Good Morning, Precalculus!When you come in, please....1. Grab your DO NOW sheet2. 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:

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

AnnouncementsThe unit 3 test is this Thursday, Nov. 15

Due tomorrow: 1. Pg. 209-210 # 5-10, #312. unit 3 test study guide - for extra

credit! FIRST THING IN THE MORNING!

Today's Agenda:1. Do Now2. Today's Objective3. Finish Up Unit 3, Objective 44. Practice with Partners5. Closing

Today's Objective:Unit 3, Obj. 4: I will be able to analyze and graph polynomial functions with and

without technology.

(Pgs. 207-212)

Analyzing & Graphing Polynomial Functions

Cont.

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)

Analyzing and Graphing Polynomial Functions

Corollary to the Fundamental Theorem of Algebra:

(Also on pg. 207)

Analyzing and Graphing Polynomial Functions

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

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)

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?

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?

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?

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?

Asymptotes (Last New Topic...Still Part

of Obj. 4!)

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

Vertical asymptote

Horizontal asymptote

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

Vertical asymptote

Horizontal 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.

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

Asymptotes What is the vertical asymptote below?

What is the horizontal asymptote below?

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

Ex: f(x) = 3x-1x-2

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

Ex: f(x) = 3x-1x-2

Practice with Partners

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

Ex: f(x) = x x-5

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

Ex: f(x) = 2x x+4

Closing

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