Rational Functions and Their Characteristics (DAY 1)

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 

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