Lesson

Hyperbolas can have either 0 or 1 \, x-intercepts. This is the point on the graph which touches the x-axis. We can find this by setting y=0 and finding the value of x. If the x-value is undefined, there is no x-intercept. For example, there is no x-intercept of y=\dfrac{1}{x}.

Similarly, hyperbolas can have either 0 or 1 \, y-intercept. This is the point on the graph which touches the y-axis. We can find this by setting x=0 and finding the value of y. If the y-value is undefined, there is no y-intercept. For example, there is no y-intercept of y=\dfrac{1}{x}.

Hyperbolas have a **vertical asymptote** which is the vertical line which the graph approaches but does not touch. For example, the vertical asymptote of y=\dfrac{1}{x} is x=0.

Hyperbolas also have a **horizontal asymptote** which is the horizontal line which the graph approaches but does not touch. For example, the horizontal asymptote of y=\dfrac{1}{x} is y=0..

Consider the function y = - \dfrac{1}{4 x}.

a

Complete the following table of values:

x | -3 | -2 | -1 | 1 | 2 | 3 |
---|---|---|---|---|---|---|

y |

Worked Solution

b

Sketch the graph.

Worked Solution

c

In which quadrants does the graph lie?

Worked Solution

Idea summary

The graph of an equation of the form y=\dfrac{a}{x-h}+k is a **hyperbola**.

Hyperbolas can have 0 or 1 **x-intercepts** and can have 0 or 1 **y-intercepts,** depending on the solutions to the equation.

Hyperbola have a **vertical asymptote** which is the vertical line that the graph approaches but does not intersect and a **horizontal asymptote** which is the horizontal line that the graph approaches but does not intersect.

A hyperbola can be **vertically translated** by increasing or decreasing the y-values by a constant number. So to translate y=\dfrac{1}{x} up by k units gives us y=\dfrac{1}{x} + k.

Similarly, a hyperbola can be **horizontally translated** by increasing or decreasing the x-values by a constant number. However, the x-value together with the translation must both be in the denominator. That is, to translate y=\dfrac{1}{x} to the left by h units we get y=\dfrac{1}{x+h}.

A hyperbola can be **scaled** by changing the value of the numerator. So to expand the hyperbola y=\dfrac{1}{x} by a scale factor of a we get y=\dfrac{a}{x}. We can compress a hyperbola by dividing by the scale factor instead. Note that for hyperbolas, vertically scaling is equivalent to horizontally scaling.

We can **reflect** a hyperbola about either axis by taking the negative of the y-values. So to reflect y=\dfrac{1}{x} about the x-axis gives us y=-\dfrac{1}{x}. Note that for hyperbolas, vertically reflecting is equivalent to horizontally reflecting.

The following applet demonstrates how a scale factor affects the shape of a hyperbola. Play with the applet below by dragging the sliders.

As the scale factor a increases in size, the hyperbola moves away from the axes. If a is negative, the hyperbola is reflected across the x-axis.

Consider the equation f(x) = \dfrac{3}{x}.

a

Sketch a graph of the function.

Worked Solution

b

Which of the following statements about the symmetry of the graph is true?

A

The graph is symmetric about the x-axis.

B

The graph is symmetric about the y-axis.

C

The graph has no symmetry.

D

The graph is rotationally symmetric about the origin.

Worked Solution

c

Find an expression for f(-x).

Worked Solution

Idea summary

Hyperbolas can be transformed in the following ways (starting with the hyperbola defined by y=\dfrac{1}{x}):

**Reflected**about the y-axis: y=-\dfrac{1}{x}**Vertically translated**by k units: y=\dfrac{1}{x} + k**Horizontally translated**by h units: y=\dfrac{1}{x-h}**Scaled**by a scale factor of a: y=\dfrac{a}{x}

Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations.