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Rational Functions

Finding Asymptotes. Rational Functions. Rational Functions. What is a rational function? That’s right, a fraction! What have we done with them already? Yep, we found the domain limits. We set the denominator equal to “0”. That is where it did not exist.

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Rational Functions

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  1. Finding Asymptotes Rational Functions

  2. Rational Functions • What is a rational function? • That’s right, a fraction! • What have we done with them already? • Yep, we found the domain limits. • We set the denominator equal to “0”. That is where it did not exist. • Does the domain have anything to do with asymptotes? • Of course, that is the vertical asymptote.

  3. Rational Functions • Vertical Asymptotes • To find the vertical asymptote, set the denominator equal to zero and solve. • Find the vertical asymptotes: • 6x2 – x – 2 = 0 • x2 – x - 12 = 0 • (x – 4)(x + 3) = 0 • (x - (x + ) = 0 • (3x – 2)(2x + 1) = 0 • x = x = - Set denominator equal to 0. Swing the 6 over. Factor the polynomial. Divide the 6 back out. Solve.

  4. Rational Functions • Horizontal Asymptotes • When looking for the horizontal asymptote, you must compare the degree of numerator to the degree of the denominator. • I will explain in a minute! • There will either be 1 asymptote or no asymptote. • There are 3 rules. • Equal • Big bootie • Big top

  5. Rational Functions • Horizontal Asymptotes • The degree is the largest exponent. You will need to compare the top to the bottom. • Here are the rules. • Big Bootie • If the degree on bottom is bigger than the top, the HA is y = 0. • Big Top • If the degree on top is bigger than the bottom, there is no HA. • Equal (The tricky one) • If the degrees are the same, then the HA is a ratio of the leading coefficients.

  6. Rational Functions • Horizontal Asymptotes • y = 0 • Lets find the horizontal asymptotes. • What is the degree on top? • 1 • What is the degree on bottom? • 2 • Which is bigger top, bottom, or neither? • Bottom • So, the HA is…….

  7. Rational Functions • Horizontal Asymptotes • No HA • Lets find the horizontal asymptotes. • What is the degree on top? • 3 • What is the degree on bottom? • 2 • Which is bigger top, bottom, or neither? • Top • So, the HA is…….

  8. Rational Functions • Horizontal Asymptotes • y = • y = 2 • Lets find the horizontal asymptotes. • What is the degree on top? • 2 • What is the degree on bottom? • 2 • Which is bigger top, bottom, or neither? • Neither • So, the HA is…… • A ratio of the leading coefficients

  9. Rational Functions • Try it on your own. • Find the vertical and horizontal asymptotes for: VA: x = 2 HA: none VA: x = 4 and x = 3 HA: y = 1

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