Parametric Graphs Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parametric Graphs.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

To sketch a parametric curve, make a table of t, x, and y values, then plot the (x, y) points and connect them in order of increasing t. Arrows on the curve show the direction of travel. Alternatively, you can sometimes eliminate t to get a familiar Cartesian equation—but you may lose information about direction and speed.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Parametric graphs carry more information than Cartesian graphs: they encode not just the shape of the curve but also the direction and speed of traversal. Eliminating the parameter recovers the shape but may lose this dynamic information.

Common stuck point: When eliminating the parameter, check whether the domain of t restricts the Cartesian equation. For example, x = t^2 means x \geq 0, so the Cartesian equation only applies for non-negative x.

Sense of Study hint: Try solving one equation for t, then substitute into the other equation to eliminate the parameter and get a Cartesian equation.

Worked Examples

Example 1

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

Solution

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

Answer

\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 t. The same rectangular curve can be traced in different directions with different parameterizations.

Example 2

medium
Describe the graph of x = 2\cos(t), y = 5\sin(t) for 0 \le t \le 2\pi, including shape, direction, and starting point.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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

Example 2

hard
Find the slope of the tangent line to the curve x = t^2 + 1, y = t^3 - 3t at the point where t = 2.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

parametric equationstrigonometric functions