Practice Parametric Equations in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A way of defining a curve by expressing both $x$ and $y$ as separate functions of a third variable (parameter), typically $t$ : $x = f(t)$ , $y = g(t)$ .

Instead of saying ' $y$ depends on $x$ ,' parametric equations say 'both $x$ and $y$ depend on time $t$ .' Imagine an ant walking on a table—at each moment $t$ , the ant has an $x$ -position and a $y$ -position. The path it traces is the parametric curve, and $t$ is the clock ticking forward.

Showing a random 20 of 50 problems.

Example 1

easy

For

x = 4t

and

y = t + 5

, find the point when

t = 2

Example 2

medium

Eliminate the parameter:

x = 2 + 3t

y = -1 + 4t

Example 3

medium

Find parametric equations for the segment from

(2,5)

(8,11)

using

t \in [0,1]

Example 4

medium

For

x = t^3

y = t^2

, find

\dfrac{dy}{dx}

t = 2

Example 5

easy

Find the point on the curve

x = 2t+1

y = t^2

when

t = -3

Example 6

medium

Find the point on

x = t - \sin t

y = 1 - \cos t

t = \pi

Example 7

hard

Eliminate the parameter from

x = \sin t

y = \sin(2t)

and state the restriction.

Example 8

medium

Eliminate the parameter:

x = e^t

y = e^{2t}

Example 9

easy

Given

x = 5

y = 2t

, describe the resulting curve.

Example 10

easy

Eliminate the parameter:

x = t + 1

y = 2t

Example 11

easy

Eliminate the parameter:

x = t

y = 3t + 5

Example 12

medium

t = \pi/3

, find the point on

x = 2\cos t

y = 2\sin t

Example 13

hard

For the curve

x = t^2 + 1

y = 2t

, eliminate the parameter.

Example 14

easy

For

x = 7

y = 2t - 3

, what kind of curve does the parametric description trace?

Example 15

medium

Eliminate the parameter:

x = t + 2

y = t^2

Example 16

challenge

The point

(x,y)

moves with

x = t^2 - 1

y = t^3 - t

. Find all parameter values where the curve passes through the origin.

Example 17

easy

Eliminate the parameter from

x = t - 4

and

y = 3t

Example 18

medium

For

x = t^2

y = t^3

, find the point(s) where the curve crosses the

x

-axis.

Example 19

easy

Given

x = 2t

y = t + 1

, find the point when

t = 3

Example 20

medium

Write parametric equations for the circle of radius

5

centered at

(2,-3)

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Related Concepts

Function Trigonometric Functions Parametric Graphs Polar Coordinates