Graphing Hyperbolas (k/x)

Lesson

Practice

Lesson

We can graph a function $\frac{k}{x}$kx by constructing a table of values having first specified a value for the parameter $k$k. The shape of the graph will be a hyperbola and the effect of changing $k$k is to change the scale of the graph. These properties are illustrated in the following diagram where the graph of $y=\frac{1}{x}$y=1x is shown in blue, $y=\frac{3}{x}$y=3x is shown in red and $y=\frac{5}{x}$y=5x is shown in green.

If we had to draw these graphs by hand, we could construct tables of values like the following. We have restricted $x$x to values between $-5$−5 and $5$5.

You should check whether the graphs above really do match the corresponding tables.

$x$`x`	$-5$−5	$-4$−4	$-3$−3	$-2$−2	$-1$−1	$\frac{1}{2}$12	$1$1	$2$2	$3$3	$4$4	$5$5
$\frac{1}{x}$1`x`	$-\frac{1}{5}$−15	$-\frac{1}{4}$−14	$-\frac{1}{3}$−13	$-\frac{1}{2}$−12	$-1$−1	$2$2	$1$1	$\frac{1}{2}$12	$\frac{1}{3}$13	$\frac{1}{4}$14	$\frac{1}{5}$15
$\frac{3}{x}$3`x`	$-\frac{3}{5}$−35	$-\frac{3}{4}$−34	$-1$−1	$-\frac{3}{2}$−32	$-3$−3	$6$6	$3$3	$\frac{3}{2}$32	$1$1	$\frac{3}{4}$34	$\frac{3}{5}$35
$\frac{5}{x}$5`x`	$-1$−1	$-\frac{5}{4}$−54	$-\frac{5}{3}$−53	$-\frac{5}{2}$−52	$-5$−5	$10$10	$5$5	$\frac{5}{2}$52	$\frac{5}{3}$53	$\frac{5}{4}$54	$1$1

Example

To be convinced that, for example, the graph of $f(x)=\frac{2}{x}$f(x)=2x has exactly the same shape as the graph of $g(x)=\frac{1}{x}$g(x)=1x, but with a different scale, we can think of a new variable $u=\frac{x}{2}$u=x2 or, equivalently, $x=2u$x=2u. Now, the natural domain of the function $f(x)=\frac{2}{x}$f(x)=2x is the set of real numbers without zero and it is clear that as $x$x varies over this domain, $u$u must vary over exactly the same set of numbers.

So, with the function $f(x)=\frac{2}{x}$f(x)=2x, we can write $f(x)=\frac{2}{2u}=\frac{1}{u}=g(u)$f(x)=22u=1u=g(u). Thus, we see that $f$f and $g$g are the same function.

This idea is illustrated in the diagram below.

The values of $g(x)$g(x) are the same as the values of $f(u)$f(u).

A glance at all of the hyperbola graphs displayed above suggests that they are symmetrical about the line $y=x$y=x. We confirm this by noting that the equation $y=\frac{k}{x}$y=kx can be written as $k=xy$k=xy and it is clear that $x$x and $y$y can change positions without affecting the relation. We could swap the positions of the $x$x- and $y$y-axes and the graph should look the same.

Worked Examples

Question 1

Consider the function $y=\frac{2}{x}$y=2x

Complete the following table of values.

$x$x $-2$−2 $-1$−1 $\frac{-1}{2}$−12 $\frac{1}{2}$12 $1$1 $2$2

$y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Plot the graph.

Loading Graph...
In which quadrants does the graph lie?
$3$3
A
$2$2
B
$1$1
C
$4$4
D

Question 2

$Ursula$ wants to sketch the graph of $y=\frac{7}{x}$y=7x, but knows that it will look similar to many other hyperbolas.

What can she do to the graph to show that it is the hyperbola $y=\frac{7}{x}$y=7x, rather than any other hyperbola of the form $y=\frac{k}{x}$y=kx?

She can label the axes of symmetry.
A
She can label a point on the graph.
B
She can label the asymptotes.
C

Question 3

A graph of the hyperbola $y=\frac{10}{x}$y=10x is shown below. Given points $C$C$\left(-4,0\right)$(−4,0) and $D$D$\left(2,0\right)$(2,0), find the length of interval $AB$AB.

Loading Graph...