Graphing Rational Functions

1. Graphing Rational Functions

2. Rational Function 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) = As x →0–,f(x) → -∞. As x →0+, f(x) → +∞.

3. 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 –. as x→ a –f(x) →+ ∞ as x→ a –f(x) →– ∞ as x→ a +f(x) →+ ∞ as x→ a +f(x) →– ∞ x x x x x = a x = a x = a x = a Vertical Asymptote

4. Example 1: Vertical Asymptote Example: Show that the line x = 2 is a vertical asymptote of the graph off(x) = . x = 2 y f(x) = 100 x 0.5 Observe that: This shows that x = 2 is a vertical asymptote. x→2–, f(x) → – ∞ x→2+, f(x) → + ∞

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

6. 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) y x = 2 x (-2, -0.25) Example 3: Vertical Asymptote Possible vertical asymptotes are x = -2 and x = +2. 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) → – ∞ andx →+2+, f(x) → + ∞. So,x = 2 is a vertical asymptote. f is undefined at -2 A hole in the graph of fat (-2, -0.25) shows a removable singularity.

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

8. 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 –. y f(x) = x y =0

9. Example:Determine the horizontal asymptotes of the graph off(x) = . Dividex2+ 1 into x2. f(x) = 1 – As x → +∞,→ 0– ; so, f(x) = 1 – →1 –. y y = 1 x Example 2: Horizontal Asymptote Similarly, as x → – ∞,f(x) →1–. Therefore, the graph of f hasy = 1 as a horizontal asymptote.

10. P(x) amxm+ lower degree terms Given a rational function: f(x) = = Q(x) bnxn+ lower degree terms • If m = n, then y = am is a horizontal asymptote. bn Finding Asymptotes for Rational Functions 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.

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

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