Rational Functions and Their Characteristics (DAY 1):
Definition of a Rational Function f( x ) A rational function is a quotient of polynomials that has the form h( x) = . g( x )
The domain of a rational function consists of all real numbers except the zeroes of the polynomial in the denominator. g(x) 0. (i.e. D:{ x喂 g( x) 0})
Over the next three days, you will examine the basic characteristics of a rational function. Understanding these functions will enable you to draw an accurate sketch of a rational function. These characteristics are: Domain Intercepts (both x and y) Holes in the function Asymptotes (Vertical, Horizontal, & Linear Oblique) Behaviour of the function close to the asymptote(s)
Let’s start with familiar concepts, domain and intercepts.
Example 1: Determine the domain and intercepts of each of the following:
7 x (a) f (x) (b) f( x) = x 2 x2 -3 x - 4
2x2 + x - 3 x 2 (c) f( x) = (d) f (x) x2 - 4 x 2 2x
The graph of a rational function usually has at least one asymptote, which may be vertical, horizontal, or oblique. An oblique asymptote is neither vertical nor horizontal. Vertical Asymptotes (V.A.): Vertical asymptotes are imaginary vertical lines that form boundaries in the graph. Vertical asymptotes can be found where the function is undefined. However, not all restrictions produce a vertical asymptote. To make sure you have a vertical asymptote, simplify the rational function fully. The restrictions that are left in the reduced function will produce the vertical asymptotes. The equation of a vertical asymptote will then be x= restriction . A graph never crosses a vertical asymptote because that x – value can never occur in the domain of the function.
Holes: Sometimes a rational function has a hole in the graph. This is a point in the function where the graph gets really close to but never equals it. This will occur at the x – value of a restriction that is cancelled out during simplification. To get the y – value of the hole, substitute the x – value into the simplified rational function. Note: The x – value you are substituting in is a restriction and does not exist in the domain of the function. This is why the co-ordinates that you are finding are for a hole in the function.
Example 2: Determine the vertical asymptotes and or holes of the following rational functions AND sketch on a Cartesian plane.
2x2 + x - 3 x 2 a) f( x) = b) f (x) x2 - 4 x 2 2x Horizontal Asymptotes (H.A.): Horizontal asymptotes are imaginary horizontal lines that the graph will approach as x approaches very large positive values (x ) and/or very large negative values (x � ) . The following are general rules for finding a horizontal asymptote:
Given that the numerator and denominator in the rational function are polynomials in x of degree n and m , respectively.
1. If n m, then the horizontal asymptote is y = 0 . c o e ffic ie n t o f x n te rm 2. If n m, then the horizontal asymptote is y . c o e ffic ie n t o f x m te r m 3. If n m, there is no horizontal asymptote. NOTE: Since a horizontal asymptote is only a barrier as x gets to be very large positive values (x ) and/or very large negative values (x � ) , the graph may cross the horizontal asymptote in the central area of the graph. You will examine the behaviour of the curve in the next lesson.
Example 3: Determine the equation of the horizontal asymptote, if it exists. x x3 - 7 x a) f( x) = b) h( x) = x2 -3 x - 4 x - 2
( x+3)( 2 x + 1)2 c) g( x) = 4- 3x2 - 5 x 3 x2 Example 4: Given f( x) = . Find the domain, intercepts, and vertical and x3-2 x 2 - 5 x + 6 horizontal asymptotes. Then use this information to sketch what you now know about your function. ASSIGNED EXERCISES: Do # 1 – 5, 7, 8, 11 –
Knowledge
For question 1 – 6, refer to the following functions. Answer questions 1 to 5 without graphing technology. 2 x x 2 (a) f (x) (b) g(x) x 7 x 2 5x 6
x 2 x 6 x 2 9 (c) h(x) (d) k(x) x 2 x 3 4x 2 x 4
1. Determine the x- and y-intercepts of each function.
2. State the domain for each function.
3. Will the graph of each function have a vertical asymptote? If so, determine the equation for each vertical asymptote.
4. Will the graph of each function have a horizontal asymptote? Give reasons for you answers. Determine the equation of each asymptote.
5. Use the information from questions 1 to 5 to sketch what you know about the graph of each function.
6. Confirm your answers to questions 1 to 5 by graphing each function using graphing technology.
Application
7. Functions R(x) = -2x2 + 8x and C(x) = 3x + 2 are the estimated revenue and cost functions for the manufacture of a new product. Determine the average profit function P(x) AP(x) = . Express this function in two different forms. Explain what can be x determined from each form. Restrict the domain of the function to represent the context. What are the break-even quantities?
8. Repeat question 7 for R(x) = -x2 + 30x and C(x) = 17x + 36.
9. The model for the concentration y of a drug in the bloodstream, x hours after it is taken 7x orally, is y = . What is the domain of y in this context? What do you know about x 2 2 the graph of y just by looking at the equation? Graph the function. Describe what happens to the concentration of the drug over 24 consecutive hours. Does the model seem reasonable?
10. A rectangular garden, 21 m2 in area, will be fenced to keep out rabbits and skunks. Find the dimensions that will require the least amount of fencing if a barn already protects one side of the garden.
Communication/Thinking & Inquiry 11. What is a rational function? How is the graph of a rational function different from the graph of a polynomial function?
12. Use specific examples of your own to describe the condition for the graph of a rational function to have vertical and horizontal asymptotes.
13. For each case, create a function that has a graph with the given features.
(a) a vertical asymptote x = 1 and a horizontal asymptote y = 0
(b) two vertical asymptotes x = -1 and x = 3, horizontal asymptote y = -1, and x-intercepts – 2 and 4. ( x a ) ( x b ) 14. Explain the difference between the graph of f ( x ) and that of g(x) x a. ( x b )
15. Find constants a and b that guarantee that the graph of the function defined by a x 2 7 3 h ( x ) will have a vertical asymptote at * x = and a horizontal asymptote at 9 b x 2 5 y 2.
Answers: 1. (a) x-int (-2,0), y-int (0,-2/7) (b) x-int (2,0), y-int: (0,-1/3) (c) x-int: (2,0), (-3,0), y-int: (0,-3) (d) x-int(3,0),(-3,0), y-int: (0,9/4)
2. (a) x 7 (b) x -3,-2 (c) x -2 (d) x -4,-1,1
3. (a) x = 7 (b) x = 4 (c) x = -2 (d) x = -4,-1,1
4. (a) y = 1 (b) y=0 (d) y = 0
5. Graphs
6. Graphing Calculator (2x 1)(x 2) 2 7. AP(x) = or 2x 5 , D: x>0. Break even: x = ½,2 (zeroes) x x (x 9)(x 4) 36 8. AP(x) = or x 13 . D: x>0, Break even: x= 4 or 9 (zeroes) x x
9. D: x 0. Function increases to a maximum of (1.4, 2.5) Model is reasonable.
10. 6.5m X 3.2 m
1 x 2 2x 8 13. (a) y (b) y x 1 x 2 2x 3
14. The function f ( x ) is not defined at the value b which is represented by a hollow dot. The function g ( x ) is a linear graph.
15. a 50, b 25