`f(x)=(x^2-x-6 )/(x-6)` (a) Find the slant asymptote of the graph of the rational function and (b) use the slant asymptote to graph.   

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lemjay | High School Teacher | (Level 3) Senior Educator

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`f(x) = (x^2-x-6)/(x-6)`

A rational function has a slant asymptote if the degree of the numerator is exactly one degree higher than the degree in the denominator. Since the above function satisties this condition, f(x) has a slant asymptote.

To determine the slant asymptote, divide the numerator by the denominator using long division.

         `x``+` `5`

`x - 6` `| bar(x^2-x-6)`

  `(-)` `x^2 - 6x`


               `5x - 6`

       `(-)`  `5x - 30`



So `f(x) = (x^2-x-6)/(x-6) = x+5 + 24/(x-5)` .

Note that the slant asymptote is the polynomial part of the quotient, not the remainder.

Hence, slant asymptote is `y = x+5` .

(b) To graph a rational function, check if it has a vertical, horizintal or slant asypmpote.

As indicated above, f(x) has a slant asymptote which is y=x+5.

The vertical asymptote is the value of x that result to zero denominator. To solve for VA, set x-6 equal to zero.



Hence, f(x) has a vertical asymptote  which is x=6.

And a rational function has a slant asymptote if the degree of the numerator is less than  or equal to the degree of the denominator. Otherwise, it has no horizontal asympotote.

Since the degree of the numerator og the given f(x) is greater than that of the denominator, hence f(x) has no horizontal asymptote.

So the graph of f(x) is:

(Note: Blue - graph of f(x); Green - graph of vertical asymptote; and Purple - graph of slant asympote.)


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