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Solving for Discontinuities Algebraically

Solving for Discontinuities Algebraically. 16 – 17 November 2010. Always Factor!. The 1 st step → always factor the numerator and the denominator!!! Goal: Get matching factors in numerator and denominator. Vertical Asymptotes. Occur when the denominator equals zero .

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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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