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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Solving for the Discontinuities of Rational Equations

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Solving for the discontinuities of rational equations

Solving for the Discontinuities of Rational Equations


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


Examples

Examples


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.


Solving for vertical asymptotes and holes cont

Solving for Vertical Asymptotes and Holes, cont.

Does the linear factor:


Example 1

Example #1


Example 2

Example #2


Example 3

Example #3


Example 4

Example #4


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?


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