1. Graphing Rational Functions

2. 2 xf(x)f(x) 20.5 11 2 0.110 0.01100 0.0011000 xf(x)f(x) -2-0.5 -0.5-2 -0.1-10 -0.01-100 -0.001-1000 As x → 0 –, f(x) → -∞. As x → 0 +, f(x) → +∞. 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. f(x) = Example: f (x) = is defined for all real numbers except x = 0.

3. 3 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) → – ∞ 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 –. Vertical Asymptote

4. 4 Example: Show that the line x = 2 is a vertical asymptote of the graph of f(x) =. xf(x)f(x) 1.516 1.9400 1.9940000 2- 2.0140000 2.1400 2.516 Observe that: x→2 –, f (x) → – ∞ x→2 +, f (x) → + ∞ This shows that x = 2 is a vertical asymptote. Example 1: Vertical Asymptote y x 100 0.5 f (x) = x = 2

5. 5 Set the denominator equal to zero and solve. Solve the quadratic equation x 2 + 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 →1 +, f(x) → + ∞. As x → -5 –, f(x) → + ∞. As x →-5 +, f(x) → – ∞. x = -5 is a vertical asymptote.x = 1 is a 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) =.

6. 6 1. Find the roots of the denominator.0 = x 2 – 4 = (x + 2)(x – 2) 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) → – ∞ and x →+2 +, f(x) → + ∞. So, x = 2 is a vertical asymptote. f is undefined at -2 A hole in the graph of f at (-2, -0.25) shows a removable singularity. x = 2 Example 3: Vertical Asymptote Example: Find the vertical asymptotes of the graph of f(x) =. x y (-2, -0.25)

7. 7 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 + 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

8. 8 xf(x)f(x) 100.1 1000.01 10000.001 0 – -10-0.1 -100-0.01 -1000-0.001 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 –. Example 1: Horizontal Asymptote Example: Show that the line y = 0 is a horizontal asymptote of the graph of the function f(x) =. x y f(x) = y = 0

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

10. 10 Finding 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 = a m is a horizontal asymptote. bnbn Asymptotes for Rational Functions Given a rational function: f (x) = P(x) a m x m + lower degree terms Q(x) b n x n + lower degree terms =

11. 11 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. 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. Horizontal and Vertical Asymptotes Example: Find all horizontal and vertical asymptotes of f (x) =. y = 3 x = 2 x y

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