Parametric Equations Plane Curves Parametric Equations Let x

Parametric Equations Plane Curves

Parametric Equations • Let x = f(t) and g(t), where f and g are two functions whose common domain is some interval I. The collection of points defined by • (x, y) = (f(t), g(t)) • is called a plane curve.

Parametric Equations • The equations • x = f(t) y = g(t) • where t is in I, are called parametric equations of the curve. The variable t is called a parameter.

Parametric Equations • Parametric equations are used to describe movement along a curve. • Arrows are drawn along the curve in order to show direction or orientation along the curve as t varies from a to b.

Discussing a Curve • Discuss the curve defined by the parametric equations • x = 3 t 2 y = 2 t • -2 ≤ t ≤ 2

Graphing Parametric Equations Using a Graphing Calculator • 1. Set the mode to PARametric. • 2. Enter x(t) and y(t). • 3. Select the viewing window. In addition to setting Xmin, Xmax, Xscl, and so on, the viewing window in parametric mode requires values for the parameter t and an increment setting for t (Tstep) • 4. Graph

Graphing Parametric Equations Using a Graphing Calculator • Graph • x = 2 cos t • 0≤t≤p y = 3 sin t

Finding the Rectangular Equation of a Curve • Find the rectangular equation of the curve whose parametric equations are • x = a cos t y = a sin t • where a > 0 is a constant.

Finding the Rectangular Equation of a Curve • When given trig functions, we use the identity • cos 2 t + sin 2 t = 1 • (x/a)2 + (y/a)2 = 1 • x 2 + y 2 = a 2 • As the parameter t increases, the corresponding points are traced in a counterclockwise direction around the circle.

Finding the Rectangular Equation of a Curve • If the function is not a trig function: • 1. Solve both equations for t • 2. Set the two equations equal to each other (transitive property of equality) • 3. Solve for y.

Projectile Motion • We can use parametric equations to describe the motion of an object (curvilinear motion). • When an object is propelled upward at an inclination q to the horizontal with initial speed v 0, the resulting motion is called projectile motion.

Projectile Motion • • Parametric equations of a projectile: x = (v 0 cos q)t y = -½gt 2 + (v 0 sin q)t + h where t is the time and g is the constant acceleration due to gravity (32 ft/sec 2 or 9. 8 m/sec 2).

Projectile Motion • Examples p. 724 and 725. • Talladega 500 example

Finding Parametric Equations • Find parametric equations for the equation: • y = x 2 – 4 • Remember there are two equations. • Let x = t • y = t 2 – 4 (Yes it is that easy)