Parametric Graphs Math Example 1

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

Example 1

easy
Sketch the direction of motion for the parametric curve x=tx = t, y=t2y = t^2 as tt increases from −2-2 to 22.

Solution

  1. 1
    Create a table of values: t=−2:(−2,4)t = -2: (−2, 4); t=−1:(−1,1)t = -1: (−1, 1); t=0:(0,0)t = 0: (0, 0); t=1:(1,1)t = 1: (1, 1); t=2:(2,4)t = 2: (2, 4).
  2. 2
    The rectangular equation is y=x2y = x^2 (a parabola opening upward).
  3. 3
    As tt increases from −2-2 to 22, the point moves from left to right along the parabola: starting at (−2,4)(-2, 4), descending to the vertex (0,0)(0, 0), then ascending to (2,4)(2, 4).

Answer

Parabola y=x2, traced left to right as t increases\text{Parabola } y = x^2 \text{, traced left to right as } t \text{ increases}
Parametric graphs include direction of motion (orientation), which rectangular equations do not provide. Arrows on the curve indicate the direction of increasing tt. The same rectangular curve can be traced in different directions with different parameterizations.

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 2 medium

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

Example 3 medium

What is the difference between the graphs of (a) [formula], [formula] and (b) [formula], [formula]?

Example 4 hard

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

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