Parametric equations
Use one parameter to describe a point, trace a curve and model a path.
Start with paired coordinates, elimination, circles and ellipses. Then explore curve sketching. Then study intersections, tangents and chords. Apply the ideas to straight-line, projectile and periodic motion. Compare crossing paths and collisions. Finish with fixed shape models, dimensions and mean gradients.
- Understanding parametric equationsUnderstand parametric equations as two coordinate rules for one parameter. Build a table, trace a curve manually and distinguish a point, its parameter and the direction of travel.
- Eliminating the parameterConvert parametric equations to Cartesian form using substitution, powers, exponentials and logarithms. Preserve the correct branch, coordinate ranges, endpoints and excluded points.
- Parametric circles and ellipsesEliminate sine and cosine to identify circles and ellipses. Find centres, radii, semiaxes, restricted arcs, direction and circular arc length using original worked examples.
- Eliminating trigonometric parametersEliminate trigonometric parameters using double-angle, triple-angle, compound-angle and reciprocal identities. Select the correct radical branch and preserve every original domain restriction.
- Sketching parametric curvesSketch parametric curves using domains, intercepts, symmetry, direction and limiting behaviour. Explore a self-intersecting loop and separate rational branches, with original worked practice.
- Parametric curve intersectionsFind where parametric curves meet axes, lines, circles and other curves. Substitute both coordinate rules, solve for the parameter, filter its domain and count distinct intersection points.
- Tangents and chords of parametric curvesFind chord equations from parametric points and use a quadratic discriminant to find tangents. Explore parallel lines, intersection counts and cases where repeated parameter roots need care.
- Parametric models of straight-line motionModel constant-velocity motion with parametric equations. Distinguish position, displacement, distance and speed; find a path equation, valid time interval and direction with original worked practice.
- Projectile models with parametric equationsUse given parametric projectile equations to find landing time, horizontal range, maximum height and obstacle clearance. Keep ground height, launch displacement and the valid flight interval distinct.
- Periodic parametric motionModel circular and figure-of-eight motion using parametric equations. Find full periods, return times and circular distances, and distinguish revisiting a point from repeating the whole motion.
- Intersecting paths and particle collisionsDistinguish crossing paths from simultaneous collisions. Solve parametric position equations with separate parameters or a shared clock, check start delays and domains, and calculate separation.
- Parametric shape modelsUse parametric equations to model fixed shapes and profiles. Find widths, heights, depth, mean gradients and physical restrictions for trigonometric, logarithmic and exponential models.