RATIONAL

FUNCTIONSA rational function is a function of the form:

xqxp

xR where p and q are polynomials

xqxp

xR What would the domain of a rational function be?

We’d need to make sure the denominator 0

x

xxR

3

5 2

Find the domain. 3: xx

22

3

xx

xxH 2,2: xxx

45

12

xx

xxF

If you can’t see it in your head, set the denominator = 0 and factor to find “illegal” values.

014 xx 1,4: xxx

The graph of looks like this: 2

1

xxf

Since x 0, the graph approaches 0 but never crosses or touches 0. A vertical line drawn at x = 0 is called a vertical asymptote. It is a sketching aid to figure out the graph of a rational function. There will be a vertical asymptote at x values that make the denominator = 0

If you choose x values close to 0, the graph gets close to the asymptote, but never touches it.

Let’s consider the graph x

xf1

We recognize this function as the reciprocal function from our “library” of functions.

Can you see the vertical asymptote?

Let’s see why the graph looks like it does near 0 by putting in some numbers close to 0.

10

1011

10

1

f

100

10011

100

1

f

10

101

1

10

1

f 100

1001

1

100

1

f

The closer to 0 you get for x (from positive

direction), the larger the function value will be Try some negatives

Does the function have an x intercept? x

xf1

There is NOT a value that you can plug in for x that would make the function = 0. The graph approaches but never crosses the horizontal line y = 0. This is called a horizontal asymptote.

A graph will NEVER cross a vertical asymptote because the x value is “illegal” (would make the denominator 0)

x

10

A graph may cross a horizontal asymptote near the middle of the graph but will approach it when you move to the far right or left

Graph x

xQ1

3

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

31

x

vertical translation,

moved up 3

x

xf1

x

xQ1

3

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

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

Finding AsymptotesVER

TIC

AL A

SYM

PTO

TE

S

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

43

522

2

xx

xxxR

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.

Hole (in the graph)

• If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote.

• The exact point of the hole can be found by plugging b into the function after it has been simplified.

• Huh???? Let’s look at an example or two.

Find the domain and identify vertical asymptotes & holes.

2

1( )

2 3

xf x

x x

Find the domain and identify vertical asymptotes & holes.

2( )

4

xf x

x

Find the domain and identify vertical asymptotes & holes.

2

5( )

2 3

xf x

x x

Find the domain and identify vertical asymptotes & holes.

2

2

3 2( )

2

x xf x

x x

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

xx

xxR

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

43

5422

2

xx

xxxR

1

2y

43

5322

23

xx

xxxxR

If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique (a/k/a slant) 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

OBLIQUE (SLANT) ASYMPTOTES

degree of top = 3

532 23 xxx432 xx

remainder a 5x

Oblique (slant) asymptote at y = x + 5

SUMMARY OF HOW TO FIND ASYMPTOTES

Vertical Asymptotes are the values that are NOT in the domain. To find them, set the denominator = 0 and solve.

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 (slant) asymptote found by long division.