3.4 Rational Functions and Their

Graphs

SUMMARY OF HOW TO FIND ASYMPTOTESVertical Asymptotes are the values that are NOT in the domain. To find them, set the denominator = 0 and solve.

“WHAT VALUES CAN I NOT PUT IN THE DENOMINATOR????”

To determine horizontal or oblique asymptotes, compare the degrees of the numerator and denominator.

1. If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0)

2. If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom

3. If the degree of the top > the bottom, oblique asymptote found by long division.

Finding AsymptotesVERTICAL ASYMPTOTES

There will be a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0

4352

2

2

xxxxxR

Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be.

014 xx

So there are vertical asymptotes at x = 4 and x = -1.

If the degree of the numerator is less than the degree of the denominator, (remember degree is the highest power on any x term) the x axis is a horizontal asymptote.

If the degree of the numerator is less than the degree of the denominator, the x axis is a horizontal asymptote. This is along the line y = 0.

We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes.

43

522

xxxxR

degree of bottom = 2

HORIZONTAL ASYMPTOTES

degree of top = 1

1

1 < 2

If the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at:

y = leading coefficient of top

leading coefficient of bottom

degree of bottom = 2

HORIZONTAL ASYMPTOTES

degree of top = 2

The leading coefficient is the number in front of the highest powered x term.

horizontal asymptote at:

1

2

43542

2

2

xxxxxR

12

y

43

5322

23

xxxxxxR

If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder.

degree of bottom = 2

SLANT ASYMPTOTES

degree of top = 3

532 23 xxx432 xx

remainder a 5x

Oblique asymptote at y = x + 5

The graph of looks like this:

2

1x

xf

Graph x

xQ 13

This is just the reciprocal function transformed. We can trade the terms places to make it easier to see this.

31x

vertical translation,

moved up 3

x

xf 1

x

xQ 13

The vertical asymptote remains the same because in either function, x ≠ 0

The horizontal asymptote will move up 3 like the graph does.

Strategy for Graphing a Rational Function

1. Graph your asymptotes2. Plot points to the left and right of each

asymptote to see the curve

Sketch the graph of

10532)(

xxxf

• The vertical asymptote is x = -2

• The horizontal asymptote is y = 2/5

10532)(

xxxf

10532)(

xxxf

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Sketch the graph of:

€

g(x) =1x −1

Vertical asymptotes at??

x = 1

Horizontal asymptote at??

y = 0

Sketch the graph of:

€

f (x) =2x

Vertical asymptotes at??

x = 0

Horizontal asymptote at??

y = 0

Sketch the graph of:

€

h(x) =−4x

Vertical asymptotes at??

x = 0

Horizontal asymptote at??

y = 0

Sketch the graph of:

€

y =1x + 3

− 2

Vertical asymptotes at?? x = 1

Horizontal asymptote at?? y = 0

Hopefully you remember,y = 1/x graph and it’s asymptotes:

Vertical asymptote: x = 0Horizontal asymptote: y = 0

Or…We have the function:

€

y =1x + 3

− 2

But what if we simplified this and combined like terms:

€

y =1x + 3

−2(x + 3)x + 3

y =1 − 2x − 6x + 3

y =−2x − 5x + 3

Now looking at this:Vertical Asymptotes??

x = -3

Horizontal asymptotes??

y = -2

Sketch the graph of:

€

h(x) =x 2 + 3xx

Hole at??

x = 0€

h(x) =x(x + 3)x

Find the asymptotes of each function:

€

y =x 2 + 3x − 4

x

€

y =x 2 + 3x − 28x 3 −11x 2 + 28x

€

y =x 2

x+

3xx

−4x

€

y = x + 3 −4x

Slant Asymptote:

y = x + 3

Vertical Asymptote:

x = 0

€

y =(x + 7)(x − 4)x(x − 7)(x − 4)

Hole at x = 4

Vertical Asymptote:

x = 0 and x = 7

Horizontal Asymptote:

y = 0