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3.7 Rational functions  (Page 7/16)

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Find the vertical and horizontal asymptotes of the function:

f ( x ) = ( 2 x − 1 ) ( 2 x + 1 ) ( x − 2 ) ( x + 3 )

Vertical asymptotes at   x = 2   and   x = – 3 ;   horizontal asymptote at   y = 4.

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Intercepts of rational functions

A rational function    will have a y -intercept when the input is zero, if the function is defined at zero. A rational function will not have a y -intercept if the function is not defined at zero.

Likewise, a rational function will have x -intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, x -intercepts can only occur when the numerator of the rational function is equal to zero.

Finding the intercepts of a rational function

Find the intercepts of   f ( x ) = ( x − 2 ) ( x + 3 ) ( x − 1 ) ( x + 2 ) ( x − 5 ) .

We can find the y -intercept by evaluating the function at zero

f ( 0 ) = ( 0 − 2 ) ( 0 + 3 ) ( 0 − 1 ) ( 0 + 2 ) ( 0 − 5 )          = − 6 10          = − 3 5           = − 0.6

The x -intercepts will occur when the function is equal to zero:

0 = ( x − 2 ) ( x + 3 ) ( x − 1 ) ( x + 2 ) ( x − 5 ) This is zero when the numerator is zero . 0 = ( x − 2 ) ( x + 3 ) x = 2 ,   − 3

The y -intercept is   ( 0 , –0.6 ) ,   the x -intercepts are   ( 2 , 0 )   and   ( –3 , 0 ) .   See [link] .

Graph of f(x)=(x-2)(x+3)/(x-1)(x+2)(x-5) with its vertical asymptotes at x=-2, x=1, and x=5, its horizontal asymptote at y=0, and its intercepts at (-3, 0), (0, -0.6), and (2, 0).
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Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the x - and y -intercepts and the horizontal and vertical asymptotes.

For the transformed reciprocal squared function, we find the rational form.   f ( x ) = 1 ( x − 3 ) 2 − 4 = 1 − 4 ( x − 3 ) 2 ( x − 3 ) 2 = 1 − 4 ( x 2 − 6 x + 9 ) ( x − 3 ) ( x − 3 ) = − 4 x 2 + 24 x − 35 x 2 − 6 x + 9

Because the numerator is the same degree as the denominator we know that as   x → ± ∞ ,   f ( x ) → − 4 ;   so   y = – 4   is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is   x = 3 ,   because as   x → 3 , f ( x ) → ∞ .   We then set the numerator equal to 0 and find the x -intercepts are at   ( 2.5 , 0 )   and   ( 3.5 , 0 ) .   Finally, we evaluate the function at 0 and find the y -intercept to be at   ( 0 , − 35 9 ) .

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Graphing rational functions

In [link] , we see that the numerator of a rational function reveals the x -intercepts of the graph, whereas the denominator reveals the vertical asymptotes of the graph. As with polynomials, factors of the numerator may have integer powers greater than one. Fortunately, the effect on the shape of the graph at those intercepts is the same as we saw with polynomials.

The vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side of the vertical asymptote the graph heads towards positive infinity, and on the other side the graph heads towards negative infinity. See [link] .

Graph of y=1/x with its vertical asymptote at x=0.

When the degree of the factor in the denominator is even, the distinguishing characteristic is that the graph either heads toward positive infinity on both sides of the vertical asymptote or heads toward negative infinity on both sides. See [link] .

Graph of y=1/x^2 with its vertical asymptote at x=0.

For example, the graph of   f ( x ) = ( x + 1 ) 2 ( x − 3 ) ( x + 3 ) 2 ( x − 2 )   is shown in [link] .

Graph of f(x)=(x+1)^2(x-3)/(x+3)^2(x-2) with its vertical asymptotes at x=-3 and x=2, its horizontal asymptote at y=1, and its intercepts at (-1, 0), (0, 1/6), and (3, 0).
  • At the x -intercept   x = − 1   corresponding to the   ( x + 1 ) 2   factor of the numerator, the graph bounces, consistent with the quadratic nature of the factor.
  • At the x -intercept   x = 3   corresponding to the   ( x − 3 )   factor of the numerator, the graph passes through the axis as we would expect from a linear factor.
  • At the vertical asymptote   x = − 3   corresponding to the   ( x + 3 ) 2   factor of the denominator, the graph heads towards positive infinity on both sides of the asymptote, consistent with the behavior of the function   f ( x ) = 1 x 2 .
  • At the vertical asymptote   x = 2 ,   corresponding to the   ( x − 2 )   factor of the denominator, the graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side, consistent with the behavior of the function   f ( x ) = 1 x .

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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