Parametric Graphs Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
What is the difference between the graphs of (a) x=tx = t, y=t2y = t^2 and (b) x=sin⁡(t)x = \sin(t), y=sin⁡2(t)y = \sin^2(t)?

Solution

  1. 1
    Both satisfy y=x2y = x^2, but (a) has x=t∈(−∞,∞)x = t \in (-\infty, \infty), tracing the entire parabola.
  2. 2
    (b) has x=sin⁡t∈[−1,1]x = \sin t \in [-1, 1], so only the portion of y=x2y = x^2 with −1≤x≤1-1 \le x \le 1 is traced. Additionally, the point oscillates back and forth rather than moving in one direction.

Answer

(a) Full parabola, (b) Only −1≤x≤1 portion, oscillating\text{(a) Full parabola, (b) Only } -1 \le x \le 1 \text{ portion, oscillating}
Different parameterizations of the same rectangular equation can produce different graphs due to restricted domains or different traversal patterns. The parametric form carries more information than the rectangular equation — it specifies which portion of the curve is traced and how.

About Parametric Graphs

Plotting and analyzing curves defined by parametric equations x=f(t)x = f(t), y=g(t)y = g(t), including eliminating the parameter, determining direction of motion, and finding tangent lines.

Learn more about Parametric Graphs →

More Parametric Graphs Examples

Example 1 easy

Sketch the direction of motion for the parametric curve [formula], [formula] as [formula] increases

Example 2 medium

Describe the graph of [formula], [formula] for [formula], including shape, direction, and starting p

Example 4 hard

Find the slope of the tangent line to the curve [formula], [formula] at the point where [formula].

← All Parametric Graphs Examples Parametric Graphs Concept