Graphing Rational Functions

Example: f (x) = is defined for all real numbers except x = 0. A rational function is a function of the form f(x) = , where P(x) and Q(x) are polynomials and Q(x) = 0. Example: f (x) = is defined for all real numbers except x = 0. f(x) = x f(x) 2 0.5 1 0.1 10 0.01 100 0.001 1000 x f(x) -2 -0.5 -1 -0.1 -10 -0.01 -100 -0.001 -1000 As x → 0+, f(x) → +∞. As x → 0–, f(x) → -∞. Rational Function

The line x = a is a vertical asymptote of the graph of y = f(x), if and only if f(x) → + ∞ or f(x) → – ∞ as x → a + or as x → a –. x x = a as x → a – f(x) → + ∞ x x = a as x → a – f(x) → – ∞ x x = a as x → a + f(x) → + ∞ x x = a as x → a + f(x) → – ∞ Vertical Asymptote

Example 1: Vertical Asymptote Example: Show that the line x = 2 is a vertical asymptote of the graph of f(x) = . y x 100 0.5 x = 2 x f(x) 1.5 16 1.9 400 1.99 40000 2 - 2.01 2.1 2.5 f (x) = Observe that: x→2–, f (x) → – ∞ This shows that x = 2 is a vertical asymptote. x→2+, f (x) → + ∞ Example 1: Vertical Asymptote

Example 2: Vertical Asymptote A rational function may have a vertical asymptote at x = a for any value of a such that Q(a) = 0. Example: Find the vertical asymptotes of the graph of f(x) = . Example 2: Vertical Asymptote Set the denominator equal to zero and solve. Solve the quadratic equation x2 + 4x – 5. (x – 1)(x + 5) = 0 Therefore, x = 1 and x = -5 are the values of x for which f may have a vertical asymptote. As x →1– , f(x) → – ∞. As x → -5–, f(x) → + ∞. As x →1+, f(x) → + ∞. As x →-5+, f(x) → – ∞. x = 1 is a vertical asymptote. x = -5 is a vertical asymptote.

Example 3: Vertical Asymptote Example: Find the vertical asymptotes of the graph of f(x) = . 1. Find the roots of the denominator. 0 = x2 – 4 = (x + 2)(x – 2) Possible vertical asymptotes are x = -2 and x = +2. Example 3: Vertical Asymptote 2. Calculate the values approaching -2 and +2 from both sides. x → -2, f(x) → -0.25; so x = -2 is not a vertical asymptote. x → +2–, f(x) → – ∞ and x →+2+, f(x) → + ∞. So, x = 2 is a vertical asymptote. x y (-2, -0.25) x = 2 f is undefined at -2 A hole in the graph of f at (-2, -0.25) shows a removable singularity.

The line y = b is a horizontal asymptote of the graph of y = f(x) if and only if f(x) → b + or f(x) → b – as x → + ∞ or as x → – ∞. Horizontal Asymptote y y = b as x → + ∞ f(x) → b – y y = b as x → – ∞ f(x) → b – y y = b as x → + ∞ f(x) → b + y y = b as x → – ∞ f(x) → b +

Example 1: Horizontal Asymptote Example: Show that the line y = 0 is a horizontal asymptote of the graph of the function f(x) = . As x becomes unbounded positively, f(x) approaches zero from above; therefore, the line y = 0 is a horizontal asymptote of the graph of f. As f(x) → – ∞, x → 0 –. x y x f(x) 10 0.1 100 0.01 1000 0.001 – -10 -0.1 -100 -0.01 -1000 -0.001 f(x) = y = 0 Example 1: Horizontal Asymptote

Example 2: Horizontal Asymptote Example: Determine the horizontal asymptotes of the graph of f(x) = . Example 2: Horizontal Asymptote Divide x2 + 1 into x2. f(x) = 1 – As x → +∞, → 0– ; so, f(x) = 1 – →1 –. y x Similarly, as x → – ∞, f(x) →1–. y = 1 Therefore, the graph of f has y = 1 as a horizontal asymptote.

Asymptotes for Rational Functions Finding Asymptotes for Rational Functions Given a rational function: f (x) = P(x) am xm + lower degree terms Q(x) bn xn + lower degree terms = Asymptotes for Rational Functions If c is a real number which is a root of both P(x) and Q(x), then there is a removable singularity at c. If c is a root of Q(x) but not a root of P(x), then x = c is a vertical asymptote. If m > n, then there are no horizontal asymptotes. If m < n, then y = 0 is a horizontal asymptote. If m = n, then y = am is a horizontal asymptote. bn

Horizontal and Vertical Asymptotes Example: Find all horizontal and vertical asymptotes of f (x) = . Horizontal and Vertical Asymptotes Factor the numerator and denominator. The only root of the numerator is x = -1. The roots of the denominator are x = -1 and x = 2 . x = 2 Since -1 is a common root of both, there is a hole in the graph at -1 . x y Since 2 is a root of the denominator but not the numerator, x = 2 will be a vertical asymptote. y = 3 Since the polynomials have the same degree, y = 3 will be a horizontal asymptote.

A slant asymptote is an asymptote which is not vertical or horizontal. Example: Find the slant asymptote for f(x) = . Divide: x y y = 2x - 5 x = -3 As x → + ∞, → 0+. As x → – ∞, → 0–. Therefore as x → ∞, f(x) is more like the line y = 2x – 5. The slant asymptote is y = 2x – 5. Slant Asymptote