Rational Functions HONORS PRECALCULUS :: MR. VELAZQUEZ

Definition of Rational Functions Rational Functions are defined as the quotient of two polynomial functions. This means any rational function can be expressed as a ratio: f x = p x q x Where p and q are polynomial functions, and q x 0. The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero.

Finding the Domain Find the domain of the rational function f x = x2 16 x 4 The domain will include all real numbers except the values that would make the denominator zero. So we simply make the denominator equal to zero and solve for x. x 4 = 0 x = 4 Therefore: x (, 4) 4,

Finding the Domain Find the domain of the rational function f x = x x 2 36 x (, 6) 6, 6 (6, )

Vertical Asymptotes

Vertical Asymptotes y The equation f x = 1 x Vertical Asymptote on the y-axis x

Vertical Asymptotes The equation f x = 1 x 2 Vertical Asymptote on the y-axis

Arrow Notation

Definition of Vertical Asymptotes

Finding Vertical Asymptotes

Finding Vertical Asymptotes Find the vertical asymptotes, if any, of the rational function: f( x) x 2 x 36 Vertical asymptotes will occur at the zeros of the denominator: x 2 36 = 0 x 6 x + 6 = 0 x 6 = 0 x + 6 = 0 x = ±6 Vertical asymptotes of f(x) occur at x = 6 and x = 6

Finding Vertical Asymptotes Find the vertical asymptotes, if any, of the rational function: f( x) x 6 2 x 36 Vertical asymptote of f(x) occurs at x = 6

Finding Vertical Asymptotes Find the vertical asymptotes, if any, of the rational function: f x = x 2 x 2 4x TRY IT! (no need to graph)

But Wait! Consider the function f x = x2 4. The denominator in this x 2 case would be zero when x = 2, so you would expect there to be an asymptote. However, there is a reduced form of f(x) which cancels the denominator altogether:

Instead of an asymptote, the resulting graph will have a hole (called a removable discontinuity) corresponding to the denominator s zero. CAUTION: Your graphing calculator will not show removable discontinuities, unless you know exactly where to look for them.

Horizontal Asymptotes

Finding Horizontal Asymptotes Let f be the rational function defined as f x = a nx n + a n 1 x n 1 + + a 1 x + a 0 b m x m + b m 1 x m 1 + + b 1 x + b 0, a n 0, b n 0 The degree of the numerator is n. The degree of the denominator is m. 1. If n < m, the horizontal asymptote of f is just y = 0, the x-axis. 2. If n = m, the horizontal asymptote of f is the line y = a n b m. 3. If n > m, there are no horizontal asymptotes of f. WRITE THIS DOWN! You ll use it a lot.

Finding Horizontal Asymptotes Notice how the horizontal asymptote intersects the graph.

Finding Horizontal Asymptotes Find the horizontal asymptote, if any, of the rational function: Degree of numerator: n = 1 f( x) 3x 2 x 1 Degree of denominator: m = 2 n < m, therefore f(x) has a horizontal asymptote at y = 0

Finding Horizontal Asymptotes Find the horizontal asymptote, if any, of the rational function: Degree of numerator: n = 2 f( x) 2 6x 2 x 1 Degree of denominator: m = 2 n = m, and the leading coefficients are a n = 6 and b m = 1, therefore f(x) has a horizontal asymptote at y = a n b m = 6 1 = 6

Graphing Rational Functions

Graphing by Transformation

Graphing by Transformation Use the graph of f x = 1 to graph x g x = 1 4 x 3 y x

Graphing by Transformation Use the graph of f x = 1 x 2 to graph g x = 1 x 2 4 + 2 y x

Graphing Strategy for Rational Functions

Applying the Graphing Strategy Use the 7-step strategy to graph f x = 5x x 2 y x

Applying the Graphing Strategy Use the 7-step strategy to graph f x = 2x2 x 2 25 y x

Oblique Asymptotes

Oblique Asymptotes The graph of a rational function will have an oblique asymptote (or slant asymptote) if the degree of the numerator is one more than the degree of the denominator. The equation of the slant asymptote can be found by division:

Oblique Asymptotes Find the oblique asymptote of the function f x = x2 +6x+2 x

Oblique Asymptotes Find the oblique asymptote of the function f x = x3 1 x 2 +2x+2

Exit Ticket Due Today! Find (but do not graph) all horizontal, vertical and/or oblique asymptotes of the following three rational functions: 1. f x = x2 2x+3 x 2 x 6 2. g x = 8x3 x x 2 4x 5 3. h x = x3 2x 2 x+2 x 2 1 HOMEWORK Due 12/4 Pg. 355, #24-76 Multiples of 4