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# Good Morning, Precalculus! - PowerPoint PPT Presentation

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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1. Grab your DO NOW sheet

Do Now:

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

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

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

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

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!

1. Do Now

2. Today's Objective

3. Finish Up Unit 3, Objective 4

4. Practice with Partners

5. Closing

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

(Pgs. 207-212)

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)

Corollary to the Fundamental Theorem of Algebra:

(Also on pg. 207)

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

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)

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?

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?

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?

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?

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

Vertical asymptote

Horizontal asymptote

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--> -∞.

What is the vertical asymptote below?

What is the horizontal asymptote below?

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

Ex: f(x) = 3x-1

x-2

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

Ex: f(x) = 3x-1

x-2

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

Ex: f(x) = x

x-5

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

Ex: f(x) = 2x

x+4

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