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A constant of variation, also known as a constant of proportionality, is a non-zero constant that relates two variables.
In a direct variation, the constant of variation k represents the ratio of two variables. The ratio remains unchanged when the variables x and y change. y/x=k, [0.7em] where x≠ 0 and y≠ 0 When the variable y is isolated on the left-hand side, the constant of variation k defines the slope of the line through the origin that represents the relationship between x and y. y/x=k ⇔ y=kx
In an inverse variation, the constant of variation k is the product of two variables. The product remains the same when the variables x and y change.
xy=k, [0.7em] where x≠ 0 and y≠ 0Direct variation, also known as direct proportionality or proportional relationship, occurs when two variables, x and y, have a relationship that forms a linear function passing through the origin.
y=kx
| Examples of Direct Variation | ||
|---|---|---|
| Example | Rule | Comment |
| The circumference of a circle. | C=π d | Here, d is the diameter of the circle and the constant of variation is π. |
| The mass of an object. | m= V | Here, is the constant density of the object and V is the volume. |
| Distance traveled at a constant rate. | d=rt | The constant of variation r is the rate and t is the time spent traveling. |
Direct variation is closely related to other types of 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.
xy=k or y=k/x
| 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. | P=nRT/V | The variables are the pressure P and the volume V. The amount of gas n, temperature T, and universal gas constant R are fixed values. Therefore, the constant of variation is nRT. |
| The time it takes to travel a given distance at various speeds. | t=d/s | The constant of variation is the distance d and the variables are the time t and the speed s. |
If the ordered pairs (x_1,y_1) and (x_2,y_2) are solutions to an inverse variation, then the products x_1 y_1 and x_2 y_2 are equal.
y_1=k/x_1 and y_2=k/x_2 ⇓ x_1 y_1 = x_2 y_2
If (x_1,y_1) and (x_2,y_2) are solutions to an inverse variation with constant of variation k, then both of the products x_1 y_1 and x_2 y_2 are equal to k. y_1=k/x_1 &⇒ x_1 y_1 = k [0.8em] y_2=k/x_2 &⇒ x_2 y_2 = k Since the products x_1 y_1 and x_2 y_2 are both equal to the same constant, by the Transitive Property of Equality they are also equal to each other. x_1 y_1 = x_2 y_2
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.
z=kxy
Here, the variable z varies jointly with x and y, and k is the constant of variation. Here are some examples of joint variation.
| Examples of Joint Variation | ||
|---|---|---|
| Example | Rule | Comment |
| The area of a rectangle | A=l w | Here, l is the rectangle's length, w its width, and the constant of variation k is 1. |
| The volume of a pyramid | V=1/3l w h | Here, l and w are the length and the width of the base, respectively, while h is the pyramid's height. The constant of variation k is 13. |
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.
z=kx/y
The variable z varies directly with x and inversely with y, and k 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 | F=G m_1 m_2/d^2 | The gravitational force F varies directly as the masses of the objects m_1 and m_2, and inversely as the square of the distance d^2 between the objects. The gravitational constant G is the constant of variation. |
| The Ideal Gas Law | P=nRT/V | The pressure P varies directly as the number of moles n and the temperature T, and inversely as the volume V. The universal gas constant R is the constant of variation. |