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Graphing Rational Functions
A function f is a rational function if
f x
h x g x
where g(x) and h(x) are polynomials. The domain consist of all real values of x, except for those values where g(x) = 0.
Recall the domain of a rational expression. For instance, letโs consider
2 1 3 4
x x
Domain: All reals, except where 3x! 4 = 0, this mean x โ 4.
The same thing would apply for rational function. For instance, if f x x x
Domain: All reals, except where x^2! 4 = 0, this means x โฆ !2 or 2 and x = !2 and 2 are called the vertical asymptotes of the function. However, it is important to answer a couple of questions when graphing rational functions.
- What is the behavior of the function as x get closer and closer to the vertical asymptote?
- What is the behavior of the function as x get larger through positive values or smaller through negative values?
- When using the calculator to graph y = , is calculator sketching in the
x asymptotes?
- What is a way calculators can correct this problem?
2nd WINDOW
Table
A. Horizontal Asymptotes of a Rational Function
Let us graph f x using the graphing calculator. We use the standard x
window (push zoom 6) and obtain a graph similar to the one below:
One notices that the curve is getting close to the x-axis (line y = 0). When x is large (whether it is positive or negative). The graphing calculator actually is not able to show the curve distinctively for values greater than 6 or smaller than 6 when we use zoom 6. However, an inspection of the input/output table [2nd^ table] shows that large values of x correspond to y-values that are certainly not zero.
Let us set the table to start at x = 100, x = 1,000 and x = 10,00. Push for TBLSET and change the value for TblStart to 100. We can also
change )TBL which is the value of increments between any two inputs.
Let us obtain a screen such that
TblStart = 100
)Tbl = 500 and
!5 E-
!3 E-
!3 E-
!2 E-
!2 E-
!1 E-
!1 E-
Clearly the outputs are getting closer and closer to zero. In this case, though, as the absolute value of x increases, the output y is increasing to get closer and closer to zero
Summary
As x get larger and stays positive, As x gets larger and stays negative f(x) is decreasing toward zero f(x) is increasing toward zero
QUESTION
Is the curve crossing the x-axis anywhere?
Is this possible? The answer is NO
ANSWER
Well, if there existed an x-intercept (that is a point of intersection between the curve
y = and the x-axis), this point would have a y-coordinate equal to 0.
x
That is 0 = Is this possible?
x
Could we divide 1 by a number and obtain zero?
No we cannot!
Therefore, the curve does not cross the x-axis.
PRACTICE
Use your graphing calculators to find the horizontal asymptotes with the following cases. (You can use the graph of the function and the table of inputs and outputs.)
f x x (The x-asymptote is y = 1) x
g x x (The horizontal-asymptote is y = 2) x
h x (The horizontal-asymptote is y = 1)
x x
2 2
r x (The horizontal-asymptote is y = 0)
x x
F x x^ x (The horizontal-asymptote is y = 0) x x
2 3
No horizontal asymptotes
The horizontal asymptote is the x-axis (line y = 0)
There is a horizontal asymptote y =
a b
B. A Quick Way to Find Horizontal Asymptotes of a Rational Function if the Asymptotes Exist
The quick way out is to compare the degree of the polynomial in the numerator and the degree of the polynomial in the denominator. These are the three possible situations:
- degree on top is larger degree on the bottom is smaller
Example: x x
x x
x x x x x
3 2
2 3 2
1 2
Verify that these rational expressions do not have horizontal asymptotes by graphing them using your calculators.
- degree on top is smaller degree at the bottom is larger
Example: x x x
x x (^2) x x x x
3 2
3 1 5 3 2
+ โ^3
Check that these expressions have y = 0 as a horizontal asymptote.
- degree on top and degree at the bottom are identical
Example: y x x
y x x x
2 2
y
x x x x
4 4 3 2
Where a is the coefficient of the term with the largest exponent on top, and b is the coefficient of the term with the largest exponent at the bottom.
x x
x x
x
3 2 2
x
x
x x
x
x
x
Example 2 TblStart = 0
)Tbl = 0.
x y
0 . . . .
In the case of y , the vertical asymptote x = 0 corresponds to setting the x
denominator to zero. Indeed, x = 0 is the one value that is not in the domain of the function.
In general, vertical asymptotes are obtained by setting the denominator to zero.
For instance, the function f x has a horizontal asymptote at
x x x x x
2 3 2
y = 0 and three vertical asymptotes corresponding to the equation.
It is important to note that on the graphing calculators the three vertical lines x = 0,
x = 2, x = !1 are shown.
D. Summary
In order to find horizontal asymptotes of a rational expression one just inspects the degrees of the polynomials in the numerator and the denominator.
- There are no horizontal asymptotes when the degree of the numerator is larger than the degree of the denominator.
- There is a horizontal asymptote, the line y = 0, when the degree of the denominator is larger than the degree of the numerator.
- There is a horizontal asymptote (^) y a when the degrees of the denominator b
and numerator are identical. a is the coefficient of the term with the highest exponent in the numerator and b is the coefficient of the term with the highest exponent in the denominator.
- The vertical asymptotes correspond to the values that make the denominator go to zero. They are of the form x = c where c is a zero of the denominator.
R 1 :!3 ; R 2 :!1 ; R 3 : 1 ; R 4 : 2 ; R 5 :
2
VII. Table
R 1
(! 4 , !2)
R 2
R 3
R 4
R 5
(3, 4 )
x! 1 โ โ โ + +
x! 3 โ โ โ โ +
x + 2 โ + + + +
f(x) โ + + โ +
f(x) > 0: (!2, 1) c (3, 4 ) f(x) < 0: (! 4 , !2) c (1, 3)
VIII. Graphing
a.
See Section VII
b.
VII. Table
R 1
(! 4 , !1)
R 2
R 3
R 4
(2, 4 )
x 2 + + + +
x! 2 โ โ โ +
x + 1 โ + + +
f(x) + โ โ +
f(x) > 0: (! 4 , !2) c (2, 4 ) f(x) < 0: (!1, 0) c (0, 2)
VIII. Graphing
a. In R 1 : To determine if f(x) crosses the horizontal asymptote - test a value.
f( โ ) =
In R 2 : Use the same procedure to determine if f(x) crosses the horizontal asymptote. (It doesnโt).
b.
Example: Sketch the graph of the function: f x
x x
Solution:
I. The x-intercept:
x 2! 9 = 0
(x! 3)(x + 3) = 0
x = 3,! 3
II. The y-intercept
Let x = 0: y = โ โ
III. No horizontal asymptote. Since the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, therefore an oblique asymptote exist. Letโs find it.
2
2
x x x
x
x x x x
As x โ ยฑ โ , y = x+
IV. Vertical Asymptote
2x! 4 = 0
x = 2
b.
Summary
I. As x โ 0 ,+ f x( ) โ โ As x approaches zero
from the left y increases without bound.
As (^) x โ โ , f x( ) โ 0 As x gets larger (approaches positive infinity) y is decreasing towards zero.
II. As x โ 0; f x( ) โ โ โ As x approaches zero
from the left, y decreases without bound.
As x โ โ โ , f x( ) โ 0 As x gets small (approaches negative infinity) y is decreasing toward zero.
