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 17

Compare

3

5

7

f x

x

x

f

x

x

f x

Math Composer 1. 1. 5http: / / www. mathcomposer. com

-3 -2 -1 1 2 3

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

x

y

Page 18: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 18

Examples

• Sketch the graphs of:

5

5

5

5

1

1

11

2

y x

y x

y x

y x

Page 19: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 19

The Leading Coefficient Test

• If n (the degree) is odd• The left and right go opposite directions• A positive leading coefficient means the

graph falls to the left and rises to the right– As x becomes more positive, the graph goes

up• A negative leading coefficient means

the graph rises to the left and falls to the right– As x becomes more negative, the graph

goes up

Page 20: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 20

The Leading Coefficient Test

• If n (the degree) is even• The left and right go the same direction• A positive leading coefficient means the

graph rises to the left and rises to the right– The graph opens up

• A negative leading coefficient means the graph falls to the left and falls to the right– The graph opens down

Page 21: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 21

The Leading Coefficient test

• Does not tell you how many ups and downs there are in between

• See the Exploration on page 277

Page 22: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 22

Zeros of polynomial functions

• For a polynomial function of degree n

• There are at most n real zeros• There are at most n – 1 turning

points (where the graph switches between increasing and decreasing).

• There may be fewer of either

Page 23: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 23

Finding zeros

• Factor whenever possible• Check graphically

2

3 2

5 3

6 9

20

2

examples

h t t t

f x x x x

g x x x x

Page 24: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 24

Repeated zeros

A factor , 1, yields a of .

1. If is odd, the graph the x-axis at .

2. If is even, the graph the x-axis (but doesn't cross it)

at

kx a k x a k

k crosses x a

k touches

x a

repeated zero multiplicity

Page 25: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 25

Standard form

• For a polynomial greater than degree 2– Terms are in descending order of

exponents from left to right

Page 26: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 26

Graphing polynomial functions

1. Write in standard form

2. Apply leading coefficient test

3. Find the zeros4. Plot a few

additional points5. Connect the

points with smooth curves.

2 33

example

f x x x Math Composer 1. 1. 5http: / /www. mathcomposer. com

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

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Page 27: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 27

Examples

2 35f x x x

2 212 2

4f x x x

Page 28: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 28

The Intermediate Value Theorem

• See page 282• Helps locate real zeros• Find one x value at which the

function is positive and another x value at which the function is negative

• Since the function is continuous, there must be a real zero between these two values

• Use the table on a calculator to get closer to the zero and approximate it

Page 29: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 29

Examples

• Use the intermediate value theorem and the table feature to approximate the real zeros of the functions. Use the zero or root feature to verify.

4 212

4f x x x

2 212 3 5

5f x x x

Page 30: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Polynomial and Synthetic Division

Advanced MathSection 3.3

Page 31: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 31

Long division of polynomials

• Write the dividend in standard form

• Divide– Divide each term by the leading term

of the divisor

217 5 12 4

example

x x x

Page 32: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 32

Examples

3 24 7 11 5 4 5x x x x

3 26 16 17 6 3 2x x x x

Page 33: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 33

Checking your answer

• Graph both the original division problem and your answer

• The graphs should match exactly

Page 34: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 34

Remainders

• Write remainder as a fraction with the divisor on the bottom

• Examples:

3 24 3 12 3x x x x

3 29 1x x

4 2 23 1 2 3x x x x

Page 35: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 35

Division Algorithm

• Get rid of the fraction in the remainder by multipling both sides by the denominator.

4 22

2 2

3 1 2 112 4

2 3 2 3

x x xx x

x x x x

dividend divisor quotient remainder

f x d x q x r x

Page 36: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 36

Synthetic division

• The shortcut• Works with divisors of the form x –

k , where k is a constant• Remember that x + k = x – (– k)

Page 37: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 37

Synthetic division

• Use an L-shaped division sign with k on the outside and the coefficients of the dividend on the inside

• Leave space below the dividend

• Add the vertical columns, then multiply diagonally by k

24 10 21

5

example

x x

x

Page 38: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 38

Examples

3 23 17 15 25 5x x x x

3 75 250 10x x x

2 35 3 2 1x x x x

Page 39: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 39

Remainder Theorem

• If a polynomial f(x) is divided by x – k , then the remainder is r = f(k)

• The remainder is the value of the function evaluated at k

Page 40: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 40

Examples

• Write the function in the form f(x) = (x – k)q(x) + r for the given value of k , and demonstrate that f(k) = r

3 25 11 8

2

f x x x x

k

3 22 5 4

5

f x x x x

k

Page 41: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 41

If the remainder is zero…

• (x – k) is a factor of the dividend• (k, 0) is an x-intercept of the graph

Page 42: Polynomial Functions Advanced Math Chapter 3. Quadratic Functions and Models Advanced Math Section 3.1

Advanced Math 3.1 - 3.3 42

Example

• Show that (x + 3) and (x – 2) are factors of f(x) = 3x3 + 2x2 – 19x + 6.

• Write the complete factorization of the function

• List all real zeros of the function