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Solving for Discontinuities Algebraically 16 – 17 November 2010
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Always Factor! The 1 st step → always factor the numerator and the denominator!!! Goal: Get matching factors in numerator and denominator
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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 =
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Example:
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Your Turn: Complete problems 1 – 5 on the “Solving for the Discontinuities of Rational Equations” handout.
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Removable Discontinuities Occur when Shortcut! Factors that occur in both the numerator and the denominator
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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 =
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Example:
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Your Turn: Complete problems 6 – 10 on the “Solving for the Discontinuities of Rational Equations” handout.
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Vertical Asymptote vs. Removable Discontinuity Algebraically, they act similarly Consider:
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Vertical Asymptote vs. Removable Discontinuity, cont.
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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.
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Vertical Asymptote vs. Removable Discontinuity, cont.
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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
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Vertical Discontinuity vs. Removable Discontinuity, cont. Common Factor: Multiplicity Greater in Numerator or Denominator? Type of Discontinuity:
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Your Turn: Complete problems 11 – 15 on the “Solving for the Discontinuities of Rational Equations” handout.
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Homework In Precalculus textbook, pg. 290: 7 – 12 Hint! You will need to use the quadratic formula for #8.
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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
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Example 1 If n = m → HA: If n < m → HA: y = 0 If n > m → HA doesn’t exist
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Example 2 If n = m → HA: If n < m → HA: y = 0 If n > m → HA doesn’t exist HA: none
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Example 3 If n = m → HA: If n < m → HA: y = 0 If n > m → HA doesn’t exist
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Your Turn: Complete problems 11 – 15 on the “Solving for the Discontinuities of Rational Equations” handout.
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Solving for Multiple Discontinuities Rational equations can have more than one type of discontinuity Vertical Asymptote Removable Discontinuity
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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
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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
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Homework Finish problems 21 – 30 on the “Solving for the Discontinuities of Rational Equations” Handout In Precalculus textbook Pg. 320: 40 – 43
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