Rational Functions.

1

Rational Function

What is a rational function?

A fraction made of two polynomials P(x)Q(x)Q(x)P(x)​ where Q(x)≠0Q(x)=0.

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Simplify

What do you always do first?

Simplify the expression — cancel common factors.

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V.A.

How do you find vertical asymptotes?

Set denominator = 0 after simplification.

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V.A. rule

When is a value NOT a vertical asymptote?

When it’s a hole (factor cancels).

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H.A. rule 1

If degree top < degree bottom?

y=0y=0.

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H.A. rule 2

If degree top = degree bottom?

y=leading coeff numeratorleading coeff denominatory=leading coeff denominatorleading coeff numerator​.

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H.A. rule 3

If degree top > degree bottom?

No H.A. (check for slant).

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S.A.

When does a slant asymptote occur?

When degree of numerator = degree of denominator + 1.

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S.A. method

How do you find a slant asymptote?

Use long division (quotient = asymptote).

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X-Int

How do you find x-intercepts?

Set numerator = 0 (after simplification).

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X-Int check

What must you check after solving numerator = 0?

That x doesn’t make denominator = 0.

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Y-Int

How do you find the y-intercept?

Plug in x=0x=0.

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Holes

When do holes occur?

When a factor cancels between numerator and denominator.

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Holes (x)

How to find x of a hole?

Set the cancelled factor = 0.

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Holes (y)

How to find y of a hole?

Plug x of hole into simplified function.

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Step order

What’s the best order to analyze?

Simplify → Holes → V.A. → H.A./S.A. → X-Int → Y-Int.

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Degrees

What does “degree” mean?

The highest power of x in the polynomial.

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Coefficients

What is the “leading coefficient”?

The number in front of the highest-power x term.

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Graph meaning

What do asymptotes represent?

Lines the graph approaches but never touches.

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Domain reminder

What x-values are excluded?

Any x that makes the denominator 0.