Solving for the Discontinuities of Rational Equations

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# Solving for the Discontinuities of Rational Equations - PowerPoint PPT Presentation

Solving for the Discontinuities of Rational Equations. Review: 3 Types of Discontinuities. Vertical Asymptotes (VAs) Horizontal Asymptotes (HAs) Holes. Degree. The greatest exponent of an expression Examples: f(x) = x 6 – x 2 + 3 f(x) = x 4 – x 9 + x 11 – x 2 + 5 f(x) = 8x + 4

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### Solving for the Discontinuities of Rational Equations

Review: 3 Types of Discontinuities
• Vertical Asymptotes (VAs)
• Horizontal Asymptotes (HAs)
• Holes
Degree
• The greatest exponent of an expression
• Examples:
• f(x) = x6 – x2 + 3
• f(x) = x4 – x9 + x11 – x2 + 5
• f(x) = 8x + 4
• f(x) = 7
• The coefficient of the term with the largest degree
• Examples:
• f(x) = x6 – x2 + 3
• f(x) = x4 – x9 + x11 – x2 + 5
• f(x) = 8x + 4
• f(x) = 7
Horizontal Asymptotes Investigation
• Remember:
• The horizontal asymptote describes how the graph behaves AT ITS ENDS
• Look for the graph to taper to the same y-value on both ends of the graph
• Look for dashed, horizontal lines
• We DON’T DRAW dashed lines on the X-AXIS or the Y-AXIS!!!
Investigation Conclusion Questions:

4. What observations can you make about a rational equation’s horizontal asymptote when the degree of the numerator and the denominator are the same?

8. What observations can you make about a rational equation’s horizontal asymptote when the degree of the denominator is greater than the degree of the numerator?

12. What observations can you make about a rational equation’s horizontal asymptote when the degree of the numerator is greater than the degree of the denominator?

*Horizontal Asymptotes
• Depend on the degree of the numerator and the denominator
• Degree of Numerator < Degree of Denominator
• HA: y = 0
• Degree of Numerator = Degree of Denominator
• HA: y = ratio of leading coefficients
• Degree of Numerator > Degree of Denominator
• HA: doesn’t exist
• For problems 1 – 4 on the Introduction to Solving Rational Equations Practice, solve for the horizontal asymptote.

1. 2.

3. 4.

Solving for Vertical Asymptotes and Holes
• Always factor the numerator and the denominator 1st!
• Identify linear factors in the denominator
• Figure out where the linear factors in the denominator occur the most to decide if you have a vertical asymptote or a hole
• Set the linear factors from step 2 equal to zero and solve for x.