Direct Variation Chapter 8 Section 8.9

Objective Students will use direct variation to solve problems

Vocabulary Direct variation Constant of variation Constant of proportionality

Concept The table below shows the mass m of a gold bar whose volume is V cubic centimeters. You can see that m = 19.3V. This equation defines a linear function. Note that if the volume of the bar is doubled, the mass is doubled; if the volume is tripled, the mass is tripled, and so on. You can say that the mass varies directly as the volume. This function is an example of direct variation. Volume V Mass in grams 1 19.3 2 38.6 3 57.9 4 77.2 5 96.5

Concept A direct variation is a function defined by an equation of the form y = kx, where k is a nonzero constant You can say that y varies directly as x The constant k is called the constant of variation

Concept When the domain is the set of real numbers, the graph of a direct variation is a straight line with slope k that passes through the origin.

Example Given that m varies directly as n and that m = 42 when n = 2, find the following. The constant of variation The value of m when n = 3

Example Find the constant of variation y varies directly as x and y = 18 when x = 6

Concept Suppose (x1, y1) and (x2, y2) are two ordered pairs of a direct variation defined by y = kx and that neither x1 nor x2 is zero. Since (x1, y1) and (x2, y2) must satisfy y = kx, you know that y1 = kx1 and y2 = kx2 From those equations you can write the ratios y1 / x1 = k and y2 / x2 = k Since each ratio equals k, the ratios are equal y1 / x1 = y2 / x2

Concept This equation, which states that two ratios are equal, is a proportion. For this reason, k is sometimes called the constant of proportionality, and y is said to be directly proportional to x. When you use a proportion to solve a problem, you will find it helpful to recall that the product of the extremes equals the product of the means.

Find the missing value x1 = 104 x2 = ? y1 = 1300 y2 = 1800 Example Find the missing value x1 = 104 x2 = ? y1 = 1300 y2 = 1800

Find the missing value x1 = 6 y1 = 5 x2 = 3 y2 = ? Example Find the missing value x1 = 6 y1 = 5 x2 = 3 y2 = ?

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