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:
Note: In this course you can use technology to graph hyperbolas, find stationary points, intercepts, and points of intersection.
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.