Aim: What are the rational function and asymptotes?

Do Now: Graph xy = 4

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-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

xy

4

and determine the domain

A rational function is defined as any function that is in the form of p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not 0.

To graph xy = 4, we first need to rewrite xy = 4 into

xy

4 that is a rational function

Note: q(x) can equal 1 so any polynomial function is actually a rational function; however we usually use the term only when the function is in the form of a fraction.

x can not be 0, therefore, the asymptote of xy = 4 is the line x = 0 (the y-axis)

x = 0 is the vertical asymptote,

What is the vertical asymptote of ?1

12)(

x

xxf

To find the vertical asymptote, we simplify let the denominator equal to zero then solve for x.

Therefore, the vertical asymptote of f(x) is x = –1

For the functionsx

xf4

)( and1

12)(

x

xxf

Is there any horizontal asymptotes?

The horizontal asymptote of y = 8/x is y = 0 (again through observation and experience).The horizontal asymptote of f(x) = 2x + 1 appears to be y = 2. x + 1 Is there any other way we can figure out the horizontal asymptote of any rational functions besides looking at the graphs?

There are some simple rules for finding the horizontal asymptote:

To do this let’s say the degree of p(x) = n and degree of q(x) = m.

1. If n < m , then the x-axis (y = 0) is the horizontal asymptote. 2. If n = m , then the horizontal asymptote is y = lead. coeff of p(x) / lead. coeff of q(x).

3. If n > m , then there is NO horizontal asymptote.

If the degree of numerator is exactly one degree higher than the denominator, then there is a slant asymptote

How do we find the equation of the slant asymptote?

To find the equation of a slant asymptote, use long division and the quotient is the equation of a slant asymptote

1)(

2

x

xxxf

1

22

x

x

Therefore, the slant asymptote is y = x – 2

Drill: State the domain of each rational function, then use your calculator and the rules to find the vertical and

horizontal asymptotes; if either does not exist say so 1) f(x) = 4

x2+1

2) f(x) = 3x3 + 7x2 + 2 -4x3 + 5

3) f(x) = x + 10 |x| + 2

4) f(x) = 2 + x 2 - x