Assignment 10

by

Jackson Huckaby

Parametric Equations

1. Graph x=cos(t) and y=sin(t) for 0<t<6.28

How can we manipulate this equation to explore other graphs?

If you have graphing calculator you can follow this link to explore on your own.

One method is to insert slider values into the equations by using n as a variable.

To start with I have inserted an n inside function seen here:

With n increasing from 1 to 10:

What if we put the slider in the sin function?

With n increasing now from 1 to 10:

What changed? What we know about parametric curves is that our x value is attached to the cos function and the y value is attached to the sin function.

We start with a circle. When manipulating the cos value we have a rapidly changing x value causing the horizontal squiggles.

When manipulating our sin function we have a rapidly changing y value causing vertical squiggles.

So what if we change what we manipulate and manipulate a and b as such:

We can use the same approach and input sliders into graphing calculator. We will start with manipulating our value of a while leaving b as 1.

So once again, with n as 1, 3, and 5:

What we notice first is that as n increases, a horizontal stretch happens to our original curve.

If we look closer at our graphs we can note the values specifically. When n is 1 then our x intercepts are 1 and -1. When n is 3 our x intercepts are 3 and -3,

when n is 5 our x intercepts are 5 and -5.

we could predict that by manipulating b we will be simply changing the y intercept.

Lets test.

This time lets try using values of 1, 2, and 3.

We can see that we have now achieved a vertical stretch!

Write parametric equations of a line segment through (7,5) with a slope of 3. Graph the line segment using your equations.

First we can find an equation for this line using our standard slope intercept form:

y=mx+b

Plugging in we get that

5=3(7)+b

5=21+b

-16=b

This is our y intercept so our line would be y=3x-16.

Now we have to transform our equation into a parametric equation:

Step 1 is to set x=t.

Step 2 is substituting t into the second equation for x and we get y=3t-16.

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