Graphing Equations and Inequalities

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Chapter 10

Graphing Equations and

Inequalities


10.8

Direct and Inverse Variation

Martin-Gay, Developmental Mathematics, 2e 33


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.



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



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



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.



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



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



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



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



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



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



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



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



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.

.

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Graphing Equations and Inequalities