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Good Morning, Precalculus! When you come in, please.... 1. Grab your DO NOW sheet 2. Place the homework that was due today at the front of your desk: Pg.

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Presentation on theme: "Good Morning, Precalculus! When you come in, please.... 1. Grab your DO NOW sheet 2. Place the homework that was due today at the front of your desk: Pg."— Presentation transcript:

1 Good Morning, Precalculus! When you come in, please Grab your DO NOW sheet 2. Place the homework that was due today at the front of your desk: Pg #5-10, #31 3. Begin your DO NOW! Do Now: 1. Write a polynomial of least degree with the roots 3, 5, and Name the horizontal and vertical asymptotes in the graph below.

2 Do Now: 1. Write a polynomial of least degree with the roots 3, 5, and -5.

3 Do Now: 2. Name the horizontal and vertical asymptotes in the graph below.

4 Announcements T he unit 3 test is TOMORROW!!!! You will be allowed a cheat sheet for the test. Remember, you can put ANYTHING on both sides of the index cards! The sky's the limit!

5 Today's Agenda: 1. Do Now 2. HW Review 3. Today's Objectives 4. Practice with Asymptotes (Cleaning up Objective 4) 5. Obj. 4 Exit Slip 6. Study Guide Review/Time to work on Cheat Sheets 7. Closing

6 HW Review: Pg #5-10, #31

7 HW Review:

8 Today's Objective: Unit 3, Obj. 4: I will be able to analyze and graph polynomial functions with and without technology. I will be able to review for the unit 3 test, which is tomorrow!

9 Cleaning Up Obj. 4: Asymptotes

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

11 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--> -∞.

12 Asymptotes What is the vertical asymptote below? What is the horizontal asymptote below?

13 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

14 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

15 Asymptotes With a partner, determine where the functions below have vertical asymptotes. Ex 1: f(x) = x x-5 Ex 2: f(x) = 2x x+4

16 Exit Slip

17 Objective: Unit 3, Objective 4 Determine the vertical asymptote for the function below: f(x) = 4x x-1

18 Study Guide Review

19 Jump discontinuity Infinite discontinuityPoint discontinuity Yes; f(x) is continuous at x = 1 because f(1) = -3, and the graph of f(x) approaches -3 from both the left and the right. No, f(x) is discontinuous at x = -9 because f(- 9) is undefined.

20 Study Guide Review This function appears to be continuous at x = 10 because f(10) = 20 and f(x) approaches 20 from both sides. This function is discontinuous at x = 10 because although f(x) = 85.1 and is defined, f(x) does not appear to be approaching 85.1 from both sides. It actually appears as if there is jump discontinuity.

21 Study Guide Review This function appears to be continuous at x = 10 because f(10) = 20 and f(x) approaches 20 from both sides. This function is discontinuous at x = 10 because although f(x) = 85.1 and is defined, f(x) does not appear to be approaching 85.1 from both sides. It actually appears as if there is jump discontinuity.

22 Study Guide Review an is negative n is even so, f(x) --> -∞ as x--> ∞ and f(x) --> -∞ as x--> -∞ an is positive n is odd so, f(x) --> ∞ as x--> ∞ and f(x) --> -∞ as x--> -∞ an is negative n is odd so, f(x) --> -∞ as x--> ∞ and f(x) --> ∞ as x--> -∞ <-- this is just one example of a possible graph!

23 Study Guide Review Absolute minimum Inflection point Relative maximum Relative minimum Inflection point x = 0 is a maximum because f(0) = 16, f(-0.1) = 15.9, and f(0.1) = x = 4 is an inflection point because f(4) = -48, f(3.9) = , and f(4.1) =

24 Study Guide Review Degree is 7, leading coefficient ix 5. Degree is 5, so p(x) has 5 zeros. Yes, because m(7) = 0. Start off with f(x) = (x-5)(x-3)(x+1) f(x) = x3 - 7x2 + 7x + 15 The graph has a vertical asymptote at x = 5. The graph has a horizontal asymptote at y = 3.

25 Obj. 4 Exit Slip Review

26 Exit Slip Objective: Unit 3, Objective 4 Determine the vertical asymptote for the function below: f(x) = 4x x-1 The definition of a vertical asymptote says: 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. As x-->1, x-1 (the denominator) gets smaller and smaller, and closer and closer to 0. This would result in f(x) getting larger and larger, and approaching ∞. Therefore, there is a vertical asymptote at x=1.

27 Closing

28 Summarize, in your own words, how to find the asymptote of an equation.


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