**Parametric Equations**

**By**

**Jeffrey R. Frye**

In this assignment, I explored the general parametric
equation shown below. Initially, I
varied all variables **a**, **b**, and **k**. By exploring what
occurred when these changes were made, I was able to see that the **a** value changes the location of the x
value, the **b** changes the location of
the y value, and changing the **k**
value changes the slope of the curve graphed by these parametric equations.

Compare the graphs shown below. In the first illustration, a=2, b=1, and k=2.

In the second illustration, a=3, b=4, and k=3.

When **a** and **b** are kept constant and the **k** is varied, a set of curves that go
through a common point are graphed. This
common point is shown to be (a, b) when t=0.
Each of the curves has the slope that is the corresponding value of **k**.
The curves shown are for a=2 and b=3.
The value for **k **is varied
from 1 to 4. The illustration shows that
the curves all intersect at point (2, 3) and have slopes that correspond to the
value for **k**.

An equation for each of these curves can be derived from the
parametric equations, by using the point slope form. For example, where the **k** is 4, y-3=4(x-2) will result in the linear equation of
y=4x-5. From the graph we can see that
the y intercept for the green curve is -5 and the slope of the line is 4.

To explore using Graphing Calculator, click **here.**

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