Rational Functions and Asymptotes

Arrow notation:

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x→ a+

x→ a−

x→ +∞

x→ −∞

Meaning:

x approaches a from the right

x approaches a from the left

x is approaching infinity, increasing forever

x is approaching infinity, decreasing forever

Vertical Asymptotes

The line x = a is a vertical asymptote of a function if f(x) increases or decreases without bound as x approaches a.

If as

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x→ a+ or x→ a−

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f x( ) → +∞ or f x( ) → −∞

then x = a is vertical asymptote of the function.

Horizontal Asymptotes

The line y = b is a horizontal asymptote of a function if f(x) approaches b as x increases or decreases without bound.

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x→ +∞ or x→ −∞If as

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f x( ) →b+ or f x( ) →b−

Vertical asymptotes are a result of a domain issue.Find these by setting the denominator equal to zero.

Horizontal asymptotes guide the end behavior of graph.

If the degree of the numerator < degree of the denominator then the horizontal asymptote will be y = 0.

If the degree of the numerator = degree of denominator then the horizontal asymptote will be

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y =leading coefficientleading coefficient

If the degree of the numerator > degree of the denominatorthen there is no horizontal asymptote.

Graphing a rational function.

1. Find the vertical asymptotes if they exist.2. Find the horizontal asymptote if it exists.3. Find the x intercept(s) by finding f(x) = 0.4. Find the y intercept by finding f(0).5. Find any crossing points on the horizontal asymptote

by setting f(x) = the horizontal asymptote.6. Graph the asymptotes, intercepts, and crossing

points.7. Find additional points in each section of the graph as

needed.8. Make a smooth curve in each section through known

points and following asymptotes and end behavior.

Homework: Practice # 1 p. 342 9-35 odds, 49 - 69 eoo