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

Polynomial Functions

Advanced MathChapter 3

Quadratic Functions and Models

Advanced MathSection 3.1

Advanced Math 3.1 - 3.3 3

Quadratic function

• Polynomial function of degree 2

2f x ax bx c


Parabola

• “u”-shaped graph of a quadratic function

• May open up or down


Axis of symmetry

• Vertical line through the center of a parabola

• Vertex: where the axis intersects the parabola

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vertex

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

-3

-2

-1

1

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x

y


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


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


Example

• Sketch the graph of the quadratic function

2 4 1f x x x


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


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)


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


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

Polynomial Functions of Higher Degree



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.


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


Compare

2

4

6

f x

x

x

f

x

x

f x

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-1

1

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x

y


Compare

3

5

7

f x

x

x

f

x

x

f x

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-8

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1

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x

y


Examples

• Sketch the graphs of:

5

5

5

5

1

1

11

2

y x

y x

y x

y x


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


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


The Leading Coefficient test

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

• See the Exploration on page 277


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


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


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


Standard form

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

exponents from left to right


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

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Examples

2 35f x x x

2 212 2

4f x x x


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


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

Polynomial and Synthetic Division



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


Examples

3 24 7 11 5 4 5x x x x

3 26 16 17 6 3 2x x x x


Checking your answer

• Graph both the original division problem and your answer

• The graphs should match exactly


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


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


Synthetic division

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

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


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


Examples

3 23 17 15 25 5x x x x

3 75 250 10x x x

2 35 3 2 1x x x x


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


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


If the remainder is zero…

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


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

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

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