Transcript 10.8 - DRS & Company

Chapter 10
Graphing
Equations and
Inequalities
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
10.8
Direct and Inverse
Variation
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Direct Variation
y varies directly as x, or y is directly proportional
to x, if there is a nonzero constant k such that
y = kx
The number k is called the constant of variation or
the constant of proportionality.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
3
Direct Variation
Suppose that y varies directly as x. If y = 5
when x = 30, find the constant of variation
and the direct variation equation.
y = kx
5 = k • 30
1
k=6
So the direct variation equation is
1
y  x.
6
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
4
Example
Suppose that y varies directly as x, and y = 48 when
x = 6. Find y when x = 15.
y = kx
48 = k • 6
8=k
So the equation is y = 8x.
y = 8 ∙ 15
y = 120
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
5
Direct Variation
Direct Variation: y = kx
• There is a direct variation relationship between x and y.
• The graph is a line.
• The line will always go through the origin (0, 0). Why?
Let x = 0. Then y = k ∙ 0 or y = 0.
• The slope of the graph of y = kx is k, the constant of
variation. Why? Remember that the slope of an equation
of the form y = mx + b is m, the coefficient of x.
• The equation y = kx describes a function. Each x has a
unique y and its graph passes the vertical line test.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
6
Example
The line is the graph of a direct
variation equation. Find the
constant of variation and the
direct variation equation.
To find k, use the slope formula
and find slope.
slo p e 
k 
y
(4, 1)
(0 0)
x
1 0
1

40 4
1 and the variation equation is
4
1
y  x.
4
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
7
Inverse Variation
y varies inversely as x, or y is inversely proportional
to x, if there is a nonzero constant k such that
y=k
x
The number k is called the constant of variation or the
constant of proportionality.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
8
Example
Suppose that y varies inversely as x. If y = 63 when x
= 3, find the constant of variation k and the inverse
variation equation.
k
y 
x
k
63 
3
k = 63·3
k = 189
So the inverse variation
equation is y  1 8 9 .
x
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
9
Powers of x
Direct and Inverse Variation as nth Powers of x
y varies directly as a power of x if there is a nonzero
constant k and a natural number n such that
y = kxn
y varies inversely as a power of x if there is a
nonzero constant k and a natural number n such that
y
k
x
n
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
10
Example
At sea, the distance to the horizon is directly
proportional to the square root of the elevation of the
observer. If a person who is 36 feet above water can
see 7.4 miles, find how far a person 64 feet above the
water can see. Round your answer to two decimal
places.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
continued
11
continued
d k e
Translate the problem into an equation.
7 . 4  k 36
Substitute the given values for the elevation
and distance to the horizon for e and d.
7 .4  6 k
Simplify.
k 
7 .4
Solve for k, the constant of proportionality.
6
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
continued
12
continued
So the equation is d 
d 
7 .4
6
7 .4
e .
64
Replace e with 64.
6
7.4

(8)
6
Simplify.
59.2

 9.87 m iles
6
A person 64 feet above the water can see about 9.87 miles.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
13
Example
The maximum weight that a circular column can
hold is inversely proportional to the square of its
height.
If an 8-foot column can hold 2 tons, find how much
weight a 10-foot column can hold.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
continued
14
continued
w
2
k
Translate the problem into an equation.
2
h
k
8
2
k  128

k
Substitute the given values for w and h.
64
Solve for k, the constant of proportionality.
So the equation is w 
128
2
h .
128
128
w

 1 . 28 tons
2
10
100
Let h = 10.
A 10-foot column can hold 1.28 tons.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
15