good morning, precalculus!
DESCRIPTION
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: - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/1.jpg)
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.
![Page 2: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/2.jpg)
Do Now: Determine if x = -2 is a zero of the function below:
f(x) = -3x3 - 8x2 - 2x + 4.
![Page 3: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/3.jpg)
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!
![Page 4: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/4.jpg)
Today's Agenda:1. Do Now2. Today's Objective3. Finish Up Unit 3, Objective 44. Practice with Partners5. Closing
![Page 5: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/5.jpg)
Today's Objective:Unit 3, Obj. 4: I will be able to analyze and graph polynomial functions with and
without technology.
(Pgs. 207-212)
![Page 6: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/6.jpg)
Analyzing & Graphing Polynomial Functions
Cont.
![Page 7: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/7.jpg)
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)
![Page 8: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/8.jpg)
Analyzing and Graphing Polynomial Functions
Corollary to the Fundamental Theorem of Algebra:
(Also on pg. 207)
![Page 9: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/9.jpg)
Analyzing and Graphing Polynomial Functions
In short, the Corollary to the Fundamental Theorem of Algebra (above) states that:
![Page 10: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/10.jpg)
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)
![Page 11: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/11.jpg)
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?
![Page 12: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/12.jpg)
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?
![Page 13: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/13.jpg)
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?
![Page 14: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/14.jpg)
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?
![Page 15: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/15.jpg)
Asymptotes (Last New Topic...Still Part
of Obj. 4!)
![Page 16: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/16.jpg)
Asymptotes An asymptote (defined on pg. 180) is a line that a function approaches but never touches.
Vertical asymptote
Horizontal asymptote
![Page 17: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/17.jpg)
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--> -∞.
![Page 18: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/18.jpg)
Asymptotes What is the vertical asymptote below?
What is the horizontal asymptote below?
![Page 19: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/19.jpg)
Asymptotes To determine if a rational function has a vertical asymptote (recall the definition of a vertical asymptote):
Ex: f(x) = 3x-1x-2
![Page 20: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/20.jpg)
Asymptotes To determine if a rational function has a horizontal asymptote (recall the definition of a horizontal asymptote):
Ex: f(x) = 3x-1x-2
![Page 21: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/21.jpg)
Practice with Partners
![Page 22: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/22.jpg)
Asymptotes Determine where the graph has a vertical and a horizontal asymptote.
Ex: f(x) = x x-5
![Page 23: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/23.jpg)
Asymptotes Determine where the graph has a vertical and a horizontal asymptote.
Ex: f(x) = 2x x+4
![Page 24: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/24.jpg)
Closing
![Page 25: Good Morning, Precalculus!](https://reader036.vdocuments.net/reader036/viewer/2022070420/56815f14550346895dcdde44/html5/thumbnails/25.jpg)
ClosingSummarize, in your own words, how to find the vertical and horizontal asymptotes of an equation.