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Rational Functions

* Inverse/Direct

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Direct Variation

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

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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 1 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 2 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 3 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 4 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 5 Using a Proportion

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

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Inverse Variation

What you’ll learn …• To use inverse variation• To use combined 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 1 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 2 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 1 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 2 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