Basic Graph Solving Equations Characteri stics Rational Functions

Basic Graph Solving Equations Characteri stics Rational Functions Asymptotes vs Holes Transform ations Math 30 -1 1

9. 1 Exploring Rational Functions 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. 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. For example, the domain of the rational function p(x). q(x). These Non-permissible values for x, have a significance on the graph of a function. is the set of all real numbers except 0, 2, and -5. Math 30 -1 2

Basic Rational Function Characteristic Non-permissible value x=0 Behaviour near the As x approaches 0, the |y| becomes very non-permissible large. value As |x| becomes very End behaviour large, y approaches 0 Intercepts Asymptotes Key Points Domain {x | x ≠ 0, x R} Range {y | y ≠ 0, y R} Equation of the vertical asymptote x=0 Equation of the horizontal asymptote y=0 Math 30 -1 3

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 - y y f f x a a x=a x x=a Thus, f (x) or f(x) - as x approaches a from either the left or the right. Math 30 -1 4

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. 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. y y=b y=b f f x f y=b x x f (x) b as x 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. Math 30 -1 5

Graphing Rational Functions Using Transformations vertical stretch a f(x) = +k x–h (–a indicates a reflection in the x-axis) vertical translation horizontal translation Transformations of Rational Functions Mc. Graw Hill Teacher Resource DVD N 18_9. 1_434_IA Math 30 -1 6

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

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

Graphing Rational Functions Using Transformations, 2 steps Graph: f(x) = 1 +4 x– 3 Compared to the graph of , the graph of the function has been translated 3 units right and 4 up. Vertical Asymptote: x = 3 Horizontal Asymptote: y = 4

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

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. Vertical Asymptote: x = – 3 Horizontal Asymptote: y = – 6

Graphing Rational Functions Using Transformations Characteristic Non-permissible value Behaviour near the nonpermissible value Intercepts: x-intercept at -2. 2 y- intercept at -4. 3 x = – 3 As x approaches – 3, the |y| becomes very large. End behaviour As |x| becomes very large, y approaches – 6 Domain {x | x ≠ – 3, x R} Range {y | y ≠ – 6, y R} Equation of the vertical asymptote x = – 3 Equation of the horizontal asymptote y = – 6

What does the graph of look like? Math 30 -1 13

. Write the equation of each function in the form Math 30 -1 14

Graph What are the equations for the asymptotes?

Graphing Rational Functions 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 Math 30 -1 16

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. Assignment Page 442 1, 3, 4 a, c, 6, 7, 8, 12, 13, 14, 16 Math 30 -1 17

Graphing Rational Functions an Investigative Approach Graph: The graph is undefined for x = 0. Begin at x = 1 and consider the domain x > 1. x f(x) 1 2 3 4 10 1000 ZERO As x approaches +∞, f(x) approaches_______. There is. Math a 30 -1 horizontal asymptote at y = 0. 18

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

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

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