12.1: Inverse Variation Models
What if you were paid $500 per week regardless of the number of hours you worked? The more hours you worked in a week (increasing quantity), the less your hourly rate (decreasing quantity) would be. How could you write and solve a function to model this situation? After completing Concept, you'll be able to write inverse variation equations and solve inverse variation applications like this one.
Watch This
Foundation: 1201S nverse Variation Models
Watch this video to see some more variation problems worked out, including problems involving joint variation.
Mathphonetutor: Algebra: Direct, Inverse, Joint Variation Problem
Guidance
Many variables in realworld problems are related to each other by variations. A variation is an equation that relates a variable to one or more other variables by the operations of multiplication and division. There are three different kinds of variation: direct variation, inverse variation and joint variation.
Distinguish Direct and Inverse Variation
In direct variation relationships, the related variables will either increase together or decrease together at a steady rate. For instance, consider a person walking at three miles per hour. As time increases, the distance covered by the person walking also increases, at the rate of three miles each hour. The distance and time are related to each other by a direct variation:
Since the speed is a constant 3 miles per hour, we can write:
The general equation for a direct variation is
You can see from the equation that a direct variation is a linear equation with a
A second type of variation is inverse variation. When two quantities are related to each other inversely, one quantity increases as the other one decreases, and vice versa.
For instance, if we look at the formula
If we keep the distance constant, we see that as the speed of an object increases, then the time it takes to cover that distance decreases. Consider a car traveling a distance of 90 miles, then the formula relating time and speed is:
The general equation for inverse variation is
In this chapter, we’ll investigate how the graphs of these relationships behave.
Another type of variation is a joint variation. In this type of relationship, one variable may vary as a product of two or more variables.
For example, the volume of a cylinder is given by:
In this example the volume varies directly as the product of the square of the radius of the base and the height of the cylinder. The constant of proportionality here is the number
In many application problems, the relationship between the variables is a combination of variations. For instance Newton’s Law of Gravitation states that the force of attraction between two spherical bodies varies jointly as the masses of the objects and inversely as the square of the distance between them:
In this example the constant of proportionality is called the gravitational constant, and its value is given by
Graph Inverse Variation Equations
We saw that the general equation for inverse variation is given by the formula
Example A
Graph an inverse variation relationship with the proportionality constant
Solution



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1 



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10 

Here is a graph showing these points connected with a smooth curve.
Both the table and the graph demonstrate the relationship between variables in an inverse variation. As one variable increases, the other variable decreases and vice versa.
Notice that when
Similarly, as the value of
Write Inverse Variation Equations
As we saw, an inverse variation fulfills the equation
Example B
If \begin{align*}y\end{align*}
Solution
\begin{align*}\text{Since} \ y \ \text{is inversely proportional to} \ x, \text{then:} \qquad \qquad \qquad \qquad y =\frac {k}{x}\!\\ \\ \text{Plug in the values} \ y = 10 \ \text{and} \ x = 5: \qquad \qquad \ \qquad \qquad \qquad 10 =\frac {k}{5}\!\\ \\ \text{Solve for} \ k \ \text{by multiplying both sides of the equation by} \ 5: \ \ \ k =50\!\\ \\ \text{The inverse relationship is given by:} \qquad \qquad \qquad \qquad \ \qquad \ \ y =\frac {50}{x}\!\\ \\ \text{When} \ x = 2: \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \qquad \ \ \qquad y=\frac {50}{2} \ \text{or} \ y=25\end{align*}
Compare Graphs of Inverse Variation Equations
Inverse variation problems are the simplest example of rational functions. We saw that an inverse variation has the general equation: \begin{align*}y= \frac{k}{x}\end{align*}. In most realworld problems, \begin{align*}x\end{align*} and \begin{align*}y\end{align*} take only positive values. Below, we will show graphs of three inverse variation functions.
Example C
On the same coordinate grid, graph inverse variation relationships with the proportionality constants \begin{align*}k = 1, k = 2,\end{align*} and \begin{align*}k = \frac{1}{2}\end{align*}.
Solution
We’ll skip the table of values for this problem, and just show the graphs of the three functions on the same coordinate axes. Notice that for larger constants of proportionality, the curve decreases at a slower rate than for smaller constants of proportionality. This makes sense because the value of \begin{align*}y\end{align*} is related directly to the proportionality constants, so we should expect larger values of \begin{align*}y\end{align*} for larger values of \begin{align*}k\end{align*}.
Watch this video for help with the Examples above.
CK12 Foundation: Inverse Variation Models
Vocabulary
 The general equation for a direct variation is \begin{align*}y = kx\end{align*}, where \begin{align*}k\end{align*} is called the constant of proportionality.
 The general equation for inverse variation is \begin{align*}y= \frac{k}{x}\end{align*}, where \begin{align*}k\end{align*} is the constant of proportionality.
Guided Practice
If \begin{align*}p\end{align*} is inversely proportional to the square of \begin{align*}q\end{align*}, and \begin{align*}p = 64\end{align*} when \begin{align*}q = 3\end{align*}, find \begin{align*}p\end{align*} when \begin{align*}q = 5\end{align*}.
Solution
\begin{align*}\text{Since} \ p \ \text{is inversely proportional to} \ q^2, \text{then:} \qquad \qquad \qquad \qquad \ \ p =\frac {k}{q^2}\!\\ \\ \text{Plug in the values} \ p = 64 \ \text{and} \ q = 3: \qquad \qquad \quad \qquad \qquad \qquad \ \ 64 =\frac {k}{3^2} \ \text{or} \ 64=\frac {k}{9}\!\\ \\ \text{Solve for} \ k \ \text{by multiplying both sides of the equation by} \ 9: \qquad \ k =576\!\\ \\ \text{The inverse relationship is given by:} \qquad \qquad \qquad \qquad \qquad \qquad \ p =\frac {576}{q^2}\!\\ \\ \text{When} \ q = 5: \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \qquad p =\frac {576}{25} \ \text{or} \ y=23.04\end{align*}
Practice
For 14, graph the following inverse variation relationships.
 \begin{align*}y= \frac{3}{x}\end{align*}
 \begin{align*}y= \frac{10}{x}\end{align*}
 \begin{align*}y= \frac{1}{4x}\end{align*}
 \begin{align*}y= \frac{5}{6x}\end{align*}
 If \begin{align*}z\end{align*} is inversely proportional to \begin{align*}w\end{align*} and \begin{align*}z = 81\end{align*} when \begin{align*}w = 9\end{align*}, find \begin{align*}w\end{align*} when \begin{align*}z = 24\end{align*}.
 If \begin{align*}y\end{align*} is inversely proportional to \begin{align*}x\end{align*} and \begin{align*}y = 2\end{align*} when \begin{align*}x = 8\end{align*}, find \begin{align*}y\end{align*} when \begin{align*}x = 12\end{align*}.
 If \begin{align*}a\end{align*} is inversely proportional to the square root of \begin{align*}b\end{align*}, and \begin{align*}a = 32\end{align*} when \begin{align*}b = 9\end{align*}, find \begin{align*}b\end{align*} when \begin{align*}a = 6\end{align*}.
 If \begin{align*}w\end{align*} is inversely proportional to the square of \begin{align*}u\end{align*} and \begin{align*}w = 4\end{align*} when \begin{align*}u = 2\end{align*}, find \begin{align*}w\end{align*} when \begin{align*}u = 8\end{align*}.
 If \begin{align*}a\end{align*} is proportional to both \begin{align*}b\end{align*} and \begin{align*}c\end{align*} and \begin{align*}a = 7\end{align*} when \begin{align*}b = 2\end{align*} and \begin{align*}c = 6\end{align*}, find \begin{align*}a\end{align*} when \begin{align*}b = 4\end{align*} and \begin{align*}c = 3\end{align*}.
 If \begin{align*}x\end{align*} is proportional to \begin{align*}y\end{align*} and inversely proportional to \begin{align*}z\end{align*}, and \begin{align*}x = 2\end{align*} when \begin{align*}y = 10\end{align*} and \begin{align*}z = 25\end{align*}, find \begin{align*}x\end{align*} when \begin{align*}y = 8\end{align*} and \begin{align*}z = 35\end{align*}.
 If \begin{align*}a\end{align*} varies directly with \begin{align*}b\end{align*} and inversely with the square of \begin{align*}c\end{align*}, and \begin{align*}a = 10\end{align*} when \begin{align*}b = 5\end{align*} and \begin{align*}c = 2\end{align*}, find the value of \begin{align*}a\end{align*} when \begin{align*}b = 3\end{align*} and \begin{align*}c = 6\end{align*}.
 If \begin{align*}x\end{align*} varies directly with \begin{align*}y\end{align*} and \begin{align*}z\end{align*} varies inversely with \begin{align*}x\end{align*}, and \begin{align*}z = 3\end{align*} when \begin{align*}y = 5\end{align*}, find \begin{align*}z\end{align*} when \begin{align*}y = 10\end{align*}.
Constant of Proportionality
The constant of proportionality, commonly represented as is the constant ratio of two proportional quantities such as and .Direct Variation
When the dependent variable grows large or small as the independent variable does.Inverse Variation
Inverse variation is a relationship between two variables in which the product of the two variables is equal to a constant. As one variable increases the second variable decreases proportionally.Joint Variation
Variables exhibit joint variation if one variable varies directly as the product of two or more other variables.Image Attributions
Description
Learning Objectives
Here you'l learn how to graph inverse variation equations. You'll also learn how to write and solve such equations to find unknown values.