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Variations

Constant of Variation

A constant of variation, also known as a constant of proportionality, is a non-zero constant that relates two variables.

Direct Variation

In a direct variation, the constant of variation represents the ratio of two variables. The ratio remains unchanged when the variables and change.
When the variable is isolated on the left-hand side, the constant of variation defines the slope of the line through the origin that represents the relationship between and

Inverse Variation

In an inverse variation, the constant of variation is the product of two variables. The product remains the same when the variables and change.

Direct Variation

Direct variation, also known as direct proportionality or proportional relationship, occurs when two variables, and have a relationship that forms a linear function passing through the origin where and

The constant is the constant of variation. It defines the slope of the line. When the relationship is not in direct variation. In the example below, the constant of variation is
The constant of variation may be any real number except It is worth noting that the quotient of and is the constant of variation.
Here are some examples.
Examples of Direct Variation
Example Rule Comment
The circumference of a circle. Here, is the diameter of the circle and the constant of variation is
The mass of an object. Here, is the constant density of the object and is the volume.
Distance traveled at a constant rate. The constant of variation is the rate and is the time spent traveling.

Direct variation is closely related to other types of variation.

Inverse Variation

An inverse variation, or inverse proportionality, occurs when two non-zero variables have a relationship such that their product is constant. This relationship is often written with one of the variables isolated on the left-hand side.

The constant is the constant of variation. When the relationship is not an inverse variation. In the following example, the constant of variation is
The constant of variation may be any real number except Here are some examples of inverse variations.
Examples of Inverse Variation
Example Rule Comment
The gas pressure in a sealed container if the container's volume is changed, given constant temperature and constant amount of gas. The variables are the pressure and the volume The amount of gas temperature and universal gas constant are fixed values. Therefore, the constant of variation is
The time it takes to travel a given distance at various speeds. The constant of variation is the distance and the variables are the time and the speed
Inverse variation is closely related to other types of variation.

Product Rule for Inverse Variations

If the ordered pairs and are solutions to an inverse variation, then the products and are equal.

Proof

If and are solutions to an inverse variation with constant of variation then both of the products and are equal to
Since the products and are both equal to the same constant, by the Transitive Property of Equality they are also equal to each other.

Joint Variation

A joint variation, also known as joint proportionality, occurs when one variable varies directly with two or more variables. In other words, if a variable varies directly with the product of other variables, it is called joint variation.

Here, the variable varies jointly with and and is the constant of variation. Here are some examples of joint variation.

Examples of Joint Variation
Example Rule Comment
The area of a rectangle Here, is the rectangle's length, its width, and the constant of variation is
The volume of a pyramid Here, and are the length and the width of the base, respectively, while is the pyramid's height. The constant of variation is
Lastly, it is important to note that joint variation is closely related to other types of variation.

Combined Variation

A combined variation, or combined proportionality, occurs when one variable depends on two or more variables, either directly, inversely, or a combination of both. This means that any joint variation is also a combined variation.

The variable varies directly with and inversely with and is the constant of variation. Therefore, this is a combined variation. Here are some examples.

Examples of Combined Variation
Example Rule Comment
Newton's Law of Gravitational Force The gravitational force varies directly as the masses of the objects and and inversely as the square of the distance between the objects. The gravitational constant is the constant of variation.
The Ideal Gas Law The pressure varies directly as the number of moles and the temperature and inversely as the volume The universal gas constant is the constant of variation.
Combined variation is closely related to other types of variation.