Rational Functions A function f represented by where p(x) and q(x) are polynomials and q(x) ≠ 0, is a rational function.

Rational Function The domain of a rational function includes all real numbers except the zeros of the denominator q(x). The graph of a rational function is continuous except at x-values where q(x) = 0.

Example: Identifying rational functions For each rational function. Determine domain and range a) b) c)

Example: Identifying rational functions Solution a) b) Is a rational function - both numerator and denominator are polynomials; domain is all real numbers; x2 + 1 ≠ 0 Is NOT a rational function Denominator is not a polynomial; domain is {x | x > 0}

Example: Identifying rational functions Is a rational function - both numerator and denominator are polynomials; domain is {x | x ≠1, x ≠ 2} because (x – 1)(x – 2) = 0 when x = 1 and x = 2.

Vertical Asymptotes The line x = k is a vertical asymptote of the graph of f if f(x) g ∞ or f(x) g –∞ as x approaches k from either the left or the right.

Horizontal Asymptotes The line y = b is a horizontal asymptote of the graph of f if f(x) g b as x approaches either ∞ or –∞.

Finding Vertical & Horizontal Asymptotes Let f be a rational function given by written in lowest terms. Vertical Asymptote To find a vertical asymptote, set the denominator, q(x), equal to 0 and solve. If k is a zero of q(x), then x = k is a vertical asymptote. Caution: If k is a zero of both q(x) and p(x), then f(x) is not written in lowest terms, and x – k is a common factor.

Finding Vertical & Horizontal Asymptotes (a) If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is a horizontal asymptote. (b) If the degree of the numerator equals the degree of the denominator, then y = a/b is a horizontal asymptote, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.

Finding Vertical & Horizontal Asymptotes (c) If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes.

Example: Identifying rational functions For each rational function, determine any horizontal or vertical asymptotes. a) b) c)

Example: Identifying rational functions Solution a) Degree of numerator and denominator are both 1. Since the ratio of the leading coefficients is 6/3, the horizontal asymptote is y = 2. When x = –1, the denominator, 3x + 3, equals 0 and the numerator, 6x – 1 does not equal 0, so the vertical asymptote is x = 1

Example: Identifying rational functions Here’s a graph of f(x).

Example: Identifying rational functions b) Degree of numerator is one less than the degree of the denominator so the x-axis, or y = 0, is a horizontal asymptote. When x = ±2, the denominator, x2 – 4, equals 0 and the numerator, x + 1 does not equal 0, so the vertical asymptotes are x = 2 and x = 2.

Example: Identifying rational functions b) Here’s a graph of g(x).

Example: Identifying rational functions Degree of numerator is greater than the degree of the denominator so there are no horizontal asymptotes. When x = –1, both the numerator and denominator equal 0 so the expression is not in lowest terms: g(x) = x – 1, x ≠ –1. There are no vertical asymptotes.

Example: Identifying rational functions Here’s the graph of h(x). A straight line with the point (–1, –2) missing.

Slant, or Oblique, Asymptotes A third type of asymptote, which is neither vertical nor horizontal, occurs when the numerator of a rational function has degree one more than the degree of the denominator.

Slant, or Oblique, Asymptotes The line y = x + 1 is a slant asymptote, or oblique asymptote of the graph of f.

Graphs and Transformations of Rational Functions Graphs of rational functions can vary greatly in complexity. We begin by graphing and then use transformations to graph other rational functions.

Example: Analyzing the graph of Sketch a graph of and identify any asymptotes. Solution Vertical asymptote: x = 0 Horizontal asymptote: y = 0

Example: Using transformations to graph a rational function Use the graph of to sketch a graph of Include all asymptotes in your graph. Write g(x) in terms of f(x).

Example: Using transformations to graph a rational function Solution g(x) is a translation of f(x) left 2 units and then a reflection across the x-axis. Vertical asymptote: x = –2 Horizontal asymptote: y = 0 g(x) = –f(x + 2)

Example: Analyzing a rational function with technology Let a) Use a calculator to graph f. Find the domain of f. b) Identify any vertical or horizontal asymptotes. c) Sketch a graph of f that includes the asymptotes.

Example: Analyzing a rational function with technology Solution a) Here’s the calculator display using “Dot Mode.” The function is undefined when x2 – 4 = 0, or when x = ±2. The domain of f is D = {x|x ≠ 2, x ≠ –2}.

Example: Analyzing a rational function with technology Solution b) When x = ±2, the denominator x2 – 4 = 0 (the numerator does not), so the vertical asymptotes are x = ±2. Degree of numerator = degree of denominator, ratio of leading coefficients is 2/1 = 2, so the horizontal asymptote is y = 2.

Example: Analyzing a rational function with technology c) Here’s another version of the graph.

Graphing Rational Functions by Hand Let define a rational function in lowest terms. To sketch its graph, follow these steps. STEP 1: Find all vertical asymptotes. STEP 2: Find all horizontal or oblique asymptotes. STEP 3: Find the y-intercept, if possible, by evaluating f(0).

Graphing Rational Functions by Hand STEP 4: Find the x-intercepts, if any, by solving f(x) = 0. (These will be the zeros of the numerator p(x).) STEP 5: Determine whether the graph will intersect its nonvertical asymptote y = b by solving f(x) = b, where b is the y-value of the horizontal asymptote, or by solving f(x) = mx + b, where y = mx + b is the equation of the oblique asymptote.

Graphing Rational Functions by Hand STEP 6: Plot selected points as necessary. Choose an x-value in each interval of the domain determined by the vertical asymptotes and x-intercepts. STEP 7: Complete the sketch.

Example: Analyzing a rational function with technology Graph Solution STEP 1: Vertical asymptote: x = 3 STEP 2: Horizontal asymptote: y = 2 STEP 3: f(0) = , y-intercept is

Example: Analyzing a rational function with technology STEP 4: Solve f(x) = 0 The x-intercept is

Example: Analyzing a rational function with technology STEP 5: Graph does not intersect its horizontal asymptote, since f(x) = 2 has no solution. STEP 6: The points are on the graph. STEP 7: Complete the sketch (next slide)

Example: Analyzing a rational function with technology STEP 7

Direct, Inverse and Joint Variation

Direct Variation y varies directly as x if there is some nonzero constant k such that y = kx. k is called the constant of variation

Example If y varies directly as x and y = 9 when x is -15, find y when x = 21.

Inverse Variation y varies inversely as x if there is some nonzero constant k such that xy = k or y = k/x

Example If y varies inversely as x and y = 4, when x = 12, find y when x = 5.

Joint Variation y varies jointly as x and z if there is some number k such that y = kxz, where and

Example The area A of a trapezoid varies jointly as the height h and the sum of its bases b1 and b2. Find the equation of joint variation if A = 48 in^3, h = 8 in., b1 = 5 in. and b2 = 7 in.

Practice If y varies jointly as x and z and y = 45 when x = 9 and z = 15, find y when x = 25 and z = 12

Practice If y varies inversely as x and y = ¼ when x = 24, find y when x = ¾ y = 8

TRY! If m varies directly as w and m = -15 when w = 2.5, find m when w = 12.5 m = ?

Example The power P in watts of an electrical circuit varies jointly as the resistance R and the square of the current I. For a 600-watt microwave oven that draws a current of 5.0 amperes, the resistance is 24 ohms. What is the resistance of a 200-watt refrigerator that draws a current of 1.7 amperes?

TRY The LEM (Lunar Exploration Module) used by astronauts to explore the moon’s surface during the Apollo space missions weighs about 30,000 pounds on Earth. On the moon there is less gravity so it weighs less, meaning that less fuel is needed to lift off from the moon’s surface. The force of gravity on Earth is about 6 times as much as that on the moon. How much does the LEM weigh on the moon?

Answer y =kx 30,000 = 6x x = 5,000 lbs