Graphing General Rational Functions

What’s the big idea? We’re always looking for shortcuts to make sketching easier. The next two questions shed some light on a function’s graph: What happens to f(x) as x as gets closer and closer to where the denominator is zero? What happens to f(x) as x goes way right (positive infinity) or way left (negative infinity)? The answer to the first question deals with asymptotes. The answer to the second question deals with the end behavior of a function

Talkin’ bout Asymptotes. Let’s look at f(x) = 1/(x-2) We know that x ≠ 2. But what happens as x gets close to 2? If x = 1.9999, f(x) = -10,000 If x = 2.0001, f(x) = 10,000. Ah! As x approaches the asymptote from the left, the function goes to -∞ And as x approaches the asymptote from the right, the function goes to ∞ We write that as: x  2- f(x)  -∞ x  2+ f(x)  ∞

Vertical Asymptotes The line x = a is a vertical asymptote of f(x) if f(x)  ∞ or f(x)  -∞ as x approaches a from the left or right side. NOTE: We are not guaranteed a vertical asymptote simply because we have a restricted domain value. Example: f(x) = 3x/x Notice how the x just cancels off? When that happens, you get a hole instead of a vertical asymptote. So the equation becomes y = 3 but with a hole at x = 0 (the domain restriction)

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 Example: 7x5-3x2+2x-1; degree = 5

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

Graphing a rational function where m = n 3x2 x2-4 Graph y = x-intercepts: Set top = 0 and solve! 3x2 = 0  x2 = 0  x = 0. Vertical asymptotes: Set bottom = 0 and solve x2 - 4 = 0  (x - 2)(x+2) = 0  x= ±2 Degree of p(x) = degree of q(x)  top and bottom tie  divide the leading coefficients Y = 3/1 = 3. Horizontal Asymptote: y = 3

You’ll notice the three branches. Here’s the picture! x y -4 4 -3 5.4 -1 1 3 You’ll notice the three branches. This often happens with overlapping horizontal and vertical asymptotes. The key is to test points in each region! Domain: x ≠ ±2 Range: y > 3 & y ≤ 0

Graphing a Rational Function where m < n Example: Graph y = State the domain and range. x-intercepts: None; p(x) = 4 and 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 (x-axis) 4 x2 + 1

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 Graph y = 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 x2- 2x - 3 x + 4

Picture time! x y -12 -20.6 -9 -19.2 -6 -22.5 -2 2.5 -0.75 2 -0.5 6 -0.75 2 -0.5 6 2.1 Not a lot of pretty points on this one. This graph actually has a special type of asymptote called “slant.” It’s drawn in purple. Don’t worry, I’ll show you how it works.

Slant Asymptotes Equation of Asymptote! Remainder is Irrelevant These happen when the top polynomial overpowers the bottom polynomial. “Overpowers” means has a higher degree. The equation of the slant asymptote is just the result when you divide the two polymomials out. Prior example: Remainder is Irrelevant

The Big Ideas x-intercepts (where numerator = 0) Always be able to find: x-intercepts (where numerator = 0) Vertical asymptotes (where denominator = 0) Horizontal asymptotes: If bottom wins: x-axis asymptote If they tie: divide leading coeff. If top wins: No horizontal asymp. Sketch branch in each region (plot points)

Practice Problems 6 x2 + 3 No x-intercepts No vertical asymptote Graph y = 6 x2 + 3 x y 2 -1 1.5 1 -2 6/7 No x-intercepts No vertical asymptote H.A.: y = 0

Practice Problems x2 - 4 x-int: x2 – 4 = 0; x = ±2 Graph y = x2 - 4 x + 1 x-int: x2 – 4 = 0; x = ±2 V.A.: x + 1 = 0  x = -1 H.A.: Top wins: No H.A. Slant: y = x - 1 x y -4 -3 -5/2

Practice Problems 2x2 x-int: 2x2 = 0  x = 0 x2 - 1 Graph y = x-int: 2x2 = 0  x = 0 V.A.: x2 – 1 = 0  x = ±1 H.A.: Tie  y = 2/1 = 2 x y 2 8/3 3 9/4 -2 -3 Bounce back, baby!