Graphing Hyperbolas (k/x)
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
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.
AShe can label a point on the graph.
BShe 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.