Twelfth Grade · Math · week 31

Parametric equations

Graph a curve defined parametrically, eliminate the parameter to find a rectangular equation, and apply parametric equations to projectile motion.

A parametric curve gives both x and y as separate functions of a third variable, the parameter (usually t, often standing for time): x=f(t), y=g(t). Each value of t produces one point (x,y), and letting t sweep through its domain traces out the whole curve -- with the advantage that the curve can show WHEN the object was where, not just where the path went.

Eliminating the parameter means solving one equation for t and substituting into the other, producing a single rectangular equation in x and y alone -- useful for recognizing the curve's familiar shape, though it throws away the timing information. Projectile motion is the signature application: x(t) = (v0*cos(theta))*t and y(t) = (v0*sin(theta))*t - 16t^2 (in feet, with t in seconds), where v0 is launch speed and theta is launch angle above the horizontal.

📋 For the teacher
Teach it (10–15 min):
  • Plot a handful of (x,y) points at increasing t-values by hand first, so the curve's direction of travel (not just its shape) becomes visible.
  • Practice eliminating the parameter by always solving the SIMPLER of the two equations for t first, whichever that happens to be.
  • Walk the projectile formula piece by piece, naming v0, theta, and the constant 16 (half of Earth's gravitational acceleration in feet) before plugging in numbers.
  • Find time of flight BEFORE range or max height every time -- both later quantities depend on knowing when the flight ends or reaches its midpoint.
Talk about it:
  • What information does a parametric curve preserve that a rectangular equation for the same path loses?
  • Why does eliminating the parameter sometimes hide important information about the motion?
  • Why does maximum height occur at exactly HALF the total time of flight for projectile motion launched and landing at the same height?

Mastery looks like: They graph and eliminate the parameter from a parametric curve, and set up and solve projectile motion problems for time of flight, range, and maximum height.

Common stumbles: Eliminating the parameter by solving the harder equation for t first; forgetting that maximum height occurs at half the total flight time, not at the start.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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