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Solving for the Discontinuities of Rational Equations

Solving for the Discontinuities of Rational Equations. Review: 3 Types of Discontinuities. Vertical Asymptotes (VAs) Horizontal Asymptotes (HAs) Holes. Degree. The greatest exponent of an expression Examples: f(x) = x 6 – x 2 + 3 f(x) = x 4 – x 9 + x 11 – x 2 + 5 f(x) = 8x + 4

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Solving for the Discontinuities of Rational Equations

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  1. Solving for the Discontinuities of Rational Equations

  2. Review: 3 Types of Discontinuities • Vertical Asymptotes (VAs) • Horizontal Asymptotes (HAs) • Holes

  3. Degree • The greatest exponent of an expression • Examples: • f(x) = x6 – x2 + 3 • f(x) = x4 – x9 + x11 – x2 + 5 • f(x) = 8x + 4 • f(x) = 7

  4. Leading Coefficient • The coefficient of the term with the largest degree • Examples: • f(x) = x6 – x2 + 3 • f(x) = x4 – x9 + x11 – x2 + 5 • f(x) = 8x + 4 • f(x) = 7

  5. Horizontal Asymptotes Investigation • Remember: • The horizontal asymptote describes how the graph behaves AT ITS ENDS • Look for the graph to taper to the same y-value on both ends of the graph • Look for dashed, horizontal lines • We DON’T DRAW dashed lines on the X-AXIS or the Y-AXIS!!!

  6. Investigation Conclusion Questions: 4. What observations can you make about a rational equation’s horizontal asymptote when the degree of the numerator and the denominator are the same? 8. What observations can you make about a rational equation’s horizontal asymptote when the degree of the denominator is greater than the degree of the numerator? 12. What observations can you make about a rational equation’s horizontal asymptote when the degree of the numerator is greater than the degree of the denominator?

  7. *Horizontal Asymptotes • Depend on the degree of the numerator and the denominator • Degree of Numerator < Degree of Denominator • HA: y = 0 • Degree of Numerator = Degree of Denominator • HA: y = ratio of leading coefficients • Degree of Numerator > Degree of Denominator • HA: doesn’t exist

  8. Examples

  9. Your Turn: • For problems 1 – 4 on the Introduction to Solving Rational Equations Practice, solve for the horizontal asymptote. 1. 2. 3. 4.

  10. Solving for Vertical Asymptotes and Holes • Always factor the numerator and the denominator 1st! • Identify linear factors in the denominator • Figure out where the linear factors in the denominator occur the most to decide if you have a vertical asymptote or a hole • Set the linear factors from step 2 equal to zero and solve for x.

  11. Solving for Vertical Asymptotes and Holes, cont. Does the linear factor:

  12. Example #1

  13. Example #2

  14. Example #3

  15. Example #4

  16. Your Turn: • Complete problems 5 – 12 on the Introduction to Solving for the Discontinuities of Rational Equations Practice handout . BE PREPARED TO SHARE YOUR ANSWERS!!! • Complete problems 13 – 18.

  17. 13. 14. 15. 16. 17. 18.

  18. Exit Ticket • Does have a HA? (If yes, what is it?) Why? • Does have VAs and/or holes? (If yes, what are they?) Why?

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