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Solving for the Discontinuities of Rational Equations

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Solving for the Discontinuities of Rational Equations

- Vertical Asymptotes (VAs)
- Horizontal Asymptotes (HAs)
- Holes

- 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

- Examples:

- 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

- Examples:

- 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!!!

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?

- 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

- Degree of Numerator < Degree of Denominator

- For problems 1 – 4 on the Introduction to Solving Rational Equations Practice, solve for the horizontal asymptote.
1.2.

3.4.

- 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.

Does the linear factor:

- 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.

13.14.

15.16.

17.18.

- Does have a HA? (If yes, what is it?) Why?
- Does have VAs and/or holes? (If yes, what are they?) Why?