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Good Morning, Precalculus!. When you come in, please.... 1. Grab your DO NOW sheet 2. Begin your DO NOW!. Do Now: Determine if x = -2 is a zero of the function below: f(x) = -3x3 - 8x2 - 2x + 4. Do Now: Determine if x = -2 is a zero of the function below:

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Presentation Transcript
slide1

Good Morning, Precalculus!

When you come in, please....

1. Grab your DO NOW sheet

2. Begin your DO NOW!

Do Now:

Determine if x = -2 is a zero of the function below:

f(x) = -3x3 - 8x2 - 2x + 4.

slide2

Do Now:

Determine if x = -2 is a zero of the function below:

f(x) = -3x3 - 8x2 - 2x + 4.

slide3

Announcements

The unit 3 test is this Thursday, Nov. 15

Due tomorrow:

1. Pg. 209-210 # 5-10, #31

2. unit 3 test study guide - for extra credit! FIRST THING IN THE MORNING!

slide4

Today\'s Agenda:

1. Do Now

2. Today\'s Objective

3. Finish Up Unit 3, Objective 4

4. Practice with Partners

5. Closing

slide5

Today\'s Objective:

Unit 3, Obj. 4: I will be able to analyze and graph polynomial functions with and without technology.

(Pgs. 207-212)

slide7

Analyzing and Graphing Polynomial Functions

Last time, we discussed that the Fundamental Theorem of Algebra (pg. 207) states that:

"Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers."

**Remember...a "complex number" has the form a + bi (we learned this in unit 2, objective 3)

slide8

Analyzing and Graphing Polynomial Functions

Corollary to the Fundamental Theorem of Algebra:

(Also on pg. 207)

slide9

Analyzing and Graphing Polynomial Functions

In short, the Corollary to the Fundamental Theorem of Algebra (above) states that:

slide10

Analyzing and Graphing Polynomial Functions

The Corollary to the Fundamental Theorem of Algebra (above) states that:

-The degree n of a polynomial indicates the number of possible roots of a polynomial equation.

-Each root of a polynomial (r1, r2, r3, ....rn ) is represented in the equation as a factor in the form:

P(x) = k(x - r1)(x - r2)(x - r3) .... (x - rn)

slide11

Analyzing and Graphing Polynomial Functions

What\'s the point of the Corollary of the Fundamental Theorem of Algebra???

The Corollary to the Fundamental Theorem of Algebra (previously mentioned) is useful when you must write a polynomial equation, given the roots of the equation.

Ex: Write a polynomial equation of least degree with the roots 3, -2i, 2i. How many times does the function you found cross the x-axis?

slide12

Analyzing and Graphing Polynomial Functions

The Corollary to the Fundamental Theorem of Algebra (above) is useful when you must write a polynomial equation, given the roots of the equation.

Ex (Try it on your own!): Write a polynomial equation of least degree that has the zeros -2, -4i, 4i. How many times does the function you found cross the x-axis?

slide13

Analyzing and Graphing Polynomial Functions

The Corollary to the Fundamental Theorem of Algebra (above) is useful when you must write a polynomial equation, given the roots of the equation.

Ex (Try it on your own!): Use your graphing calculator to graph the function f(x) = 9x4 - 35x2 -4. How many real zeros does the function have?

slide14

Analyzing and Graphing Polynomial Functions

Ex (Try it on your own!): Use your graphing calculator to graph the function f(x) = 9x4 - 35x2 -4. How many real zeros does the function have?

slide16

Asymptotes

An asymptote (defined on pg. 180) is a line that a function approaches but never touches.

Vertical asymptote

Horizontal asymptote

slide17

Asymptotes

An asymptote (defined on pg. 180) is a line that a function approaches but never touches.

Vertical asymptote

The line x = a is a vertical asymptote for a function f(x) if f(x) --> ∞ or f(x) --> -∞ as x --> a from either the left or the right.

Horizontal asymptote

The line y = b is a horizontal asymptote for a function f(x) if f(x) --> b as x--> ∞ or x--> -∞.

slide18

Asymptotes

What is the vertical asymptote below?

What is the horizontal asymptote below?

slide19

Asymptotes

To determine if a rational function has a vertical asymptote (recall the definition of a vertical asymptote):

Ex: f(x) = 3x-1

x-2

slide20

Asymptotes

To determine if a rational function has a horizontal asymptote (recall the definition of a horizontal asymptote):

Ex: f(x) = 3x-1

x-2

slide22

Asymptotes

Determine where the graph has a vertical and a horizontal asymptote.

Ex: f(x) = x

x-5

slide23

Asymptotes

Determine where the graph has a vertical and a horizontal asymptote.

Ex: f(x) = 2x

x+4

slide25

Closing

Summarize, in your own words, how to find the vertical and horizontal asymptotes of an equation.

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