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Warm UP!

Warm UP!. Factor the following:. Unit 7: Rational Functions. MA3A1. Students will explore rational functions.

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Warm UP!

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  1. Warm UP! Factor the following:

  2. Unit 7: Rational Functions MA3A1. Students will explore rational functions. a. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior. b. Find inverses of rational functions, discussing domain and range, symmetry, and c. Solve rational equations and inequalities analytically, graphically, and by using appropriate technology.

  3. LG 7-1 Graphs & Characteristics of Rational Functions By the end of today, you should remember how to find the following characteristics: Domain and Range x-intercepts and y-intercepts Horizontal and vertical asymptotes Holes **Important Note: Rational Functions should always be FACTORED before you do ANYTHING! If you skip this step, then you will probably do more work then you needed to!

  4. For example… You WILL be able to take the following problem and make a list of these characteristics: Domain: Range: x-intercepts: y-intercepts: Horizontal asymptotes: Vertical asymptotes: Holes:

  5. Discussion Domain What is the domain of this function? Are there any numbers that x is not allowed to equal?

  6. Practice Find the Domain.

  7. Discussion What is an x-intercept?

  8. Notes X-intercepts of Rational Function To find the x-intof Rational Functions, set the numerator equal to zero and solve for x.

  9. Practice Find all x-intercepts of each function.

  10. Discussion What is a y-intercept?

  11. Notes y-intercepts of Rational Function To find the y-intof Rational Functions, substitute 0 for x.

  12. Practice Find all y-intercepts of each function.

  13. Notes Asymptote An asymptote is a line that a function approaches but never actually reaches. The horizontal asymptote is sometimes called the “end-behavior asymptote.” Why? Vertical Asymptote Horizontal Asymptote

  14. Notes Vertical Asymptotes A rational function has a vertical asymptote at each value of x that makes only the denominator equal zero. It’s value is the numbers you used to state the domain! Example: Vertical Asymptotes:

  15. Practice Find the Vertical Asymptotes:

  16. Notes Horizontal Asymptotes • These are a little more difficult. It all depends on the degreeof the numerator and denominator. • If degree of n = d, then HA is the ratio of the coefficients. • If degree of n < d, then HA = 0 • If degreen > d, then there is no horizontal asymptote (there is a slant!)

  17. Practice Find the Horizontal Asymptotes:

  18. Notes If a number makes boththe numerator and denominator of the fraction equal zero, then the function has a hole at the x-value for that point. Find the y-value of the point by plugging in the x value. Example: Holes Hole:

  19. Practice Find the Holes:

  20. Class Work

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