Graphing Rational Functions Sect. 2.7 Graphing Rational Functions

Oblique asymptote at y = x + 5 ASYMPTOTES If the degree of the numerator is greater than the degree of the denominator by 1, then there is not a horizontal asymptote, but an oblique one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder. degree of top = 3 degree of bottom = 2 Oblique asymptote at y = x + 5

Find the oblique asymptote: Equation of Asymptote! Example 1: Find the oblique asymptote: Remainder is Irrelevant

Degree of denominator = 2 Example 2: Find the horizontal or oblique asymptote: Degree of numerator = 1 Degree of denominator = 2 Since the degree of the numerator is less than the degree of the denominator, there is NO oblique asymptote but a horizontal one at y = 0.

Degree of denominator = 1 Example 3: Find the horizontal or oblique asymptote: Degree of numerator = 2 Degree of denominator = 1 Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. The oblique asymptote is found by long division

Degree of denominator = 1 Example 4: Find the horizontal or oblique asymptote: Degree of numerator = 1 Degree of denominator = 1 Since the degree of the numerator is equal to the degree of the denominator, there is NO oblique asymptote but a horizontal one at .

Degree of denominator = 1 Example 5: Find the horizontal or oblique asymptote: Degree of numerator = 3 Degree of denominator = 1 Since the degree of the numerator is differs from the degree of the denominator by more than 1, there is NO HA and NO OA.

Graphing a Rational Function Suppose that where p(x) and q(x) are polynomial functions with no common factors. 1. Calculate the x-intercepts by setting the function = 0 and solving for x (This is the same as solving the equation p(x) = 0.). 2. Calculate the y-intercept by setting x = 0 and solving for y. 2. Find vertical asymptote(s) by factoring the numerator/ denominator, cancelling like factors, and solving the equation q(x) = 0. 3. Find holes by taking the factors that cancelled, setting them = 0, and solving for x. 4. Find the horizontal/oblique asymptote(s) using the rule based on the degrees of p(x) and q(x). 5. Plot points in between the x-intercepts and sketch the graph labeling the asymptotes.

Example 1: Sketch the graph of The vertical asymptote is x = -2. The horizontal asymptote is y = 2/5. The x-intercept is x = 3/2. The y-intercept is -3/10.

Graphing a rational function where m = n 3x2 x2-4 Ex 2: 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 Ex 3: 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 OA – use synthetic division y = x - 6 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

Ex 4 Graph y = domain: all real numbers x-intercepts: None; p(x) = 6 and 6 ≠ 0 Vertical Asymptotes: None; x2 + 3 = 0  x2 = -3. No real solutions. Degree p(x) < Degree q(x) --> Horizontal Asymptote at y = 0 (x-axis) 6 x2 + 3

x y 2 -1 1.5 1 -2 6/7 y = 6 x2 + 3

Horizontal Asymptotes: Ex 5 Domain: Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Horizontal Asymptotes: Ex 6 Domain: Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

Graph Ex 7 x y -2 -.4 -1 -.5 0 0 1 .5 2 .4 y-intercepts: 0 x-intercepts: x=0 vert. asymp.: x2+1=0 x2= -1 No vert asymp 4. horiz. asymp: 1<2 (deg. of top < deg. of bottom) y=0 x y -2 -.4 -1 -.5 0 0 1 .5 2 .4 (No real solns.)

Domain: (-∞,∞) Range:

Ex 8 Graph slant asymptote: y = –2x – 4

Horizontal Asymptotes: Ex 9 Domain: Vertical Asymptotes: Horizontal Asymptotes: Holes: Intercepts:

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

Closure Explain the process for calculating the horizontal and oblique asymptotes of a rational function.