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Polynomial and Rational Functions
Variation and Problem-Solving
ZOE Books & Concepts BOOKS Polynomial and Rational Functions Variation and Problem-Solving
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Concept Version 6
Created by Boundless

Inverse Variation

Indirect variation is used to describe the relationship between two variables when their product is constant.

Learning Objective

  • Practice solving inverse variation problems


Key Points

    • The ratio of variables in direct variation is always constant.
    • Direct variation between variables is depicted by an hyperbola.
    • The equation relating indirectly varying variables to a constant can be rearranged to hyberbolic form.

Terms

  • constant

    An identifier that is bound to an invariant value.

  • hyperbola

    A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone.


Full Text

Inverse variation is the opposite of direct variation. In the case of inverse variation, the increase of one variable leads to the decrease of another.

Consider a car driving on a flat surface at a certain speed. If the driver shifts into neutral gear, the car's speed will decrease at a constant rate as time increases, eventually coming to a stop.

Inverse variation can be illustrated, forming a graph in the shape of a hyperbola . Knowing that the relationship between the two variables is constant, we can show that their relationship is:

Interactive Graph: Graph of Indirect Variation

Graph of indirect variation with the equation y=1/x. This hyperbola shows the indirect variation of variables x and y. Notice what happens when you change the variants.

$yx=k$

where k is a constant known as the constant of proportionality. Note that as long as k is not equal to 0, neither x nor y can ever equal 0 either.

We can rearrange the above equation to place the variables on opposite sides:

$y=k/x$

Revisiting the example of the decelerating car, let's say it starts at 50 miles per hour and slows at a constant rate. If we define y as its speed in miles per hour, and x as time, the relationship between x and y can be expressed as:

$y=50/x$

Note that realistically, other factors (e.g., friction), will influence the rate of deceleration. Other constants can be incorporated into the equation for the sake of accuracy, but the overall form will remain the same.

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