1. Solving for Discontinuities Algebraically 16 – 17 November 2010

2. Always Factor! The 1 st 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 common factors 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. Think-Pair-Share 1. 30 sec – Individually think about why the equation has a vertical asymptote instead of a removable discontinuity. 2. 1 min – Talk about this with your partner. 3. Share your reasoning with the class.

13. Vertical Asymptote vs. Removable Discontinuity, cont.

14. 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. Common Factor: Multiplicity Greater in Numerator or Denominator? Type of Discontinuity:

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.