Parametric Conditions

by

Priscilla Alexander

This write-up is for students who are learning about the effects of integers on the graphs formed by parametric equations.

If given

What happens when a and b are the same positive integers

What happens when a is a positive integer and b is a negative integer (and vica versa)

What happens when a and b are the same negative integers

What happens when a and b are different integers, where

When a and b are the same a circle is formed.

When a and b are the same number but one is negative and the other is positive, a circle is still formed.

See Graph

In general when a and b are the same integers between -3 and 3 a circle will always appear that has a radius of one. This happens because the ratio of a/b in this case reduces to one.

When a and b are different integers, say a is two and b is one, then a rotated parabola is formed.

In general, when the ratio of a/b reduces to two a rotated parabola will always be formed. For example, if a equals 4 and b equals 2 then the rotated parabola will be formed.

When a and b are different integers, say a is one and b is two, then a bow tie looking image appears.

In general when the ratio of a/b reduces to 1/2 this image will always be formed.

When a and b are different integers, say a is equal to 2 and b is equal to 3, the image below is formed.

In general when the ratio of a/b reduces to 2/3 the image will always appear like the above.

When a and b are different integers, say a is equal to 1 and b is equal to 3, then an image that is reflected about the x-axis appears.

In general, when the ratio a/b reduces to 1/3, then the above image appears.

When a is equal to 3 and b is equal 1, then the graph is the same as a equals 1 and b equals 3, but is reflected about the y-axis.

In general this happens for all integers that are odd when a or b is one and the other interger is not one. For example, if a is equal to 5 and b is equal to 1, then the graph of a is equal to 1 and b is equal to five will be reflected on the opposite axis.

When a is an even integer and b is one, then the amplitude of the graph will always be

In conclusion, integers make the graphs of parametric equations act in many ways. Some of the graphs are reflected or rotated. The ratio of a and b also have an effort on the graph. In addition, odd and even integers make the graph behave in a different way then the rest.