Solving for Discontinuities Algebraically 16 17 November 2010

Solving for Discontinuities Algebraically 16 – 17 November 2010

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

Vertical Asymptotes l l l 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 =

Example:

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

Removable Discontinuities l l Occur when Shortcut! l Factors that occur in both the numerator and the denominator

Removable Discontinuities, cont. l l l 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 =

Example:

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

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

Vertical Asymptote vs. Removable Discontinuity, cont.

Vertical Asymptote vs. Removable Discontinuity, cont. l 1. 2. 3. 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.

Vertical Asymptote vs. Removable Discontinuity, cont.

Vertical Asymptote vs. Removable Discontinuity, cont. l Depends on: l l 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

Vertical Discontinuity vs. Removable Discontinuity, cont. Common Factor: Multiplicity Greater in Numerator or Denominator? Type of Discontinuity:

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

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

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

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

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

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

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