Upload
walter-hebert
View
41
Download
2
Embed Size (px)
DESCRIPTION
Chapter 10. Graphing Equations and Inequalities. Direct and Inverse Variation. 10.8. 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 - PowerPoint PPT Presentation
Citation preview
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Chapter 10
Graphing Equations and
Inequalities
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
10.8
Direct and Inverse Variation
Martin-Gay, Developmental Mathematics, 2e 33
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.
Martin-Gay, Developmental Mathematics, 2e 44
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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
k =
Direct Variation
So the direct variation equation is1 .6
y x
16
Martin-Gay, Developmental Mathematics, 2e 55
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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
Example
Martin-Gay, Developmental Mathematics, 2e 66
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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?
• 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.
Direct Variation
Let x = 0. Then y = k ∙ 0 or y = 0.
Martin-Gay, Developmental Mathematics, 2e 77
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
The line is the graph of a direct variation equation. Find the constant of variation and the direct variation equation.
Example
x
y
(0 0)(4, 1)
To find k, use the slope formula and find slope.
1 0slope 4 0
14
and the variation equation is14
k 1 .4
y x
Martin-Gay, Developmental Mathematics, 2e 88
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
y varies inversely as x, or y is inversely proportional to x, if there is a nonzero constant k such that
y = k
The number k is called the constant of variation or the constant of proportionality.
Inverse Variation
x
Martin-Gay, Developmental Mathematics, 2e 99
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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 = 63·3
k = 189
Example
So the inverse variation
equation is
kyx
633k
189.y
x
Martin-Gay, Developmental Mathematics, 2e 1010
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Direct and Inverse Variation as nth Powers of xy varies directly as a power of x if there is a nonzero constant k and a natural number n such that
y varies inversely as a power of x if there is a nonzero constant k and a natural number n such that
nx
ky
Powers of x
y = kxn
Martin-Gay, Developmental Mathematics, 2e 1111
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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.
continued
Example
Martin-Gay, Developmental Mathematics, 2e 1212
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
ekd
364.7 k
k64.7
6
4.7k
Substitute the given values for the elevation and distance to the horizon for e and d.
continued
Simplify.
Solve for k, the constant of proportionality.
Translate the problem into an equation.
continued
Martin-Gay, Developmental Mathematics, 2e 1313
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
646
4.7d
7.46
(8)
59.26
9.87 miles
So the equation is ed6
4.7
Replace e with 64.
Simplify.
A person 64 feet above the water can see about 9.87 miles.
.
continued
Martin-Gay, Developmental Mathematics, 2e 1414
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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.
continued
Example
Martin-Gay, Developmental Mathematics, 2e 1515
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
2h
kw
6482
2
kk
128k
tons28.1100
128
10
1282
w
2
128
hw So the equation is
continued
Substitute the given values for w and h.
Solve for k, the constant of proportionality.
Translate the problem into an equation.
A 10-foot column can hold 1.28 tons.
Let h = 10.
.