It is possible to graph the more generalised function \dfrac{k}{x} by constructing a table of values having first specified a value for the parameter k. The shape of the graph will be a hyperbola and the effect of changing k is to change the scale of the graph.

These properties are illustrated in the following diagram where the graph of y=\dfrac{1}{x} is shown in green, y=\dfrac{3}{x} is shown in purple and y=\dfrac{5}{x} is shown in blue.

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This diagram illustrates the properties above where the graph of y=\dfrac{1}{x} is shown in green, y=\dfrac{3}{x} is shown in purple and y=\dfrac{5}{x} is shown in blue.

If drawing these graphs by hand, it is easier to construct tables of values like the following. Notice that the x-values have been restricted to values between -5 and 5.

x	-5	-4	-3	-2	-1	\dfrac{1}{2}	1	2	3	4	5
\dfrac{1}{x}	-\dfrac{1}{5}	-\dfrac{1}{4}	-\dfrac{1}{3}	-\dfrac{1}{2}	-1	2	1	\dfrac{1}{2}	\dfrac{1}{3}	\dfrac{1}{4}	\dfrac{1}{5}
\dfrac{3}{x}	-\dfrac{3}{5}	-\dfrac{3}{4}	-1	-\dfrac{3}{2}	-3	6	3	\dfrac{3}{2}	1	\dfrac{3}{4}	\dfrac{3}{5}
\dfrac{5}{x}	-1	-\dfrac{5}{4}	-\dfrac{5}{3}	-\dfrac{5}{2}	-5	10	5	\dfrac{5}{2}	\dfrac{5}{3}	\dfrac{5}{4}	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.

Examples

Example 1

Consider the function f(x) = \dfrac{2}{x}.

Complete the following table of values:

x	-2	-1	-\dfrac{1}{2}	\dfrac{1}{2}	1	2
y

Worked Solution

Create a strategy

Substitute the values from the table into the equation.

Apply the idea

For x=-2:

\displaystyle y	\displaystyle =	\displaystyle \dfrac{2}{x}
	\displaystyle =	\displaystyle \dfrac{2}{-2}	Substitute x=-2
	\displaystyle =	\displaystyle -1	Evaluate

Similarly, by substituting the remaining x-values into y=-\dfrac{2}{x}, we get:

x	-2	-1	-\dfrac{1}{2}	\dfrac{1}{2}	1	2
y	-1	-2	-4	4	2	1

Plot the graph.

Worked Solution

Create a strategy

Plot the points in the table of values and draw the curve passing through each plotted point.

Apply the idea

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The ordered pairs of points to be plotted on the coordinate plane are (-2,-1), (-1,-2), (-\dfrac{1}{2},-4), (-\dfrac{1}{2},4), (1,2), and (2,1) which are plotted on the graph.

The curve of the hyperbola y=-\dfrac{2}{x} must pass through each of the plotted points and approaches the horizontal asymptote y=0 and the horizontal asymptote x=0.

In which quadrants does the graph lie?

Worked Solution

Apply the idea

From the graph in part(b), we can see that the hypeybola lies in quadrant 1 and quadrant 3.

Example 2

Ursula wants to sketch the graph of y=\dfrac{6}{x}, 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=\dfrac{6}{x}, rather than any other hyperbola of the form y=\dfrac{k}{x}?

She can label the axes of symmetry.

She can label the asymptotes.

She can label a point on the graph.

Worked Solution

Create a strategy

Compare the graph of y=\dfrac{7}{x} and the function \dfrac{1}{x} in terms of its axes of symmetry, asymptotes and points.

Apply the idea

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Here is the graph of y=\dfrac{1}{x} is shown in green, and y=\dfrac{6}{x} is shown in blue.

If we label the axes of symmetry, both are symmetrical at about y=x because they also have the same asymptotes. So if we label the asymptotes, we have y=0 and x=0.

So labelling the axes of symmetry and asymptotes will not show which is the hyperbola y=\dfrac{6}{x},

If we label a point on the graph, we can tell which hyperbola is which.

So the correct the answer is option B.

Example 3

The hyperbola y=\dfrac{10}{x} has been graphed. Given points C(-1,0) and D(0.5,0), find the length of interval AB.

Worked Solution

Create a strategy

Find the y-values on the curve that correspond to points C and D, then subtract the second value from the first to find the required length.

Apply the idea

To find the y-coordinate of point A, we will substitute x=-1 into the equation.y=\dfrac{10}{-1}=-10

For the y-coordinate of point B, we will substitute x=0.5 into the equation.y=\dfrac{10}{0.5}=20

So we have the length of AB:

\displaystyle \text{length}	\displaystyle =	\displaystyle 20-(-10)	Subtract the y-values
	\displaystyle =	\displaystyle 30	Evaluate

Idea summary

The function \dfrac{1}{x} is called a rectangular hyperbola, or simply a hyperbola.

Logarithmic functions

Logarithms are another useful non-linear function to be able to graph.

Logarithmic functions of the form: y = a\log_{b} x+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.

This image show a graph of logarithmic function y = log b x. Ask your teacher for more information.

The four main things to look out for are:

The vertical asymptote x = 0,
The point where x = 1,
The point where x=b, and
The direction the ends of the graph are pointing.

Also:

The value of a determines the dilation and will be negative if the graph has been vertically reflected.
The value of b is the amount the graph has been translated upwards. In this chapter, the logarithmic functions used will be base 10, that is, b=10.

Exploration

Use the following applet to explore the logarithmic graphs and to see what happens to the graph as the values for a and c are changed by dragging the sliders.

Loading interactive...

Changing c changes the steepness of the graph. Changing a changes the steepness of the graph and negative values of a flip the curve horizontally.

Note that for certain values of a and c, the graph of the logarithm cuts off because of technology's limitations - not because the graph of a logarithm actually cuts off.

Examples

Example 4

Consider the function y = \log_{4} x, the graph of which has been sketched below.

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Complete the following table of values.

x	\dfrac{1}{16}	\dfrac{1}{4}	4	16	256
y

Worked Solution

Create a strategy

Use the fact that y= \log_{b} x, if x = b^{y}.

Apply the idea

Substitute each of the x-values in the table into the equation and evaluate.

For x=\dfrac{1}{16}:

\displaystyle y	\displaystyle =	\displaystyle \log_{4} x
	\displaystyle =	\displaystyle \log_{4} \dfrac{1}{16}	Substitute x=\dfrac{1}{16}
	\displaystyle =	\displaystyle \log_{4} 4^{-2}	Write x in exponential form
	\displaystyle =	\displaystyle -2	Evaluate

Similarly, by substituting the remaining x-values into \log_{4} x, we get:

x	\dfrac{1}{16}	\dfrac{1}{4}	4	16	256
y	-2	-1	1	2	4

Determine the x-value of the x-intercept of \log_{4} x.

Worked Solution

Create a strategy

Use the graph and find the x-value in which the graph crosses the x-axis.

Apply the idea

The x-value of the x-intercept is 1.

How many y-intercept does log_{4} x have?

Worked Solution

Create a strategy

Count the number of times the graph crosses the y-axis.

Apply the idea

The graph only approaches but never crosses the y-axis. So \text{Number of } y \text{-intercepts} = 0

Determine the x-value for which \log_{4} x=1.

Worked Solution

Create a strategy

Use the fact that y= \log_{b} x, if x = b^{y}.

Apply the idea

\displaystyle 1	\displaystyle =	\displaystyle \log_{4} x
\displaystyle x	\displaystyle =	\displaystyle 4^{1}	Use y= \log_{b} x if x = b^{y}
	\displaystyle =	\displaystyle 4	Evaluate

Reflect and check

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We can also use the graph to find the xvalue.

\log_{4} x=1 means that y=1 and we can see from the graph that the corresponding xvalue for y=1 is 4. So x=4

Idea summary

Logarithmic functions of the form: y = a\log_{b} x+c are varied and can look different from one another.

The following are the things to look out for in terms the of transformations of the basic logarithmic function:

The vertical asymptote x = 0,
The point where x = 1,
The point where x=b, and
The direction the ends of the graph are pointing.
The value of a determines the dilation and will be negative if the graph has been vertically reflected.
The value of b is the amount the graph has been translated upwards. In this chapter, the logarithmic functions used will be base 10, that is, b=10.

9.02 Hyperbolic and logarithmic functions

Ideas

Hyperbolic function

Examples

Example 1

Example 2

Example 3

Logarithmic functions

Exploration

Examples

Example 4

Outcomes

U2.AoS3.5

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