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.

23. Solving for Multiple Discontinuities Rational equations can have more than one type of discontinuity Vertical Asymptote Removable Discontinuity

24. Solving for Multiple Discontinuities, cont. Step 1: Identify and solve for any horizontal asymptotes Step 2: Factor the numerator and denominator Step 3: Identify and solve for any removable discontinuities Step 4: Identify and solve for any vertical asymptotes

25. Your Turn: Complete the last two problems on the “Solving for Multiple Discontinuities” Handout Complete problems 21 – 30 on the “Solving for the Discontinuities of Rational Equations” Handout

26. Homework Finish problems 21 – 30 on the “Solving for the Discontinuities of Rational Equations” Handout In Precalculus textbook Pg. 320: 40 – 43