UK Secondary (7-11) Graphing Hyperbolas (k/x)
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}$1x $-\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}$3x $-\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}$5x $-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

1. 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 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Plot the graph.

3. In which quadrants does the graph lie?

$3$3

A

$2$2

B

$1$1

C

$4$4

D

$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?

1. She can label the axes of symmetry.

A

She can label a point on the graph.

B

She can label the asymptotes.

C

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.