Rational Functions and Their Graphs

Example Find the Domain of this Function. Solution: The domain of this function is the set of all real numbers not equal to 3.

Arrow Notation SymbolMeaning x  a  x approaches a from the right. x  a  x approaches a from the left. x   x approaches infinity; that is, x increases without bound. x    x approaches negative infinity; that is, x decreases without bound.

The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. as x  a , f (x)    as x  a  f (x)   The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. as x  a , f (x)    as x  a  f (x)   Thus, f (x)   or f(x)    as x approaches a from either the left or the right. f a y x x = a f a y x Definition of a Vertical Asymptote

The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. Thus, f (x)   or f(x)    as x approaches a from either the left or the right. x = a f a y x f a y x as x  a  f (x)   as x  a  f (x)    Definition of a Vertical Asymptote

Locating Vertical Asymptotes If f(x) = p(x) / q(x) is a rational function,  p(x) and q(x) have no common factors,  a is a zero of q(x), the denominator, Then x = a is a vertical asymptote of the graph of f. If a common factor exists-we have a whole in the graph at that point.

The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound. f y x y = b x y f f y x as x   f (x)  b as x   f (x)  b as x  f (x)  b Definition of a Horizontal Asymptote

Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. 1.If n<m, the x-axis, or y=0, is the horizontal asymptote of the graph of f.  IF  THEN the x-axis, or y=0, is the horizontal asymptote of the graph of f. 2.If n=m, the line y = a n /b m is the horizontal asymptote of the graph of f. 3.If n>m, the graph of f has no horizontal asymptote.

Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m If n=m;  then line y = a n /b m is the horizontal asymptote of the graph of f.

Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m If n>m, then the graph of f has no horizontal asymptote.  Oblique asymptote or Slant asymptote.

Strategy for Graphing a Rational Function Suppose that f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions with no common factors:.1 Define the Domain !!!!!!! Hole vs. Asymptote 1.Find the y-intercept (if there is one) by evaluating f (0). 2.Find the x-intercepts (if there are any) by solving the equation p(x)  0. (Numerator = 0) 3.Find any vertical asymptote(s) by solving the equation q (x)  0.  Denominator =0 4.Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 5.Determine behavior at ±∞ and at vertical asymptotes.

Sketch the graph of

the graph has no symmetry

The y-intercept is (0,-3/10)

the graph has no symmetry The y-intercept is (0,-3/10) The x-intercept is (3/2, 0)

the graph has no symmetry The y-intercept is (0,-3/10) The x-intercept is (3/2, 0) The vertical asymptote is x = -2

the graph has no symmetry The y-intercept is (0,-3/10) The x-intercept is (3/2, 0) The vertical asymptote is x = -2 The horizontal asymptote is y = 2/5

the graph has no symmetry The y-intercept is (0,-3/10) The x-intercept is (3/2, 0) The vertical asymptote is x = -2 The horizontal asymptote is y = 2/5 Test points include (-3, 9/5), (0, -3/10), (2, 1/20)

The x-intercept is (3/2, 0) The vertical asymptote is x = -2 x=-2 and x=3/2 are critical points: The following intervals need investigation:

Vertical Asymptotes or Holes

This is a hole in the graph!!!

Hole in the Graph Must account for all values for x that are not in the Domain!!! This is a hole in the graph!!! =

Text Example Find the slant asymptote of Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x  3 into x 2  4x  5. The equation of the slant asymptote is y  x  1. Using our strategy for graphing rational functions, the graph is shown Vertical asymptote: x = 3 Slant asymptote: y = x - 1

Rational Functions and Their Graphs