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?

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.

Return Home