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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. 209-210 #5-10, #31 3. Begin your DO NOW! Do Now: 1. Write a polynomial of least degree with the roots 3, 5, and -5. 2. Name the horizontal and vertical asymptotes in the graph below.

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

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

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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!

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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

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HW Review: Pg. 209-210 #5-10, #31

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HW Review:

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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!

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Cleaning Up Obj. 4: Asymptotes

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

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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--> -∞.

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

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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

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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

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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

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Exit Slip

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

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Study Guide Review

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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.

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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.

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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.

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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!

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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) = 15.9. x = 4 is an inflection point because f(4) = -48, f(3.9) = - 46.4, and f(4.1) = -49.56

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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.

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Obj. 4 Exit Slip Review

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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.

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Closing

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

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