Unit 11: Parametric Equations
Tracing a curve as a moving particle with time as its own separate parameter, and switching that motion back into a single x-y equation.
5 topics · 20 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Defining Curves Parametrically
- How it worksHow do two separate equations x(t) and y(t) trace out a single curve as t varies?
- Worked exampleFind the point traced by x(t) = t2 and y(t) = 2t + 1 when t = 3.
- Common mistakeWhy can two different parametrizations trace the exact same curve at completely different speeds?
Your turn1 of 2
Find the y-coordinate of the point traced by x(t) = 4t2 and y(t) = 6t + 8 when t = 2.Eliminating the Parameter
- How it worksHow do you solve one parametric equation for t and substitute it into the other to eliminate the parameter?
- Worked exampleFind the rectangular equation for x(t) = t + 1 and y(t) = t2 - 2 by eliminating the parameter.
- Common mistakeWhy does eliminating the parameter sometimes lose information about which direction the curve was traced?
- Ties togetherHow does eliminating the parameter from x(t) = cos(t) and y(t) = sin(t) connect to the circle parametrized next?
Your turn1 of 2
Find y in terms of x for x(t) = t + 4 and y(t) = t2 + 6, by eliminating the parameter.Parametrizing Lines and Circles
- How it worksHow do x(t) = cos(t) and y(t) = sin(t) trace a full circle exactly once as t runs from 0 to 2pi?
- Worked exampleFind a parametrization for the line segment from (1,2) to (5,8).
- Worked exampleFind a parametrization for a circle of radius 3 centered at (2,-1).
- Common mistakeWhy isn't there only one correct way to parametrize the same circle or line?
Your turn1 of 2
Find the y-coordinate of a parametrization for the line segment from (3,3) to (12,8) at t = 0, using x(t) = x1 + t·(x2-x1), y(t) = y1 + t·(y2-y1).Parametric Motion and Projectile Paths
- How it worksHow do horizontal motion at constant speed and vertical motion under gravity combine into one parametric path?
- Worked exampleFind the parametric equations for a ball launched at 40 m/s at 30 degrees above horizontal, and find its position at t = 2 seconds.
- Worked exampleFind the time when that projectile lands, and how far it traveled horizontally.
- Common mistakeWhy doesn't the ball's horizontal speed change at all during the entire flight, even as it rises and falls?
- Going furtherWhat happens to the projectile's parametric path when the launch angle increases from 30 to 60 degrees with the same speed?
Your turn1 of 2
A ball is launched at 38 m/s at 55 degrees above horizontal. Write its height as a function of time t (use g=9.8).Direction and Speed Along a Parametric Path
- How it worksHow do the derivatives dx/dt and dy/dt reveal the direction a parametric curve is being traced?
- Worked exampleFind the direction of motion at t = 1 for x(t) = t2 and y(t) = t3.
- Common mistakeWhy doesn't the shape of a parametric curve alone tell you which direction it's being traced?
- Ties togetherHow do dx/dt and dy/dt connect to the horizontal and vertical motion of a projectile in flight?
Your turn1 of 2
Write dy/dx (the direction's slope) as a function of t, for x(t) = 4t2 and y(t) = 4t3.