Inverse proportion, or inverse variation, means that as one amount increases the other amount decreases.
Mathematically, we write this as:
$y$y | $\propto$∝ | $\frac{1}{x}$1x |
For example, speed and travel time vary inversely because the faster you go, the shorter your travel time.
We express these inverse variation relationships generally in the form
$y=\frac{k}{x}$y=kx
where $k$k is the constant of proportionality and $x$x and $y$y are any variables
It is also possible to write the equation in the form:
$xy=k$xy=k or $x=\frac{k}{y}$x=ky
and it is clear that $k$k can be any number other than $0$0.
The graph of an inverse relationship in the $xy$xy-plane is called a hyperbola. Let's see what inverse variation looks like in a table of values.
This table shows the relationship $y=\frac{1}{x}$y=1x:
$x$x | $-4$−4 | $-2$−2 | $-1$−1 | $-0.5$−0.5 | $-0.25$−0.25 | $0.25$0.25 | $0.5$0.5 | $1$1 | $2$2 | $4$4 |
---|---|---|---|---|---|---|---|---|---|---|
$y$y | $-0.25$−0.25 | $-0.5$−0.5 | $-1$−1 | $-2$−2 | $-4$−4 | $4$4 | $2$2 | $1$1 | $0.5$0.5 | $0.25$0.25 |
Notice that:
Here are some hyperbolas with equations of the form $y=\frac{k}{x}$y=kx (or $xy=k$xy=k).
Notice the following features:
Which of the following equations represent inverse variation between $x$x and $y$y?
Select all correct answers.
$y=\frac{7}{x}$y=7x
$y=6x+8$y=6x+8
$y=-\frac{9}{x}$y=−9x
$y=\frac{8}{x^2}$y=8x2
$y=2x^2-7x-4$y=2x2−7x−4
$y=3-x$y=3−x
In the table of values below, $m$m is proportional to $\frac{1}{p}$1p.
$p$p | $2$2 | $4$4 | $5$5 | $x$x |
---|---|---|---|---|
$m$m | $140$140 | $y$y | $56$56 | $40$40 |
Determine the constant of proportionality, $k$k.
Using the solution to part (a), find the unknowns in the table:
$x$x$=$=$\editable{}$
$y$y$=$=$\editable{}$
Consider the graph of $y=\frac{2}{x}$y=2x.
For positive values of $x$x, as $x$x increases $y$y approaches what value?
$0$0
$1$1
$-\infty$−∞
$\infty$∞
As $x$x takes small positive values approaching $0$0, what value does $y$y approach?
$\infty$∞
$0$0
$-\infty$−∞
$\pi$π
What are the values that $x$x and $y$y cannot take?
$x$x$=$=$\editable{}$
$y$y$=$=$\editable{}$
The graph is symmetrical across two lines of symmetry. State the equations of these two lines.
$y=\editable{},y=\editable{}$y=,y=
Consider the hyperbola that has been graphed.
Fill in the gap to complete the statement.
Every point $\left(x,y\right)$(x,y) on the hyperbola is such that $xy$xy$=$=$\editable{}$.
Considering that the relationship between $x$x and $y$y can be expressed as $xy=6$xy=6, which of the following is true?
If $x$x increases, $y$y must increase.
If $x$x increases, $y$y must decrease.
Which of the following relationships can be modelled by a function of the form $xy=a$xy=a?
The relationship between the number of people working on a job and how long it will take to complete the job.
The relationship between the number of sales and the amount of revenue.
The relationship between height and weight.