Rational Functions and Asymptotes

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Presentation transcript:

Rational Functions and Asymptotes Arrow notation: 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 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. If as

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

Graphing a rational function. Find the vertical asymptotes if they exist. Find the horizontal asymptote if it exists. Find the x intercept(s) by finding f(x) = 0. Find the y intercept by finding f(0). Find any crossing points on the horizontal asymptote by setting f(x) = the horizontal asymptote. Graph the asymptotes, intercepts, and crossing points. Find additional points in each section of the graph as needed. 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