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Solving for Discontinuities Algebraically

This lesson focuses on solving for discontinuities in rational equations by emphasizing the importance of factoring both the numerator and denominator. Students will learn to identify vertical asymptotes and removable discontinuities and understand the differences between them. The step-by-step approach includes factoring, setting the denominator to zero, and solving for x. Practice problems are provided to reinforce the concepts, along with discussions on how multiplicity affects the classification of discontinuities. Homework assignments will further solidify these essential algebraic techniques.

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Solving for Discontinuities Algebraically

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  1. Solving for Discontinuities Algebraically 16 – 17 November 2010

  2. Always Factor! • The 1st step → always factor the numerator and the denominator!!! • Goal: Get matching factors in numerator and denominator

  3. Vertical Asymptotes • Occur when the denominator equals zero. • Step 1: Factor the numerator and the denominator • Step 2: Set the denominator equal to zero • Step 3: Solve for x • Step 4: Write your answers in the form x =

  4. Example:

  5. Your Turn: • Complete problems 1 – 5 on the “Solving for the Discontinuities of Rational Equations” handout.

  6. Removable Discontinuities • Occur when • Shortcut! • Factors that occur in both the numerator and the denominator

  7. Removable Discontinuities, cont. • Step 1: Factor the numerator and the denominator • Step 2: Identify factors that occur in both the numerator and the denominator • Step 3: Set the commonfactors equal to zero • Step 4: Solve for x • Step 5: Write your answers in the form x =

  8. Example:

  9. Your Turn: • Complete problems 6 – 10 on the “Solving for the Discontinuities of Rational Equations” handout.

  10. Vertical Asymptote vs. Removable Discontinuity • Algebraically, they act similarly • Consider:

  11. Vertical Asymptote vs. Removable Discontinuity, cont.

  12. Vertical Asymptote vs. Removable Discontinuity, cont. • Think-Pair-Share • 30 sec – Individually think about why the equation has a vertical asymptote instead of a removable discontinuity. • 1 min – Talk about this with your partner. • Share your reasoning with the class.

  13. Vertical Asymptote vs. Removable Discontinuity, cont.

  14. Vertical Asymptote vs. Removable Discontinuity, cont. • Depends on: • How many times a factor occurs • Where the factor occurs • Removable Discontinuity → the multiplicity of the factor in the numerator ≥ the multiplicity of the factor in the denominator • Vertical Asymptote → the multiplicity of the factor in the numerator < the multiplicity of the factor in the denominator

  15. Vertical Discontinuity vs. Removable Discontinuity, cont.

  16. Your Turn: • Complete problems 11 – 15 on the “Solving for the Discontinuities of Rational Equations” handout.

  17. Homework • In Precalculus textbook, pg. 290: 7 – 12 • Hint! You will need to use the quadratic formula for #8.

  18. Horizontal Asymptotes • Occurs when the degree of the numerator ≤ the degree of the denominator • If n = m → HA: • If n < m → HA: y = 0 • If n > m → HA doesn’t exist

  19. Example 1 • If n = m → HA: • If n < m → HA: y = 0 • If n > m → HA doesn’t exist

  20. Example 2 • If n = m → HA: • If n < m → HA: y = 0 • If n > m → HA doesn’t exist HA: none

  21. Example 3 • If n = m → HA: • If n < m → HA: y = 0 • If n > m → HA doesn’t exist

  22. Your Turn: • Complete problems 11 – 15 on the “Solving for the Discontinuities of Rational Equations” handout.

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