Math Rational Functions Basic Graph Characteri stics Transforma tions Asymptotes vs Holes Solving Equations

Rational functions are quotients of polynomial functions. This means that rational functions can be expressed as where p(x) and q(x) are polynomial functions and q(x)  0. Lesson 12 Exploring Rational Functions The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero. These values are non-permissible for the variable x. p(x). q(x). is the set of all real numbers except 0, 2, and -5. For example, the domain of the rational function Math These Non-permissible values for x, have a significance on the graph of a function.

Basic Rational Function Characteristic Non-permissible value Behaviour near the non-permissible value End behaviour Domain Range Equation of the vertical asymptote Equation of the horizontal asymptote Intercepts Asymptotes Key Points x = 0 As x approaches 0, the |y| becomes very large. As |x| becomes very large, y approaches 0 {x | x ≠ 0, x  R} {y | y ≠ 0, y  R} x = 0 y = 0 Math 30-13

Vertical Asymptotes of Rational Functions If is a rational function in which p(x) and q(x) have no common factors and a is a zero of function q(x), the denominator, then x  a is a vertical asymptote of the graph of f(x). The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. f (x)   as x  a  f (x)   as x  a  The line x  a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a. f (x)   as x  a  f (x)   as x  a  f a y x x = a f a y x Thus, f (x)   or f(x)    as x approaches a from either the left or the right. Math 30-14

The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound. Horizontal Asymptotes of Rational Functions A rational function may have several vertical asymptotes, but it can have at most one horizontal asymptote. The graph of a function does not cross a vertical asymptote, however, the graph may cross a horizontal asymptote. f y x y = b x y f f y x f (x)  b as x   f (x)  b as x   f (x)  b as x   Math If the degree of the denominator is equal to the degree of the numerator, the location of the horizontal asymptote is determined by divided the leading coefficients.

f(x) = + k a x – h vertical stretch vertical translation horizontal translation Graphing Rational Functions Using Transformations (–a indicates a reflection in the x-axis) Math McGraw Hill Teacher Resource DVD N18_9.1_434_IA

Graph: f(x) = 1 x + 4 Vertical Asymptote: x = –4 Horizontal Asymptote: y = 0 Graphing Rational Functions Using Transformations Compared to the graph of, the graph of the function has been translated 4 units to the left.

Characteristic Non-permissible value Behaviour near the non-permissible value End behaviour Domain Range Equation of the vertical asymptote Equation of the horizontal asymptote Identifying Characteristics of the Graph x = – 4 As x approaches –4, the |y| becomes very large. As |x| becomes very large, y approaches 0. {x | x ≠ – 4, x  R} {y | y ≠ 0, y  R} x = – 4 y = 0 Algebraically determine Intercepts: y-intercept at 1/4

Graph: f(x) = x – 3 Vertical Asymptote: x = 3 Horizontal Asymptote: y = 4 Graphing Rational Functions Using Transformations, 2 steps Compared to the graph of, the graph of the function has been translated 3 units right and 4 up.

Characteristic Non-permissible value Behaviour near the non-permissible value End behaviour Domain Range Equation of the vertical asymptote Equation of the horizontal asymptote Identifying Characteristics x = 3 As x approaches 3, the |y| becomes very large. As |x| becomes very large, y approaches 4. {x | x ≠ 3, x  R} {y | y ≠ 4, y  R} x = 3 y = 4 Algebraically determine Intercepts: x-intercept at 11/4 y-intercept at 11/3

Vertical Asymptote: x = – 3 Horizontal Asymptote: y = – 6 Graphing Rational Functions – 3 steps Compared to the graph of, the graph of the function has been vertically stretched by a factor of 5,translated 3 units right and 4 up.

Characteristic Non-permissible value Behaviour near the non-permissible value End behaviour Domain Range Equation of the vertical asymptote Equation of the horizontal asymptote Graphing Rational Functions Using Transformations x = –3 As x approaches –3, the |y| becomes very large. As |x| becomes very large, y approaches –6 {x | x ≠ –3, x  R} {y | y ≠ –6, y  R} x = –3 y = –6 Intercepts: x-intercept at -2.2 y- intercept at -4.3

Math What does the graph of look like?

Math Write the equation of each function in the form

Graph What are the equations for the asymptotes? Graph What are the equations for the asymptotes?

Graph Vertical asymptotes are at x = -2 and x = 2. The line y = 0 is the horizontal asymptote. The domain is the set of real numbers, but x ≠ -2 and x ≠ 2. The range is y > 0 and Graphing Rational Functions Math

Math Frank is on the yearbook committee and is analyzing the cost of the yearbook. A printing company charges a $150 set up fee and $3 per book. Represent the average cost per book as a function of the number of booklets printed. Page 442 1, 3, 4a,c, 6, 7, 8, 12, 13, 14, 16

Graphing Rational Functions an Investigative Approach xf(x)f(x) As x approaches +∞, f(x) approaches_______. Graph: The graph is undefined for x = 0. Begin at x = 1 and consider the domain x > 1. ZERO 18 There is a horizontal asymptote at y = 0. Math 30-1

xf(x)f(x) 11 1/2 1/3 1/4 1/6 1/10 1/100 1/1000 Behaviour near a non-permissible value. As x approaches 0 from the right. f(x) approaches _______ Graph: The graph is undefined for x = 0. Domain interval (0, 1) +∞ 19Math 30-1

Graphing Rational Functions xf(x)f(x) –1 –2 –3 –4 –10 –100 –1000 As x approaches –∞, f(x) approaches _______ Graph: Plot x = –1 and consider the domain x < -1. ZERO 20 There is a horizontal asymptote at y = 0. Math 30-1

xf(x)f(x) –½ –1/3 –1/4 –1/6 –1/10 –1/100 –1/1000 Behaviour near a non-permissible value. As x approaches 0 from the left, f(x) approaches _______ Graph: The graph is undefined for x = 0. Domain interval (-1, 0) –∞ 21Math 30-1