Hyperbolas Page 3a

Page 3a

Using the Standard Equations to Describe the Asymptotes
of Hyperbolas Centred at the Origin

Starting here, we will spend a fair bit of studying the lines of asymptote and what affects the slope of these lines. We do this because when it comes time to graph a hyperbola given its standard equation, knowing the slope of the asymptotes is necessary before you can accuratly draw the hyperbola.

In the standard equation of a hyperbola centred at the origin, we know that the values of a and b affect the slope of the asymptotes.

How can we use the standard equation to describe the asymptotes?

Just like the ellipse, the hyperbola has two axes of symmetry:
the horizontal axis of symmetry which is parallel to the x-axis and
the vertical axis of symmetry which parallel to the y-axis.

Depending on the standard equation, the vertices of a hyperbola can be located on the horizontal axis of symmetry or on the vertical axis of symmetry.

When the vertices of the hyperbola are on the horizontal axis, you have the following:
the distance between the centre of the hyperbola and one of the vertices is |a|.
there exists points on the vertical axis such that the distance between each point and the centre of the hyperbola is |b|.
When the vertices of the hyperbola are on the vertical axis:
the distance between the centre of the hyperbola and one of the vertices is |b|.
there exists points on the horizontal axis such that the distance between each point and the centre of the hyperbola is |a|.

After labeling these points and distances on the graph we can draw a special rectangle of width 2a and height 2b with the same centre as the hyperbola. See the figures below.

What do you notice about the asymptotes and the rectangle?
That's right! The asymptotes intersect the rectangle at its corners.
Drawing the rectangle shows us how to precisely determine the slopes of the asymptotes for any hyperbola:
the slope of line 1 is always rise/run = b/a
the slope of line 2 is always rise/run = b/-a or rise/run = -b/a

Check out how the standard equation of a hyperbola can be used to find the slope of the asymptotes algebraically.

Starting here, we will spend a fair bit of studying the lines of asymptote and what affects the slope of these lines. We do this because when it comes time to graph a hyperbola given its standard equation, knowing the slope of the asymptotes is necessary before you can accuratly draw the hyperbola.
In the standard equation of a hyperbola centred at the origin, we know that the values of a and b affect the slope of the asymptotes.
How can we use the standard equation to describe the asymptotes?
Just like the ellipse, the hyperbola has two axes of symmetry: the horizontal axis of symmetry which is parallel to the x-axis and the vertical axis of symmetry which parallel to the y-axis.
Depending on the standard equation, the vertices of a hyperbola can be located on the horizontal axis of symmetry or on the vertical axis of symmetry.
When the vertices of the hyperbola are on the horizontal axis, you have the following: the distance between the centre of the hyperbola and one of the vertices is \|a\|. there exists points on the vertical axis such that the distance between each point and the centre of the hyperbola is \|b\|.	When the vertices of the hyperbola are on the vertical axis: the distance between the centre of the hyperbola and one of the vertices is \|b\|. there exists points on the horizontal axis such that the distance between each point and the centre of the hyperbola is \|a\|.
After labeling these points and distances on the graph we can draw a special rectangle of width 2a and height 2b with the same centre as the hyperbola. See the figures below.

What do you notice about the asymptotes and the rectangle?
That's right! The asymptotes intersect the rectangle at its corners.
Drawing the rectangle shows us how to precisely determine the slopes of the asymptotes for any hyperbola: the slope of line 1 is always rise/run = b/a the slope of line 2 is always rise/run = b/-a or rise/run = -b/a
Check out how the standard equation of a hyperbola can be used to find the slope of the asymptotes algebraically.