Rational Functions

Definition: A Rational Function is a function in the form:

f(x) =

where p(x) and q(x) are polynomial functions and q(x) ≠ 0.

In this section, p and q will have degree 1 or 0.

)x(q

)x(p

For example: x5

y x3

2xy

5x3x

y

Very Important definitions:

Vertical asymptote occurs at values of x for which the function is undefined (exception: unless there is a hole . . . we’ll talk about that later).

Horizontal asymptote occurs if the function approaches a specific value when x approaches infinity or negative infinity. Think of the warmup: what happens to y when x gets REALLY big or REALLY small?

To graph rational functions, ALWAYS figure out the asymptotes FIRST. Then you can plot specific points!!!!

Consider: x1

)x(f

Vertical asymptote will occur at x = 0.

Df = (, 0), (0, )

Horizontal asymptote will occur at y = 0.

Think what happens when you divide 1 by a VERY large number!!!!!

Show your asymptotes!!

Pick some x’s on each side of the vertical asymptote to see the graph!!!

x y x Y

−3   3  

−2   2  

−1   1  

−.5   .5  

x1

)x(f

-.3333

-.5

-1

-2

.3333

.5

1

2

Rf = (, 0), (0, ) In most cases, the range will be closely related to the horizontal asymptote . . . be sure to check the graph.

Note: the graph represents a hyperbola centered at (0, 0)

Another type of rational function: khxa

)x(f

The vertical asymptote is still x = h.

Based on our observations, the hortizontal asymptote is y = k.

Df = (, h), (h, )

So, this will be a hyperbola centered at (h, k)!!

Show your asymptotes!!

Pick some x’s on each side of the vertical asymptote to see the graph!!!

23x

2)x(f

Vertical asymptote: x = –3

Horizontal asymptote: y = 2

x y x Y

−4   –2  

−5   –1  

   

   

0

1

4

3

Df = (, 3), (3, )

Rf = (, 2), (2, )

Use more points if you want . . .

Last example:dcxbax

)x(f

Vertical asymptote will correspond to the value that makes the denominator 0.

Horizontal asymptote: y = c

a

2x

1x3)x(f

Asymptotes: x = –2 y = 3

x y x Y

−3   –1  

−4   0  .5

–28

5.5

Df = (, 2), (2, )

Rf = (, 3), (3, )

x2

)x(f x = 0; y = 0 13x

2)x(f

x = 3; y = 1

4x

x)x(f

x = 4; y = 1

Remember: Find the asymptotes FIRST. Show them on the graph!!!

Pick x values to the right and to the left of the vertical asymptote(s).

Use the points along with the asymptotes to sketch the graph!!!