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9.3 Graphing General Rational Functions The Friedland Method Rational Function: f(x) = p(x)/q(x) Let p(x) and q(x) be polynomials with no mutual factors. p(x) = amxm + am-1xm-1 + ... + a1x + a0 Meaning: p(x) is a polynomial of degree m Example: 3x2+2x+5; degree = 2 q(x) = bnxn + bn-1xn-1 + ... + b1x + b0 Meaning: q(x) is a polynomial of degree n Key Characteristics 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. Graphing a Rational Function where m < n 4 Example: Graph y = 2 x +1 State the domain and range. x-intercepts: None; p(x) = 4 ≠ 0 Vertical Asymptotes: None; q(x) = x2+ 1. But if x2+ 1 = 0 ---> x2 = -1. No real solutions. Degree p(x) < Degree q(x) --> Horizontal Asymptote at y = 0 (xaxis) Let’s look at the picture! We can see that the domain is ALL REALS while the range is 0 < y ≤ 4 Graphing a rational function where m = n 2 2 x 3x Graph y = 4 x-intercepts: 3x2 = 0 ---> x2 = 0 ---> x = 0. Vertical asymptotes: x2 - 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 Here’s the picture! x y -4 4 5. -3 4 -1 -1 0 0 1 -1 5. 4 4 3 4 You’ll notice the three branches. This often happens with overlapping horizontal and vertical asymptotes. The key is to test points in each region! Graphing a Rational Function where m > n Graph y = x2- 2x 3 x+4 x-intercepts: x2- 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 x y 12 20.6 -9 -19.2 -6 22.5 -2 2.5 0 0.75 2 -0.5 6 Picture time! 2.1 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. 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