Concept Byte: Oblique Asymptotes

For Use With Lesson 8-3

TECHNOLOGY

In Lesson 8-2, you saw that the graphs of some rational functions have horizontal and vertical asymptotes. The graphs of some rational functions can have oblique asymptotes. Oblique asymptotes are asymptotes that are neither horizontal nor vertical. These asymptotes only occur in rational functions in which the degree of the numerator is one greater than the degree of the denominator.

Example 1

Compare the graphs of y = 6 x 2 + 1 3 x y equals . fraction 6 , x squared , plus 1 , over 3 x end fraction  and y = 2 x y equals 2 x  using a graphing calculator.

The graph of y = 6 x 2 + 1 3 x y equals . fraction 6 , x squared , plus 1 , over 3 x end fraction  gets closer to y = 2 x y equals 2 x  as | x | gets increasingly large.

The graph of y = 2 x y equals 2 x  is an oblique asymptote of y = 6 x 2 + 1 3 x . y equals . fraction 6 , x squared , plus 1 , over 3 x end fraction . .

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Example 2

Use a spreadsheet to find the differences between f ( x ) = 6 x 2 + 1 3 x f , open x close , equals . fraction 6 , x squared , plus 1 , over 3 x end fraction  and g ( x ) = 2 x g open x close equals , 2 to the x  for the values from 1 to 10.

• Step 1 Label each column in Row 1.
• Step 2 Enter the x-values in column A.

• Step 3 Enter the formulas for f(x), g(x), and f ( x ) − g ( x ) f open x close minus g open x close  into cells B2, C2, and D2.

  B 2 = ( 6 × ( A 2 ) 2 + 1 ) ) / ( 3 × A 2 ) cap b 2 equals open 6 times . open cap a 2 close squared . plus 1 close close slash open 3 times cap a 2 close

  C 2 = 2 × A 2 cap c 2 equals 2 times cap a 2

  D 2 = B 2 − C 2 cap d 2 equals cap b 2 minus cap c 2

  	A	B	C	D
  1	x	f(x) = (6x^2 + 1)/(3x)	g(x) = 2x	f ( x ) − g ( x ) f open x close minus g open x close
  2	1	2.333	2	0.333
  3	2	4.167	4	0.167
  4	3	6.111	6	0.111

• Step 4 In columns B, C, and D fill the formulas down to find the values of f(x), g(x), and f ( x ) − g ( x ) f open x close minus g open x close  for each corresponding x-value.

As the values of x get larger, the value 6 x 2 6 x squared  becomes much larger than the constant term in the numerator. As a result, the constant term has a smaller effect on the value of the function. So, as x increases, the value of 6 x 2 + 1 3 x fraction 6 , x squared , plus 1 , over 3 x end fraction  gets closer to the value of 6 x 2 3 x = 2 x , fraction 6 , x squared , over 3 x end fraction . equals 2 x comma  for x ≠ 0 . x not equal to 0 .  The spreadsheet confirms this conclusion by showing that the difference between these two values gets closer to zero as x gets larger.

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