# EMT 668 - ASSIGNMENT 10 Parametric Curves

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## x = cos(t) y = sin(t)

for t within the interval [0,2pi].

### How would you change the parametric equations to explore other graphs?

First let us consider the original parametric curve for t within the interval [0,2Pi]:

x = cos (t)
y = sin (t)

We can consider a number of variations to this curve. First let us consider multiplying our parameter t by a constant in both x = cos (t) and y = sin (t). Let's consider 2, 3 and -1.

### x = cos (3 t) y = sin (3 t) x = cos (- t) y = sin (- t)

Let us consider putting multiples on only one of the funtions.

### x = cos (5 t) y = sin (t)

It seems that the constant multiple within x = cos (a t), for even a, creates additional dips in our graph. For odd a, we create additional complete cycles in the graph.

Multiplying by a constant within y = sin (a t) will have a different effect. Consider the following graphs.

### x = cos (t) y = sin (3 t)

This manipulation creates a "Bow Tie" graph. The larger the constant multiple, the more rotations we have.

Let's consider the addition of a constant to t. We will not consider a parametric curve where the change in both x = cos (t) and y = sin (t) are the same. Consider the following graphs:

### x = cos (t-1) y = sin (t)

By adding or subtracting a constant to t within x = cos (t + a), we create an ellipse. This investigation leads me to believe, the larger the constant you add to t, the narrower your ellipse becomes. Subtracting a constant reflects the graph about the y-axis.

Let's consider adding and subtracting a constant to t within y = sin (t +a). Consider the following:

### x = cos (t) y = sin (t-1)

It seems that adding 1 within y = sin (t+a) has the same effect as subtracting 1 within
x = cos (t - a) and subtracting 1 within y = sin (t - a) has the same effect as adding 1 within
x = cos (t+ a).

These are just a few of the many explorations that can be done on this set of parametric equations.