Heath Algebra 1 McDougal Littell JoAnn Evans

Math 8H

Heath 12.5Rational Functions

A function in the form of

is called a rational function.

polynomialf x

polynomial

In today’s lesson you’ll study rational functions whose numerators and denominators are first-degree

polynomials.

Using long division, you will transform functions from

af x k

x h

polynomial

f xpolynomial

to

The graphs of the functions today will all be

.

2 4 6 -2 -4 -6

• ••••••

•••••••

You’ve worked with hyperbolas before in Lesson

11-1 in the Glencoe book.

The graph shown here is an inverse

variation graph from that lesson.

af x k

x h

h, k

The vertical and

horizontal lines

through the center

of the hyperbola

are called the

asymptotes.

The asymptotes intersect at the point (h,

k).

x

f x

This is the graph of

1f x 0

x 0

af x k

x h

The asymptotes intersect at the point (0, 0). In this case the asymptotes are the x- and y-axes.

When h and k have values other than 0, the hyperbola

will shift vertically and horizontally.

The k shifts the hyperbola vertically.

Positive = upNegative = down

x

f x

x

f x

1f x 2

x

Hyperbola in red: When the value of k changes, watch for the vertical shift that takes place.

1f x

x

1f x 3

x

af x k

x h

When h has a value other than 0, the hyperbola will

shift horizontally.

The value of h will cause the graph

to shift horizontally.

The graph will shift horizontally the number that x must equal so that the

expression in the denominator is equal to ZERO.

x

f x

x

f x

1f x

x 3

Hyperbola in red: When the value of h changes, watch

for the horizontal shift that takes place.

1f x

x

1f x

x 4

x

f x

x

f x

1f x 3

x 2

1

f x 2x 1

k will cause the graph to shift 3 spaces UP

h will cause the graph to shift 2 spaces to the RIGHT

k will cause the graph to shift 2 spaces DOWN

h will cause the graph to shift 1 spaces to the LEFT

Find the center of a hyperbola and the asymptotes.

The center is the coordinate point

The x-value will be the value x must be so that the expression x – h is equal to zero.

The y-value is k.

h, k

1 3 2f x 4 f x 2 f x 9

x 2 x 6 x 5

2, 4

is the center

6, 2

is the center

5, 9

is the center

Asymptotes are the vertical and horizontal lines that pass through the center of the hyperbola.

The equation of the vertical line (asymptote) passing through the center will be the x-value of the center.

The equation of the horizontal line (asymptote) passing through the center will be the y-value of the center.

x

f xThe center of this

hyperbola is the point (3, 1).

What is the equation of the vertical asymptote?

x 3

What is the equation of the horizontal asymptote?

y 1

1 2 3

3

1

2

Find the center and asymptotes of the hyperbola.

4 1 3f x 3 f x 16 f x 2

x 5 x 3 x 1

5, 3

x 5

y 3

3,16

x 3

y 16

1, 2

x 1

y 2

Graph a Rational Function: 1f x 1

x 2

Step 1: Find the center of the hyperbola and lightly draw the asymptotes as dashed lines. center : 2, 1

asymptotes : x 2, y 1

x

f x

Step 2: Make a table of values. Choose values on both sides of the center.

x -5 -4 -3 -2 -1 0 1

f(x) 0

... ..

2 12

23

12

13

.

1

Step 3: Graph both sides.

1 1

1 2 3

3

1

2

Graph a Rational Function: 2f x 2

x 3

Step 1: Find the center of the hyperbola and lightly draw the asymptotes as dashed lines. center : 3, 2

asymptotes : x 3, y 2

x

f x

Step 2: Make a table of values. Choose values on both sides of the center.

x 0 1 2 3 4 5 6

f(x) 0

.....

-4 1 132

.

-2

Step 3: Graph both sides.

13

23

1 2 3

3

1

2

Using Polynomial Long Division First…

2x 1f x

x 2

This function is not in the correct form yet. Divide the numerator by the denominator.

x 2 2x 1 2

2x 4

3

3f x 2

x 2

Put in parentheses and subtract.

Remember to put the remainder over the divisor and add it to the quotient.

3f x 2

x 2

Use the commutative property of addition to put the k second, then graph.

Graph a Rational Function: 3f x 2

x 2

Step 1: Find the center of the hyperbola and lightly draw the asymptotes as dashed lines. center : 2, 2

asymptotes : x 2, y 2

x

f x

Step 2: Make a table of values. This time, try choosing values on just one side of the center.

...

...

Step 3: Make symmetrical points on the other side of the center, then graph the other part of the hyperbola.

x -2 -1 0 1

f(x) 2 1 12

1

In relation to the center, this point is 1

right, 3 down. Reverse those directions to

locate the symmetrical point. Go 1 LEFT and 3

UP to get to the symmetrical point.

1 2 3

3

1

2

Using Polynomial Long Division First…

4x 11f x

x 3

This function is not in the correct form yet. Divide the numerator by the denominator.

x 3 4x 11 4

4x 12

1

1f x 4

x 3

Put in parentheses and subtract.

Remember to put the remainder over the divisor and add it to the quotient.

1f x 4

x 3

Use the commutative property of addition to put the k second, then graph.

Graph a Rational Function: 1f x 4

x 3

Step 1: Find the center of the hyperbola and lightly draw the asymptotes as dashed lines. center : 3, 4

asymptotes : x 3, y 4

x

f x

Step 2: Make a table of values. This time, try choosing values on just one side of the center.

......

Step 3: Make symmetrical points on the other side of the center, then graph the other part of the hyperbola.

x -3 -2 -1 0

f(x) 4 3 3 23

12

3

1 2 3

3

1

2