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Graphing Rational Functions

Section 2-6. Graphing Rational Functions. Objectives. I can graph rational functions including: Vertical Asymptotes Horizontal Asymptotes Slant Asymptotes Holes Intercepts (x & y) I can determine the Domain and Range of rational functions.

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Graphing Rational Functions

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  1. Section 2-6 Graphing Rational Functions

  2. Objectives • I can graph rational functions including: • Vertical Asymptotes • Horizontal Asymptotes • Slant Asymptotes • Holes • Intercepts (x & y) • I can determine the Domain and Range of rational functions

  3. Guidelines for Analyzing Graphs of Rational Functions • If f(x) = N(x)/D(x) • 1. Simplify f(x) by factoring and reducing • 2. When reducing, like factors generate holes • 3. Find y-intercept by letting all x’s = 0 • 4. Solve N(x) = 0 to find x-intercepts • 5. Solve D(x) = 0 to find VA • 6. Find HA using rules for degree • 7. Find SA using synthetic division or long division • 8. Plot at least one point on either side of VA • 9. Use smooth curves to complete graphs. Do NOT cross asymptotes.

  4. Example:Find all horizontal and vertical asymptotes of f(x) = . x = 2 y y = 3 x Horizontal and Vertical Asymptotes Factor the numerator and denominator. The factor (x+1) is in both the numerator and denominator, so when cancelled become a “hole” at that point . Since -1 is a common root of both, there is a hole in the graph at -1 . Since 2 is a root of the denominator but not the numerator, x = 2 will be a vertical asymptote. Since the polynomials have the same degree, y = 3 will be a horizontal asymptote.

  5. Example: Find the slant asymptote for f(x) = . Divide: y x = -3 y = 2x - 5 x A slant asymptote is an asymptote occurs when degree of numerator is exactly 1 more than denominator Slant Asymptote Could also use Synthetic Division The slant asymptote is y = 2x– 5.

  6. Graph: f(x) = • Vertical asymptote: • x – 2 = 0 so at x = 2 • Dashed line at x = 2 • m = 0, n = 1 so m<n • HA at y = 0 • y-int: (0, -1)

  7. f(x) = Vertical Asymptotes: x – 2 = 0 and x + 3 = 0 x = 2, x = -3 M = 0, n = 2 m < n HA at y = 0 y-int: (0, -1) Graph on right

  8. Homework • WS 5-5

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