1. how one quantity varies in relation to another quantity
VARIATION
how one quantity varies in relation to another quantity

2. Direct Variation
When modeling if we say that y varies directly with x, or y is directly proportional to x, this means direct variation. Mathematically this is written:
y = kx
constant of proportionality
Examples:
Force is directly proportional to acceleration
Revenue varies directly with sales
When force increases, acceleration increases (and vice versa)
When sales go up, revenue goes up (and vice versa)

3. y = kx 5 = k(30) Direct Variation 45
Hooke's Law states that the distance a spring is stretched when an object is placed on it is directly proportional to the weight of the object. This is modeled by:
distance spring is stretched
weight of object
y = kx 45
To find k, we substitute observed values for x and y
constant of proportionality (depends on type of spring)
Suppose a 30-lb object stretches the spring 5 inches. How far would a 45-lb object stretch the spring?
5 = k(30)

4. Resistance is inversely proportional to the diameter of a wire
Inverse Variation
When modeling if we say that y varies inversely with x, or y is inversely proportional to x, this means inverse variation. Mathematically this is written:
constant of proportionality
Examples:
The force of attraction between two planets is inversely proportional to the square of the distance between them.
Resistance is inversely proportional to the diameter of a wire
When you have a larger diameter wire, the resistance goes down (and vice versa)
When distance decreases, force increases

5. Inverse Variation
The weight of a body above the surface of the earth varies inversely with the square of the distance from the center of the earth. This is modeled by:
constant of proportionality
weight of a body above earth
(3995)2
To find k, we substitute in observed values for x and y
square of distance from center of earth
A certain body weighs 55-lbs when it is 3960 miles from the center of the earth. How much will is weigh when it is 3995 miles from the center?

6. Joint Variation and Combined Variation
When modeling if we say that a variable is proportional to the product of two or more other variables, we say it varies jointly with these quantities.
We can also have more than two variables in some combination of variation, i.e. one may vary directly and one may vary inversely. Direct variation will multiply the proportionality constant and inversely will divide the constant.
If variable w varies jointly with x and y and inversely with z, what would be the modeling equation?