The hyperbola
The function $\frac{1}{x}$1x is called a rectangular hyperbola, or simply a hyperbola.
It is possible to graph the more generalised 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 drawing these graphs by hand, it is easier to construct tables of values like the following. Notice that the $x$x-values have been restricted 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 |
It is also useful to be able to sketch these graphs using technology. The following provides an example of how to do this using CAS.
Practice questions
Question 2
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 3
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 4
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.
Logarithmic functions
Logarithms were discussed in a previous section on magnitudes and logarithmic scales, and are another useful non-linear function to be able to graph.
Logarithmic functions of the form:
$y=a\log_bx+c$y=alogbx+c
are varied and can look different from one another. Fortunately, they all have the same basic components and can even be thought of in terms of transformations of the basic logarithmic function, shown below.
$f\left(x\right)=\log_bx$f(x)=logbx
The four main things to look out for are:
- The vertical asymptote $x=0$x=0,
- The point where $x=1$x=1,
- The point where $x=b$x=b, and
- The direction the ends of the graph are pointing.
Also:
- The value of $a$a determines the dilation and will be negative if the graph has been vertically reflected.
- The value of $b$b is the amount the graph has been translated upwards. In this chapter, the logarithmic functions used will be base $10$10, that is, $b=10$b=10.
To explore logarithmic graphs further, experiment with the applet below to see what happens to the graph as the values for $a$a and $c$c are changed.
Note that for certain values of $a$a and $c$c, the graph of the logarithm cuts off because of technology's limitations–not because the graph of a logarithm actually cuts off.
Practice Questions
question 1
Consider the function $y=\log_4x$y=log4x, the graph of which has been sketched below.
Complete the following table of values.
$x$x $\frac{1}{16}$116 $\frac{1}{4}$14 $4$4 $16$16 $256$256 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Determine the $x$x-value of the $x$x-intercept of $y=\log_4x$y=log4x.
How many $y$y-intercepts does $\log_4x$log4x have?
Determine the $x$x value for which $\log_4x=1$log4x=1.
Question 2
Use the applet below to describe the transformation of $g\left(x\right)=\log_{10}x$g(x)=log10x being transformed to $f\left(x\right)=a\log_{10}x$f(x)=alog10x, where $a>1$a>1.
$f\left(x\right)$f(x) is the result of a compression vertically towards the $x$x-axis.
A$f\left(x\right)$f(x) is the result of a dilation vertically away from the $x$x-axis.
B$f\left(x\right)$f(x) is the result of a translation vertically upwards.
C$f\left(x\right)$f(x) is the result of a translation vertically downwards.
D