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Chapter 9

Rational Functions

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In this chapter you should …

• Learn to use inverse variation and the graphs of inverse variations to solve real-world problems.

• Learn to identify properties of rational functions.

• Learn to simplify rational expressions and to solve rational equations.

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2-3 Direct Variation

What you’ll learn …• To write and interpret direct variation equations

• 1.05 Model and solve problems using direct, inverse, combined and joint variation.

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This is a graph of direct variation.  If the value of x is increased, then y increases as well.Both variables change in the same manner.  If x decreases, so does the value of y.  We say that y varies directly as the value of x. 

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Definition:

Y varies directly as x means that y = kx

where k is the constant of variation.

(see any similarities to y = mx + b?)

Another way of writing this is k = y

x

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Example 1a Identifying Direct Variation from a Table

For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation.

x y

2 8

3 12

5 20

x y

1 4

2 7

5 16

k = _______ k = _______

Equation _________________ Equation _________________

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Example 1b Identifying Direct Variation from a Table

For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation.

x y

-6 -2

3 1

12 4

x y

-1 -2

3 4

6 7

k = _______ k = _______

Equation _________________ Equation _________________

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Example 4a Using a Proportion

Suppose y varies directly with x, and x = 27 when y = -51. Find x when y = -17.

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Example 4b Using a Proportion

Suppose y varies directly with x, and x = 3 when y = 4. Find y when x = 6.

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Example 4c Using a Proportion

Suppose y varies directly with x, and x = -3 when y = 10. Find x when y = 2.

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9-1 Inverse Variation

What you’ll learn …• To use inverse variation• To use combined variation

• 1.05 Model and solve problems using direct, inverse, combined and joint variation.

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In an inverse variation, the values of the two variables change in an opposite manner - as one value increases, the other decreases.

  Inverse variation:  when one variable increases,the other variable decreases. 

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Inverse Variation

When two quantities vary inversely, one quantity increases as the other decreases, and vice versa. Generalizing, we obtain the following statement.

An inverse variation between 2 variables, y and x, is a relationship that is expressed as:

           

where the variable k is called the constant of proportionality.As with the direct variation problems, the k value

needs to be found using the first set of data.

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Example 2a Identifying Direct and Inverse Variation

Is the relationship between the variables in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations.

x 0.5 2 6

y 1.5 6 18x 0.2 0.6 1.2

y 12 4 2

x 1 2 3

y 2 1 0.5

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Example 2a Identifying Direct and Inverse Variation

Is the relationship between the variables in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations.

x 0.8 0.6 0.4

y 0.9 1.2 1.8x 2 4 6

y 3.2 1.6 1.1

x 1.2 1.4 1.6

y 18 21 24

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Example 3 Real World Connection

Zoology. Heart rates and life spans of most mammals are inversely related. Us the data to write a function that models this inverse variation. Use your function to estimate the average life span of a cat with a heart rate of 126 beats / min.

Mammal Heart Rate (beats per min)

Life Span

(min)

Mouse 634 1,576,800

Rabbit 158 6,307,200

Lion 76 13,140,000

Horse 63 15,768,000

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A combined variation combines direct and inverse variation in more complicated relationships.

Combined Variation Equation Form

y varies directly with the square of x y = kx2

y varies inversely with the cube of x y =

z varies jointly with x and y. z = kxy

z varies jointly with x and y and inversely with w.

z =

z varies directly with x and inversely with the product of w and y.

z =

kx3

kxy w

kxwy

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Example 5a Finding a Formula

The volume of a regular tetrahedron varies directly as the cube of the length of an edge. The volume of a regular tetrahedron with edge length 3 is .

Find the formula for the volume of a regular tetrahedron.

9 √ 2 4

e

e

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Example 5b Finding a Formula

The volume of a square pyramid with congruent edges varies directly as the cube of the length of an edge. The volume of a square pyramid with edge length 4 is .

Find the formula for the volume of a square pyramid with congruent edges.

32 √ 2 3

e

e

e

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9-3 Rational Functions and Their Graphs

What you’ll learn …• To identify properties of rational functions• To graph rational functions

• 2.05 Use rational equations to model and solve problems; justify results.

• Solve using tables, graphs, and algebraic properties.

• Interpret the constants and coefficients in the context of the problem.

• Identify the asymptotes and intercepts graphically and algebraically.

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Definition Rational Function

A rational function f(x) is a function that can be written as

where P(x) and Q(x) are polynomial functions and Q(x) ≠ 0.

f(x) = P(x)Q(x)

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Examples of Rational Functions

In this graph, there is no value of x that makes the denominator 0. The graph is continuous because it has no jumps, breaks, or holes in it. It can be drawn with a pencil that never leaves the paper.

y = -2xx2 + 1

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Examples of Rational Functions

In this graph, x cannot be 4 or -4 because then the denominator would equal 0.

y = 1x2 - 16

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Examples of Rational Functions

In this graph, x cannot equal 1 or the denominator would equal 0.

y = (x+2)(x-1) x - 1

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Point of Discontinuity

A function is said to have a point of discontinuity at x = a or the graph of the function has a hole at x = a, if the original function is undefined for x = a, whereas the related rational expression of the function in simplest form is defined for x = a. 

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Example of Point of Discontinuity

• Consider a function .

• This function is undefined for x = 2. But the simplified rational expression of this function, x + 3 which is obtained by canceling (x - 2) both in the numerator and the denominator is defined at x = 2. Thus we can say that the function f(x) has a point of discontinuity at x = 2.

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Example 1b Finding Points of Discontinuity

1x2 - 16

x2 - 1 x2 + 3

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Vertical Asymptotes

• An asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of q(x). Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters.

• If you can write it in factored form, then you can tell whether the graph will be asymptotic in the same direction or in different directions by whether the multiplicity is even or odd.

• Asymptotic in the same direction means that the curve will go up or down on both the left and right sides of the vertical asymptote. Asymptotic in different directions means that the one side of the curve will go down and the other side of the curve will go up at the vertical asymptote.

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Example 1a Finding Points of Discontinuity

1x2 + 2x +1

-x + 1 x2 +1

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Example 2a Finding Vertical Asymptotes

x + 1(x – 2)(x – 3)

(x – 2) (x – 1) x - 2

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Example 2b Finding Vertical Asymptotes

(x – 3)(x + 4)(x – 3)(x – 3)(x+4)

x – 2 (x - 1)(x + 3)

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Horizontal Asymptotes

• The graph of a rational function has at most one HA.• The graph of a rational function has a HA at y=0 if the

degree of the denominator is greater than the degree of the numerator .

• If the degrees of the numerator and the denominator are =, then the graph has a HA at y = , a is the coefficient of the term of the highest degree in the numerator and b is the coefficient of the term of the highest degree in the denominator.

• If the degree of the numerator is greater than the degree of the denominator, then the graph has no HA

ab

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Example 4a Sketching Graphs of HA

y = x + 2(x+3)(x-4)

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Example 4b Sketching Graphs of HA

y = x + 3(x-1)(x-5)

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Example 5 Real World Connection

The CD-ROMs for a computer game can be manufactured for $.25 each. The development cost is $124,000. The first 100 discs

are samples and will not be sold.

a. Write a function for the average cost of a salable disc. Graph the function.

b. What is the average cost if 2000 discs are produced? If 12,800 discs are produced?

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9-4 Rational Expressions

What you’ll learn …• To simplify rational expression• To multiply and divide rational expressions

• 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

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

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Example 1a Simplifying Rational Expressions

x2 + 10x + 25x2 + 9x + 20

-27x3y

9x4y

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Example 1b Simplifying Rational Expressions

-6 – 3xx2 - 6x + 8

2x2 – 3x - 2

x2 – 5x + 6

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Example 2 Real World Connection

Architecture One factor in designing a structure is the need to maximize the volume (space for working) for a given surface area (material needed for construction). Compare the ratio of the volume to surface area of a cylinder with radius r and height r to a cylinder with radius r and height 2r.

SA = 2rh + 2r2

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Multiplying Rational Expressions

Simply Put: The rule for multiplying algebraic fractions is the same as the

rule for multiplying numerical fractions.

Multiply the tops (numerators) AND

multiply the bottoms (denominators). 

If possible, reduce (cancel) BEFORE you multiply the tops and bottoms!

(It's easier than simplifying at the end!)

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Example 3a Multiplying Rational Expressions

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Example 3b Multiplying Rational Expressions

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Dividing Rational Expressions

Simply Put:The rule for dividing algebraic fractions

 is the same as the rule for dividing numerical fractions.

Change the division sign to multiplication, flip the 2nd fraction ONLY,

 and then follow the steps for "multiplying rational expressions".

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Example 4a Dividing Rational Expressions

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Example 4b Dividing Rational Expressions

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9-5 Adding and Subtracting Rational Expressions

What you’ll learn …• To add and subtract rational expressions• To simplify complex fractions

• 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

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The Basic RULE for Adding and Subtracting Fractions:

• Get a Common Denominator! (the smallest number that both denominators can divide into without remainders.) 

• With each fraction, whatever is multiplied times the bottom must ALSO be multiplied times the top.

• Do not add the common denominators.  Add only the numerators (tops).

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Adding and Subtracting Fractions with Like Denominators

2 5

3 3

4 3

7 7 + -

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Adding Expressions with Like Denominators

2 5

x + 3 x + 3

y y + 3

y – 5 y – 5 + +

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Subtracting Expressions with Like Denominators

2n + 1 3n + 4

2n + 5n – 3 2n + 5n – 3

4 5

t – 2 t – 2 - -

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Example 3a Adding Rational Expressions with Unlike Denominators

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Example 3b Adding Rational Expressions with Unlike Denominators

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Example 4a Subtracting Rational Expressions with Unlike Denominators

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More Examples

1x2 + 5x + 4

+ 5x3x + 3

7y5y2 - 125

43y + 15

-

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   A complex fraction is a fraction in which the

numerator, denominator, or both, also contain fractions.

If the complex fraction contains a variable, it is called a complex rational expression.

Simplify complex fractions by multiplying by a common denominator.

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Example 5a Simplifying Complex Fractions

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Example 5b Simplifying Complex Fractions

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9-6 Solving Rational Equations

What you’ll learn …• To solve rational expressions• To use rational equations in solving problems

2.05 Use rational equations to model and solve problems; justify results.

• Solve using tables, graphs, and algebraic properties.

• Interpret the constants and coefficients in the context of the problem.

• Identify the asymptotes and intercepts graphically and algebraically.

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• A rational equation is an equation in which one or more of the terms is a fractional one. 

• When solving these rational equations, we utilize one of two methods that will eliminate the denominator of each of the terms.   

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Method 1• If the equation is in the form of a proportion: 

• you can use "cross-multiplication" to eliminate the denominator, as in:  . 

• Then solve the resulting equation and check.

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Examples

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Method 2To solve the rational equation in this method, we:

1.   Identify the least common denominator (LCD),

2.   Multiply each side of the equation by the LCD,

simplify,

3.   Solve the resulting equation, and

4. Check the answer.

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Examples

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More Examples

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Application 1

Carlos can travel 40 mi on his motorbike in the same time it takes Paul to travel 15 mi on his bike. If Paul rides his bike 20 mi/h slower than Carlos rides his motorbike, find the speed for each bike.

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Application 2

A passenger train travels 392 mi in the same time that it takes a freight train to travel 322 mi. If the passenger train travels 20 mi/h faster than the freight train, find the speed of each train.

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Application 3

Sidney can paint a fence in 8 hours. Roy can do it in 4 hours. How long will it take them to do the job if they work together?

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Application 4

One pump can fill a tank with oil in 4 hours. A second pump can fill the same tank in 3 hours. If both pumps are used at the same time, how long will they take to fill the tank?

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In this chapter you should have…

• Learned to use inverse variation and the graphs of inverse variations to solve real-world problems.

• Learned to identify properties of rational functions.

• Learned to simplify rational expressions and to solve rational equations.