rational functions objective: finding the domain of a rational function and finding asymptotes

Rational Functions

Objective: Finding the domain of a rational function and finding

asymptotes

Domain of a Rational Function

• A rational expression is a function that is a fraction. The domain will be all real numbers except what makes the denominator zero.

Domain of a Rational Function

• A rational expression is a function that is a fraction. The domain will be all real numbers except what makes the denominator zero.

• The zeros of a rational expression are the values that make the numerator zero but not the denominator.

Ex. 1

• Find the domain of . x

xf 1)(

Ex. 1

• Find the domain of .

• We know that zero will make the denominator zero, so the domain is .

• There are no zeros of this function since the numerator can never be zero.

xxf 1)(

0x

Asymptotes

• We are now going to call all points that make the denominator zero (but not the numerator) vertical asymptotes. A vertical asymptote is a line that the function approaches, but never touches. The graph will go towards as it approaches an asymptote.

Asymptotes

• We will also have horizontal asymptotes. This is a horizontal line that the graph will approach and may or may not touch.

Asymptotes

• This is how we find horizontal asymptotes:1. If the denominator is a higher power of x than the

numerator, y = 0 (the x-axis) is the horizontal asymptote.

Asymptotes

• This is how we find horizontal asymptotes:1. If the denominator is a higher power of x than the

numerator, y = 0 (the x-axis) is the horizontal asymptote.

2. If the numerator is a higher power of x than the denominator, there are no horizontal asymptotes.

Asymptotes

• This is how we find horizontal asymptotes:1. If the denominator is a higher power of x than the

numerator, y = 0 (the x-axis) is the horizontal asymptote.

2. If the numerator is a higher power of x than the denominator, there are no horizontal asymptotes.

3. If the numerator and denominator are the same power of x, the horizontal asymptote is the quotient of the leading coefficients.

Asymptotes

• Lets look at .

• Since zero makes the denominator zero, the vertical asymptote is x = 0.

• Since the denominator is a higher power of x than the numerator, y = 0 (the x-axis) is the horizontal asymptote.

xxf 1)(

Asymptotes

• Lets look at . 132)( 2

xxxf

Asymptotes

• Lets look at .

• Since there are no zeros of the denominator, there are no vertical asymptotes.

• Since the denominator is a higher power of x, y = 0 is the horizontal asymptote.

• The zero of this function is 0, since this makes the numerator zero.

132)( 2

xxxf

Asymptotes

• Lets look at .1

2)( 2

2

xxxf

Asymptotes

• Lets look at .

• Since the zeros of the denominator are 1 and -1, the vertical asymptotes are x = 1 and x = -1.

• Since the denominator and numerator are the same power of x, we just look at the coefficients.

• The horizontal asymptote is y = 2.

12)( 2

2

xxxf

212

12)( 2

2

xxxf

Asymptotes

• Lets look at .

• Since the zeros of the denominator are 1 and -1, the vertical asymptotes are x = 1 and x = -1.

• Since the denominator and numerator are the same power of x, we just look at the coefficients.

• The horizontal asymptote is y = 2.

12)( 2

2

xxxf

Asymptotes

• You try:• Lets look at

• Find all horizontal and vertical asymptotes.• Find all zeros.

.64)( 2

2

xx

xxf

Asymptotes

• You try:• Lets look at

• Find all horizontal and vertical asymptotes.• The denominator factors to (x-3)(x+2), so the zeros are

3 and -2. x = 3 and x = -2 are the vertical asymptotes.• The numerator and denominator have the same power

of x, so y = 1 is the horizontal asymptote.

.64)( 2

2

xx

xxf

Asymptotes

• You try:• Lets look at

• Find all zeros.• There are no zeros of this function.

.64)( 2

2

xx

xxf

Asymptotes

• You try:• Lets look at

• Find all asymptotes.• Find all zeros.

.94)( 2

xxxf

Asymptotes

• You try:• Lets look at

• Find all asymptotes.• Since the denominator factors to (x + 3)(x – 3), the

vertical asymptotes are • Since the denominator is a higher power of x, the

horizontal asymptote is y = 0.

.94)( 2

xxxf

.3x

Asymptotes

• You try:• Lets look at

• Find all zeros.• The only zero of the numerator is x = -4.

.94)( 2

xxxf

Holes

• If a number is a zero of both the numerator and denominator, it is a hole in the graph, not an asymptote.

• Lets look at the graph of

• The value x = 2 is a hole and x = -2 is the zero. There are no vertical or horizontal asymptotes.

2)2()2)(2(

24)(

2

xxxx

xxxf

24)(

2

xxxf

Holes

• If a number is a zero of both the numerator and denominator, it is a hole in the graph, not an asymptote.

• Lets look at the graph of

2)2()2)(2(

24)(

2

xxxx

xxxf

24)(

2

xxxf

Holes

• You try:• Lets look at

• Find all vertical and horizontal asymptotes.

62

2

2

xxxxy

Holes• You try:• Lets look at

• Find all vertical and horizontal asymptotes.• The numerator factors to (x+2)(x-1), so the zeros are x = -2

and x = 1.• The denominator factors to (x-3)(x+2), so the zeros are x =

-2 and x = 3.• So, the zero is x = 1 and the vert. asymptote is x = 3. We

have a hole at x = -2.

)3)(2()1)(2(

62

2

2

xxxx

xxxxy

Holes

• You try:• Lets look at

• Find all vertical and horizontal asymptotes.• So, the zero is x = 1 and the vert. asymptote is x = 3.

We have a hole at x = -2.• Since the numerator and denominator are the same

power, the horizontal asymptote is y = 1.

62

2

2

xxxxy

Class work

• Page 341• 6, 8, 10, 12• Find:• Vertical asymptotes (zero of denominator)• Horizontal asymptotes (powers of x)• Zeros (zero of numerator)• Holes (zeros of both)

Homework

• Pages 341-342• 1-27 odd