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3. 3: Properties of Rational Functions

Rational Functions A rational function is a function of the form where p and q are polynomial functions. The domain consists of all real numbers except those for which the denominator is zero.

Asymptotes An asymptote is a line that a function approaches as x or y goes to ∞ or -∞. Horizontal asymptotes Vertical asymptotes The graph of a function may cross a HA The graph of a function will never cross a VA Oblique asymptotes The graph of a function may cross an OA

Vertical Asymptotes Given a rational function R(x), a vertical asymptote x = r occurs for all values of r for which the denominator is zero but the numerator is not.

Vertical Asymptotes Given a rational function R(x), a vertical asymptote x = r occurs for all values of r for which the denominator is zero but the numerator is not.

Vertical Asymptotes Find the vertical asymptotes and holes, if any, of the graphs of the following functions. State the domain of each function

Horizontal Asymptotes Given a rational function , a horizontal asymptote y = b occurs if the degrees of p and q are equal OR if the degree of q > p (in other words, if R(x) is bottom heavy) Equal: Degree q = p Bottom heavy: Degree q > p HA occurs at the quotient of the leading coefficients Always have a HA at y = 0

Oblique Asymptotes Given a rational function , an oblique asymptote y = ax + b occurs if the degrees of q < p (in other words, if R(x) is top heavy). Use long division to find the OA.

Horizontal or Oblique Asymptote? State whether the following functions contain a horizontal or oblique asymptote. Then, find it!

Recap Let’s recap. . how ‘bout it? ! Type Vertical Asymptotes x=r Horizontal Asymptotes y=b Oblique Asymptotes y = ax + b Can function cross it? When does a rational function have one? How do you find it?

3. 3: Properties of Rational Functions HW #5: p. 196 #11 -27 odd, 44, 41 -51 odd