Hyperbolas Page 2a

Page 2a

Summary of the Standard Hyperbola Equations

Let's review how we arrived at the standard equation of a hyperbola centred at the origin.

Opening Direction left and right up and down

Graphs

Begin with the equations: Ax² - By² = P
where A, B and P are positive real numbers.* -Ax² + By² = P
where A, B and P are positive real numbers.*

*Note that on the previous page A and B did not always denote positive numbers. We make this distinction here to emphasise the differences between the two equations. Now we have that -A < 0 and -B < 0.

Divide both equations by P:

Rewrite the equations:

In both cases P/A and P/B are always positive values. This means that:
We can find a real number a such that P/A = a²
We can find a real number b such that P/B = b².

Make the substitutions P/A = a² and P/B = b² to arrive at the standard equations:

Information about the graph:
The centre of the hyperbola is (0, 0)

The horizontal axis of symmetry is the x-axis; the vertical axis of symmetry is the y-axis.

The vertices are located on the x-axis at (a, 0) and (-a, 0)

a and b both affect the slope of the asymptotes.

The centre of the hyperbola is (0, 0)

The horizontal axis of symmetry is the x-axis; the vertical axis of symmetry is the y-axis.

The vertices are located on the y-axis at (0, b) and (0, -b).

a and b both affect the slope of the asymptotes.

Let's review how we arrived at the standard equation of a hyperbola centred at the origin.
Opening Direction	left and right	up and down
Graphs
Begin with the equations:	Ax² - By² = P where A, B and P are positive real numbers.*	-Ax² + By² = P where A, B and P are positive real numbers.*
*Note that on the previous page A and B did not always denote positive numbers. We make this distinction here to emphasise the differences between the two equations. Now we have that -A < 0 and -B < 0.
Divide both equations by P:
Rewrite the equations:
In both cases P/A and P/B are always positive values. This means that: We can find a real number a such that P/A = a² We can find a real number b such that P/B = b².
Make the substitutions P/A = a² and P/B = b² to arrive at the standard equations:
Information about the graph:	The centre of the hyperbola is (0, 0) The horizontal axis of symmetry is the x-axis; the vertical axis of symmetry is the y-axis. The vertices are located on the x-axis at (a, 0) and (-a, 0) a and b both affect the slope of the asymptotes.	The centre of the hyperbola is (0, 0) The horizontal axis of symmetry is the x-axis; the vertical axis of symmetry is the y-axis. The vertices are located on the y-axis at (0, b) and (0, -b). a and b both affect the slope of the asymptotes.