Solving for the discontinuities of rational equations
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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) = x 6 – x 2 + 3 f(x) = x 4 – x 9 + x 11 – x 2 + 5 f(x) = 8x + 4

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Review 3 types of discontinuities
Review: 3 Types of Discontinuities

  • Vertical Asymptotes (VAs)

  • Horizontal Asymptotes (HAs)

  • Holes


Degree
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


Leading coefficient
Leading Coefficient

  • 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
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
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
*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



Your turn
Your Turn:

  • 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
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.







Your turn1
Your Turn:

  • Complete problems 5 – 12 on the Introduction to Solving for the Discontinuities of Rational Equations Practice handout . BE PREPARED TO SHARE YOUR ANSWERS!!!

  • Complete problems 13 – 18.


Solving for the discontinuities of rational equations

13. 14.

15. 16.

17. 18.


Exit ticket
Exit Ticket

  • Does have a HA? (If yes, what is it?) Why?

  • Does have VAs and/or holes? (If yes, what are they?) Why?