Accelerated Math 3

Parametric Equations

Parametric Equations

Useful to introduce 3rd variable to represent curve in a plane (x, y and

t).

Useful for modeling the path of an object.

• Parameter is 3rd variable t

• Parametric Equation can be written with both x and y as functions of t.

• Allows us to determine time t an object is at a given point (x, y).

• Consider the path followed by an object that is propelled into the air at an angle of 45o. If the initial velocity of the object is 48 feet per second, it can be shown that the object follows the parabolic path y = -(x2/72) +x.

• This equation does not tell the whole story. Although it does tell us WHERE the object has been, it doesn’t tell us WHEN the object was at a given point (x,y) on the path.

What does all of that mean?

• To determine this time, you can introduce a third variable t, which is called a PARAMETER.

• It is possible to write both x and y as a functions of t to obtain the parametric equations

• From this set of equation you can determine that at time t=0, the object is at the point (0,0). Similarly, at time t=1, the object is at the point

t22416t- y and 224 2 tx

16)-224 ,224(

Plane Curve

• If f and g are continuous functions (graph traced without lifting pencil) of t on an interval I, the set of ordered pairs (f(t), g(t)) is a plane curve C.

• x = f(t) and y = g(t)

Parametric Equations

• Ellipse (a,b = x-and y-radii; h,k = center) x = h + a cost y = k + b sint

• Circle x = a cost y = a sint

Parametric Equations

• Hyperbola (a,b = x-andy-radii; h,k = center)Opening in x-direction:x = h + a secty = K + b tant

Opening in y-direction:x = h + a tanty = k + b sect

1a. Graph x=5cos t and y = 7sin t in your calculator.

b. Use the Pythagorean Property for Cosine and Sine to eliminate the parameter t.

c. Explain how you know that the graph is an ellipse or a circle.

2. Graph x=6+5cos t and y = -3+7sin t.

3. Write the parametric equation for the ellipses.

4a. Name the conic section. Identify all components.

b. Graph.

13

4

7

222

yx

4c. Transform the given equation into the form: ax2+bxy+cy2+dx+ey+f=0

13

4

7

222

yx

5a. Name the conic section. Identify all components. x = -2 + 3 tan t and y = 1 + 4 sect

b. Graph.

6a. Name the conic section. Identify all components. x = -6+1.5(y-3)2

b. Graph.

7a. Write the Cartesian equation.

b. Write the parametric equation.