Post on 12-Jan-2016
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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