A circle is described by x = cos(t), y = sin(t) for 0 ≤ t ≤ 2π. A student eliminates t to get x² + y² = 1 and says the two representations are equivalent. What information does the rectangular form lose?
AThe equation is only valid for positive x and y values
BThe circle's center and radius
CThe direction and starting point of traversal around the circle
DThe fact that the curve is closed
The rectangular form x² + y² = 1 describes the shape but nothing about how it is traversed. The parametric form tells you traversal starts at (1, 0) when t = 0 and proceeds counterclockwise. Direction of motion, starting point, and speed are all lost when you eliminate the parameter — which is why parametric form carries strictly more information than the rectangular equation for a curve.
Question 2 Multiple Choice
Consider two parametric curves: C1: x = t², y = t for all real t; and C2: x = t², y = t for 0 ≤ t ≤ 1. Both eliminate to y² = x. Which statement is true?
ABoth curves are identical because they have the same rectangular equation
BC1 traces the full parabola y² = x; C2 traces only the arc from (0, 0) to (1, 1)
CThe rectangular equation y² = x fails the vertical line test, so neither parametric form is valid
DC1 traces the parabola twice because t can be negative
The parameter interval restricts which portion of the curve is traced. C1 (all real t) covers y ∈ (−∞, ∞) → the full right-side parabola. C2 (t ∈ [0, 1]) covers only y ∈ [0, 1], the upper arc from origin to (1, 1). Eliminating the parameter destroys this interval information, making two geometrically different curves look algebraically identical. Note: C1 does not retrace — each t gives a unique point because y = t uniquely determines t.
Question 3 True / False
When you eliminate the parameter from a set of parametric equations to obtain a rectangular equation, the two representations generally describe exactly the same set of points.
TTrue
FFalse
Answer: False
The rectangular equation may describe more points than the parametric curve. The parameter's range restricts which portion of the implicit curve is actually traced. For example, x = t², y = t for t ≥ 0 gives only the upper half of y² = x, but the rectangular equation includes the lower half too. Elimination reveals the shape; the parameter range reveals the extent and direction of traversal.
Question 4 True / False
A parametric curve x = f(t), y = g(t) can represent shapes that would fail the vertical line test as a function y = h(x).
TTrue
FFalse
Answer: True
Because x and y are defined independently as functions of t, multiple y-values can correspond to the same x-value. A circle (x = cos t, y = sin t) fails the vertical line test — x = 0 corresponds to both y = 1 and y = −1 — yet it is perfectly described parametrically. This is one of the primary advantages of parametric form: it handles multi-valued curves, loops, and spirals that rectangular equations cannot represent as functions.
Question 5 Short Answer
Explain what additional information parametric equations provide, compared to a rectangular equation y = f(x), when describing the path of a moving object.
Think about your answer, then reveal below.
Model answer: Parametric equations encode not just the geometric path (which points are visited) but also direction of travel, speed at each moment (via dx/dt and dy/dt), the specific portion of a curve that is traced, and whether any part is retraced. A rectangular equation describes only the shape of the curve.
The parameter t acts as a timeline. Knowing x(t) and y(t) separately lets you compute velocity components, detect reversals (when dx/dt or dy/dt changes sign), find where speed is zero, and determine which branch of a multi-valued curve is being traced at each moment. For motion problems in physics and engineering, these properties are essential and are completely inaccessible from y = f(x) alone.