Direct and Inverse Variation

Algebra I

Direct Variation

• When the line passes through the origin.

• When y increases, x increases.• Equation: y = kx, where k 0.• Constant of variation – k• Say “ y varies directly with x”

Identifying the constant of variation

y = 6x

6 is the constant of variation (k)

These can be graphed on the calculator using ‘y=‘ button, to see what the graph should look like.

Examples:

y = 28, x = 7 find x when y = 52

Examples:

y = 28, x = 7 find x when y = 52

y = kx y = kx

Examples:

y = 28, x = 7 find x when y = 52

y = kx y = kx28 = k(7) 52 = 4x 7 7 4 44 = k 13 = x y = 4x

Example:

y = 27, x = 6 find x when y = 45

Example:

y = 27, x = 6 find x when y = 45

27 = k(6) 45 = 9/2x 6 6 (2/9)45 = x9/2 = k 10 = x

Y = 9/2x

Example:

y = -7, x = -14 find y when x = 20

Example:

y = -7, x = -14 find y when x = 20

-7 = k(-14) y = (½)20-14 -14 y = 10½ = k

y = ½x

Inverse (Indirect) Variation

• Line going away from the origin.• When one value (x or y) increases, the

other value (x or y) decreases.• Equation xy = k • Say “y varies inversely as x” or “y is

inversely proportional to x”.

Different forms of the equation

xy = k

Y = k/x or x = k/y

Example:

If y = 12 when x = 5, find y when x = 3

xy = k 3y = 60(5)(12) = k 3 3 60 = k y = 20

xy = 60

Example:

If y = 7, when x = -2; find y when x = 7

If y = 8.5, when x = -1; find x when y = -1

If y = 8, when x = 1.55; find x when y = -0.62

Example:

If y = 7, when x = -2; find y when x = 7 xy = -14 y = -2

If y = 8.5, when x = -1; find x when y = -1 xy = -8.5 x = 8.5

If y = 8, when x = 1.55; find x when y = -0.62 xy = 12.4 x = -20