VariationUnit 13

Direct Variation

The variable y varies directly as x if there is a nonzero constant, k, such that y = kx.

The equation y = kx is called a direct variation equation and the number k is called the constant of variation.

Steps:

1. Plug in the given information to find k.

2. Once you have found k, rewrite the direct variation equation including this value.

3. Use the found k value to evaluate the equation at another value (x or y).

Examples

Find the constant of variation, k, and the direct variation equation if y varies directly as x and y = -24 when x = 4.

Examples

Find the constant of variation, k, and the direct variation equation if y varies directly as x and y = 15 when x = 3.

Examples

A varies directly as b. If a is 2.8 when b is 7, find a when b is -4.

Examples

A varies directly as b. If a is -5 when b is 2.5, find b when a is 6.

Examples

Ohm’s law states that the voltage, V, measured in volts varies directly as the electric current, I, according to the equation V= IR. The constant of variation is the electrical resistance of the circuit, R. Ex: An iron is plugged into a 110-volt electrical outlet, creating a current of 5.5 amperes in the iron. Find the electrical resistance to the iron.

Determine whether y varies directly as x. If so, find k.

Example

If y varies directly as x and y = 8 when x = -4, find x when y = 7.

If y varies directly as x and y = 3 when x = 5, find y when x = 15.

Example

If y varies directly as and y = 10 when x = 5, find y when x = 2.

Homework

Book Page 33-34 #14-28 even and #32-36 all

Inverse Variation

Two variables, x and y, have an inverse variation relationship if there is a nonzero number, k, such that y = k/x

Joint Variation

If y=kxz, then y varies jointly as x and z, and the constant of variation is k

Example

The volume of a rectangular prism is V=lwh. Therefore, volume varies jointly as the length and the width.

Example

The variable y varies inversely as x, and y = 132 when x = 15. Find the constant of variation and write an equation for the relationship. Then find y when x is 1.5.

Example

Y varies jointly as x and z. Write the appropriate joint-variation equation and find y for the given values of x and z.

Ex: y = -108 when x = -4 and z = 3. Find y when x =6 and z = -2

Example

Z varies jointly as x and y and inversely as w. Write the appropriate combined variation equation and find z for the given values of x, y and w.

Ex: z = 3 when x = 3, y = - 2, and w = -4. Find z when x = 6, y = 7, and w = -4

Homework

Book Page 486 #13 - 33 odd

Word ProblemsThe heat loss through a glass window varies jointly as the area of the window and the difference between inside and outside temperature. The heat loss through a window with an area of 4 square meters is 820 BTU when the temperature difference is 10 degrees.

Write the joint variation equation and find the constant.