Rational Functions and Their Characteristics (DAY 1):
Definition of a Rational Function
A rational functionis a quotient of polynomials that has the form.
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. )
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:
(a) (b)
(c)(d)
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 . 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.
a) b)
Horizontal Asymptotes (H.A.):
Horizontal asymptotes are imaginary horizontal lines that the graph will approach as x approaches very large positive values and/or very large negative values. The following are general rules for finding a horizontal asymptote:
Given that the numerator and denominator in the rational function are polynomials in of degree and , respectively.
- If , then the horizontal asymptote is.
2. If , then the horizontal asymptote is.
3. If , there is no horizontal asymptote.
NOTE: Since a horizontal asymptote is only a barrier as x gets to be very large positive values and/or very large negative values , 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.
a) b)
c)
Example 4: Given. Find the domain, intercepts, and vertical and horizontal asymptotes. Then use this information to sketch what you now know about your function.
ASSIGNED EXERCISES:
For question 1 – 6, refer to the following functions. Answer questions 1 to 5 without graphing technology.
(a)(b)
(c)(d)
- Determine the x- and y-intercepts of each function.
- State the domain for each function.
- Will the graph of each function have a vertical asymptote?
If so, determine the equation for each vertical asymptote.
- Will the graph of each function have a horizontal asymptote?
Give reasons for you answers. Determine the equation of each asymptote.
- Use the information from questions 1 to 5 to sketch what you know about the graph of each function.
- Confirm your answers to questions 1 to 5 by graphing each function using graphing technology.
- 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 AP(x) =. Express this function in two different forms. Explain what can be determined from each form. Restrict the domain of the function to represent the context. What are the break-even quantities?
- Repeat question 7 for R(x) = -x2 + 30x and C(x) = 17x + 36.
- The model for the concentration y of a drug in the bloodstream, x hours after it is taken orally, is y =. What is the domain of y in this context? What do you know about 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?
- 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.
- What is a rational function? How is the graph of a rational function different from the graph of a polynomial function?
- Use specific examples of your own to describe the condition forthe graph of a rational function to have vertical and horizontal asymptotes.
- 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.
- Explain the difference between the graph of and that of .
- Find constants and that guarantee that the graph of the functiondefined by will have a vertical asymptote at * and a horizontal asymptote at .
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) x7(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
- Graphs
6.Graphing Calculator
7.AP(x) = or , D: x>0. Break even: x = ½,2 (zeroes)
- AP(x) =. D: x>0, Break even: x= 4 or 9 (zeroes)
- D: x0. Function increases to a maximum of (1.4, 2.5) Model is reasonable.
- 6.5m X 3.2 m
13.(a)(b)
14. The function is not defined at the value b which is represented by a hollow dot. The function is a linear graph.
15. ,