Lessons 33–34 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
33
Graphs of simple functions given parametrically
Textbook, Part 2: pp. 51–52
GoalGiven a parametric description x = φ(t), y = ψ(t), eliminate t and find the equation of the point’s trajectory.
New words
parameter · parametrtrajectory · trayektoriyaequation of a circle · aylana tenglamasiellipse · ellips
Explanation
The coordinates of a moving point may depend on time (or another parameter t): x = φ(t), y = ψ(t). The set of points (x, y) obtained for all t in a given interval is the graph of the parametrically given function, or the trajectory of the point. In linear cases we express t from one equation and substitute it into the other, obtaining a relation between x and y. When sine and cosine appear, we isolate sin t and cos t and use sin²t + cos²t = 1. The system x = a + R cos t, y = b + R sin t gives a circle: (x − a)² + (y − b)² = R², with centre (a, b) and radius R. The system x = p cos t, y = q sin t (p, q > 0) gives an ellipse x²/p² + y²/q² = 1 with semi-axes p and q. We can also draw the graph by giving t several values and marking the points.
Worked examples
x = 3t − 2, y = t + 4. From the first, t = (x + 2)/3; substituting into the second gives y = (x + 2)/3 + 4 = (x + 14)/3, that is x − 3y + 14 = 0 — a straight line.
x = 2 + 4 cos t, y = 1 + 4 sin t. cos t = (x − 2)/4, sin t = (y − 1)/4; from sin²t + cos²t = 1 we get (x − 2)² + (y − 1)² = 16: a circle with centre (2, 1) and radius 4.
Class activity
In your notebook mark the points of x = 5 cos t, y = 5 sin t for t = 0°, 30°, 60°, 90°, 120°, … join them with a smooth curve and name the shape.
Practice
1
What is a parametrically given function?
Both x and y are given as functions of the parameter t: x = φ(t), y = ψ(t).
2
Eliminate t from x = t + 3, y = 2t − 1 and write y in terms of x.
y = 2x − 7
3
What curve is x = 6 cos t, y = 6 sin t?
A circle centred at the origin with radius 6: x² + y² = 36.
4
Why do we use sin²t + cos²t = 1 for a circle?
Because writing x and y through cos t and sin t, the sum of their squares no longer depends on t and equals the constant R².