Chapter 9: Rational Functions

CHAPTER 9:RATIONAL FUNCTIONS

9.1 INVERSE VARIATION

Direct Variation

A function of the form y = kx as x increases y increases.

(k is the constant of variation)

Example: If x and y vary directly, and x = 6 when y = 3, write an equation.

Inverse Variation

A function has the form

(k is a constant)

As one value of x and y increase, the other decreases.

y=kx

Modeling Inverse Variation

Suppose that x and y vary inversely, and x = 3 when y = -5. Write the function that models the inverse variation.

You Try! Modeling Inverse VariationSuppose that x and y vary inversely, and x = -2 when y = -3. Write the function that models the inverse variation.

Rational Functions

Identifying from a table

x 3 6 9

y 1 1/2 1/3

• Does this look like direct or inverse variation?

• Find the value of k:

• Write the equation:

You Try! Is the relationship between the variables in each table a direct variation, an inverse variation, or neither? Write then functions.

1)

2)

3)

A Combined Variation has more than one relationship.

EX: is read as y varies directly with

x (on top) and inversely with z (on the bottom).

Write the function that models each relationship.

1. Z varies jointly with x and y. (Hint jointly means directly)

2. Z varies directly with x and inversely with the cube of y

3. Z varies directly with x squared and inversely with y

Write a function

Z varies inversely with x and y.Write a function when x = 2 and y = 4 and z = 2

9.2 Graphing Inverse Variation9.3 Rational Functions

Investigation

Graph the following:1. Y = 3/X

2. Y = -8/X

What do you notice?

Rational Functions

Rational functions in the form y = k/x is split into two parts. Each part is called a Branch.

If k is POSITIVE the branches are in Quadrants I and III

If k is NEGATIVE the branches are in Quadrant II and IV

An Asymptote is a line that the graph approaches but NEVER touches.

Asymptotes

From the form

• The Vertical Asymptote is x = b

• The Horizontal Asymptote is y = c

Identify the Asymptote from the following functions.

1.

2.

You Try! Identify the Asymptote from the following functions.

1.

2.

Translate:

1) Up 3 units and Left 2 Units

2) Down 5 units and Right 1 unit

3) Right 4 units

4) Such that it has a Vertical asymptote of x=3 and a horizontal asymptote of y= -2

y=3x

Rational Functions

A rational function can also be written in the form

where p and q are polynomials.

Asymptotes

Vertical Asymptotes are always found in the BOTTOM of a rational function.

Set the bottom equal to zero and solve.

Find the Vertical Asymptotes for the following.

1. 2.

3. 4.

Asymptotes

What are the Asymptotes?

Graph it, what do you notice?!

There is a HOLE in the graph is when (x – a) is a factor in both the numerator and the denominator.

So on the graph, there is a HOLE at 4.

•A graph is Continuous if it has no jumps, breaks or holes.

•A graph is Discontinuous if it does have holes or jumps or breaks.

Find the places of Discontinuity:

y=x2 + x−63x2 −12

Find the places of Discontinuity:

y=x2 −1

x2 −2x−3

Find the places of Discontinuity:

y=x2

x2 −9

Find the places of Discontinuity:

y=12x+24x2 +2x

9.4Simplifying, Multiplying and Dividing

Simplest Form

• A rational expression is in SIMPLEST FORM when its numerator and denominator are polynomials that have no common divisors.

• When simplifying we still need to remember HOLES as points of discontinuity.

Simplify:1.

Simplify:1. x2 +10x+ 25

x2 +9x+20

Simplify:

1.

Simplify:1.

Multiplying Rational Expressions

1. Multiply the tops and the bottoms.

2. Simplify.

Example:

Example:

Dividing Rational Expressions

Dividing by a fraction is the same thing as multiplying by the reciprocal.

1. Flip the second fraction 2. Multiply the tops and the bottoms. 3. Simplify.

x−4(3x+2)(x−2)

÷5(x−4)

(x−2)(7x−5)

Dividing Example:

Dividing Examplea2 +2a−15

a2 −16÷

a+13a−12

9.5 Adding and Subtracting Rational Expressions

Simplify:

You Try! Simplify:.

• A Common Denominator is also the LEAST COMMON MULTIPLE.

• LCM is the smallest numbers that each factor can be divided into evenly.

Finding LCM of 2 numbers

• 7: • 21:

Find the LCM of each pair of numbers

1. 4, 5:

2. 3, 8:

3. 4, 12:

Unlike Denominators 1.

2.

For Variables LCM: Take the Largest Exponent!

Find the LCM:

1. x4 and x:

2. x3 and x2:

3. 3x5 and 9x8:

1.

2.

For Expressions LCM: Include ALL factors

• 3(x + 2) and 5(x – 2)

• x(x + 4) and 3x2(x + 4)

For Expressions LCM: Include ALL factors

• (x2 + 2x - 8) and (x2 – 4)

• 7(x2 – 25) and 2(x2 + 7x + 10)

1. 5x + 3

−6

x−1

2. yy2 −y−20

+2

y+ 4

3. xx−2

−3

x2 −4

4.

A complex fraction is a fraction that has a fraction in its numerator or denominator.

To Simplify: Multiply the top and the bottom by the COMMON DENOMINATOR

Simplify:

2x3y

Simplifying Complex Fractions

1+ 23

49

Simplify:

2x+6

1y

9.6 SOLVING RATIONAL EQUATIONS

To solve proportions you cross multiply:

Example:38

=x48

Use cross multiplication to solve.

Check for Extraneous Solutions when variables are in the bottom because we cannot divide by zero!

1x2

=2x

Solve: x−210

=x−75

You Try ! Solve: yy+ 3

=6

y+9

You Try! Solve:

Eliminate the Fraction

We can eliminate the fractions all together if we multiply the whole equation by the LCM of the denominators!

12x

−25x

=12

Sum or Difference Equations

Solve the equation for x.

x3+

x2

=10

Solve:7

x2 −5x+

2x

=3

2x−10