2. The Friedland Method 9.3 Graphing General Rational Functions

3. Let p(x) and q(x) be polynomials with no mutual factors. p(x) = a m x m + a m-1 x m-1 +... + a 1 x + a 0 Meaning: p(x) is a polynomial of degree m Example: 3x 2 +2x+5; degree = 2 q(x) = b n x n + b n-1 x n-1 +... + b 1 x + b 0 Meaning: q(x) is a polynomial of degree n Example: 7x 5 -3x 2 +2x-1; degree = 5 Rational Function: f(x) = p(x) / q(x)

4. x-intercepts are the zeros of p(x) Meaning: Solve the equation: p(x) = 0 Vertical asymptotes occur at zeros of q(x) Meaning: Solve the equation: q(x) = 0 Horizontal Asymptote depends on the degree of p(x), which is m, and the degree of q(x), which is n. If m < n, then x-axis asymptote (y = 0) If m = n, divide the leading coefficients If m > n, then NO horizontal asymptote. Key Characteristics

5. Example: Graph y = State the domain and range. x-intercepts: None; p(x) = 4 ≠ 0 Vertical Asymptotes: None; q(x) = x 2 + 1. But if x 2 + 1 = 0 ---> x 2 = -1. No real solutions. Degree p(x) Horizontal Asymptote at y = 0 (x- axis) Graphing a Rational Function where m < n 4 x 2 +1

6. We can see that the domain is ALL REALS while the range is 0 < y ≤ 4 Let’s look at the picture!

7. Graph y = x-intercepts: 3x 2 = 0 ---> x 2 = 0 ---> x = 0. Vertical asymptotes: x 2 - 4 = 0 ---> (x - 2)(x+2) = 0 ---> x= ±2 Degree of p(x) = degree of q(x) ---> divide the leading coefficients ---> 3 ÷ 1 = 3. Horizontal Asymptote: y = 3 Graphing a rational function where m = n 3x 2 x2-4x2-4

8. Here’s the picture! xy -44 -3 5. 4 00 1 3 5. 4 44 You’ll notice the three branches. This often happens with overlapping horizontal and vertical asymptotes. The key is to test points in each region!

9. Graph y = x-intercepts: x 2 - 2x - 3 = 0 ---> (x - 3)(x + 1) = 0 ---> x = 3, x = -1 Vertical asymptotes: x + 4 = 0 ---> x = -4 Degree of p(x) > degree of q(x) ---> No horizontal asymptote Graphing a Rational Function where m > n x 2 - 2x - 3 x + 4

10. Not a lot of pretty points on this one. This graph actually has a special type of asymptote called “oblique.” It’s drawn in purple. You won’t have to worry about that. Picture time! xy - 12 - 20.6 -9-19.2 -6 - 22.5 -22.5 0 - 0.75 2-0.5 62.1

11. The Big Ideas Always be able to find: x-intercepts (where numerator = 0) Vertical asymptotes (where denominator = 0) Horizontal asymptotes; depends on degree of numerator and denominator Sketch branch in each region