Asymptotes.

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Presentation transcript:

Asymptotes

Rational function A rational function is one that can be written in the form where p(x) and q(x) are polynomials and q(x) is not the zero polynomial. In general, the domain of a rational function includes all real numbers except x-values which make the denominator zero.

Find domain Find the domain of each of the following:

Answer Set denominator not equal to 0

Vertical Asymptotes At the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the following.

Answer Set denominator = 0

Graphically The figure below shows the graph of The equation of the vertical asymptote is x=-2.

Summary The graph of a function has vertical asymptote(s) at the discontinuity of the function. (set the denominator = 0) If the degree of the numerator is = to the degree of the denominator, the function has a horizontal asymptote. ( y = ratio of lead coefficients) If the degree of the numerator is < the degree of the denominator, the function has a horizontal asymptote. (y = 0, the x-axis). If the degree to the numerator is > the degree of the denominator, the function does not have a horizontal asymptote.

Find asymptotes Name the vertical and horizontal asymptotes for

Answer Domain: Degree of numerator: 1 Degree of denominator: 1 HA: y = 2 VA: x = -1

Find asymptotes Name the vertical and horizontal asymptotes for

Answer Domain: Degree of numerator: 2 Degree of denominator: 2 HA: y = 5/2 VA: x = -2,2

Find asymptotes Name the vertical and horizontal asymptotes for

Answer Domain: Degree of numerator: 0 Degree of denominator: 2 HA: y = 0 VA: none

Find asymptotes Name the vertical and horizontal asymptotes for

Answer Domain: Degree of numerator: 1 Degree of denominator: 2 HA: y = 0 VA: x=3,-3

Find asymptotes Name the vertical and horizontal asymptotes for

Answer Domain: Degree of numerator: 2 Degree of denominator: 1 HA: none VA: x=-2

Find asymptotes Name the vertical and horizontal asymptotes for

Answer Domain: Degree of numerator: 3 Degree of denominator: 1 HA: none VA: x=-1/3