Unit 09 · Lesson 05 · Week 27 · Vectors, Parametric Equations & Motion
Parametric Equations and Eliminating the Parameter
Describe motion and curves with a shared parameter, eliminate parameters when possible, and interpret direction, velocity, and time. In this lesson, the central focus is parametric equations and eliminating the parameter. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.
Learning intentions
- Interpret the defining quantities and notation used in parametric equations and eliminating the parameter.
- Construct or analyze at least two representations related to parametric equations and eliminating the parameter.
- Show symbolic work with correct domains, units, intervals, or angle measures.
- Explain why the selected method is valid and check the result against another representation.
Essential question
How do vectors and parameters represent magnitude, direction, components, paths, velocity, and two-dimensional motion?
Multiple-representation model
Describe motion and curves with a shared parameter, eliminate parameters when possible, and interpret direction, velocity, and time.
x=f(t),\quad y=g(t)v_x=\frac{\Delta x}{\Delta t}v_y=\frac{\Delta y}{\Delta t}
Worked reasoning
Misconception check: The Cartesian equation may describe the same set of points but lose the parametric direction and time information.
Communication and reasoning
Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.
Khaemenes Academy · Open-Age Precalculus
Parametric Equations and Eliminating the Parameter
Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.
Name
Date
Eliminate t from x=2t+1, y=3t−4.
For x=cos t, y=sin t, identify the curve.
At t=2, find the point for x=t², y=3t.
What information is lost when eliminating a parameter?
A projectile has x=20t, y=30t−5t². Find x at t=3.
For x=4cos t, y=2sin t, identify the curve.