Parametric Equations Examples in Math

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

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

Concept 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.

Read the full concept explanation →

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: $x=f(t),\ y=g(t)$ trace a path by time, capturing direction and speed, not just shape.

Common stuck point: The procedure for parametric equations is the easy part; the trap is believing eliminating $t$ keeps everything. Asking "Are $x$ and $y$ each written as a function of a separate parameter that drives both together?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are $x$ and $y$ each written as a function of a separate parameter that drives both together?

Worked Examples

Example 1

easy

Eliminate the parameter from

x = 2t + 1

and

y = t - 3

to find the rectangular equation.

Answer

y = \frac{x - 7}{2}

First step

Solve the

x

-equation for

t

t = \frac{x - 1}{2}

Full solution

2
Substitute into the $y$ -equation: $y = \frac{x - 1}{2} - 3$ .
3
Simplify: $y = \frac{x - 1}{2} - 3 = \frac{x - 1 - 6}{2} = \frac{x - 7}{2}$ .

Eliminating the parameter converts parametric equations back to a single rectangular equation. Solve one equation for

t

and substitute into the other. The result here is a line with slope

\frac{1}{2}

and

y

-intercept

-\frac{7}{2}

Example 2

medium

Eliminate the parameter from

x = 3\cos(t)

and

y = 3\sin(t)

Example 3

medium

Eliminate the parameter from

x = \sqrt{t}

and

y = t + 1

, stating any domain restriction.

Example 4

medium

Find

\dfrac{dy}{dx}

t = 1

for

x = t^2 + t

y = t^3

Example 5

medium

For

x = t^3

y = t^2

, find

\dfrac{dy}{dx}

t = 2

Example 6

hard

Find the equation of the tangent line to

x = t^2

y = t^3 - 3t

t = 2

Example 7

hard

A projectile is launched with horizontal speed

20

m/s and vertical speed

30

m/s. Using

g = 10

m/s

^2

, write parametric equations for its position.

Example 8

challenge

Find

\dfrac{d^2 y}{dx^2}

t = 1

for

x = t^2

y = t^3

Practice Problems

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

Example 1

medium

Find parametric equations for the line through

(1, 4)

and

(5, -2)

Example 2

hard

Eliminate the parameter from

x = t^2 - 1

and

y = t^3 - t

and describe the curve.

Example 3

easy

Given

x = 2t

y = t + 1

, find the point when

t = 3

Example 4

easy

For

x = t

y = t^2

, find

y

when

x = 4

Example 5

easy

Eliminate the parameter:

x = t + 1

y = 2t

Example 6

easy

Find

x

and

y

for

x = 3t

y = t - 2

t = 0

Example 7

easy

Eliminate the parameter:

x = t

y = 3t + 5

Example 8

easy

Given

x = t^2

y = t

, what is

x

in terms of

y

Example 9

easy

For

x = \cos t

y = \sin t

, evaluate the point at

t = 0

Example 10

easy

Given

x = 5

y = 2t

, describe the resulting curve.

Example 11

medium

Eliminate the parameter for

x = 2\cos t

y = 2\sin t

Example 12

medium

Eliminate the parameter:

x = t + 2

y = t^2

Example 13

medium

A particle has

x = 1 + 2t

y = 3 - t

. Find its Cartesian equation.

Example 14

medium

For

x = t^2

y = t^3

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

x

-axis.

Example 15

medium

Find the slope

\frac{dy}{dx}

x = t^2

y = t^3

t = 1

Example 16

medium

Eliminate the parameter:

x = e^t

y = e^{2t}

Example 17

medium

For

x = \sin t

y = \cos 2t

, write

y

as a function of

x

Example 18

medium

A curve is

x = 4t

y = 16t^2

. Identify the Cartesian curve.

Example 19

medium

Find the point on

x = t - \sin t

y = 1 - \cos t

t = \pi

Example 20

challenge

Show that

x = \cos t

y = \sin t

t \in [0,\pi]

traces only the upper unit semicircle, and give its Cartesian description.

Example 21

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 22

challenge

For

x = 2t

y = t^2 + 1

, find the equation of the tangent line at

t = 2

Example 23

easy

For

x = 4t

and

y = t + 5

, find the point when

t = 2

Example 24

easy

Eliminate the parameter from

x = t - 4

and

y = 3t

Example 25

easy

Find the point on the curve

x = 2t+1

y = t^2

when

t = -3

Example 26

easy

Find parametric equations for the line through

(0,0)

with direction vector

(3,5)

Example 27

medium

Find parametric equations for the segment from

(2,5)

(8,11)

using

t \in [0,1]

Example 28

medium

Eliminate the parameter:

x = 4\cos t

y = 5\sin t

Example 29

medium

Eliminate the parameter:

x = 2 + 3t

y = -1 + 4t

Example 30

medium

Write parametric equations for the circle of radius

5

centered at

(2,-3)

Example 31

medium

Find the values of

t

where

x = t^2 - 4

y = t^3 - 4t

passes through the origin.

Example 32

medium

Find a parametrization of the line

y = 2x - 3

Example 33

hard

Eliminate the parameter from

x = \sin t

y = \sin(2t)

and state the restriction.

Example 34

hard

Find values of

t

where the tangent to

x = t^2

y = t^3 - 12t

is horizontal.

Example 35

hard

For

x = e^t

y = e^{-t}

, eliminate the parameter and state the domain.

Example 36

challenge

The curve

x = t^3 - 3t

y = t^2

has a self-intersection. Find the point.

Example 37

challenge

A ladybug moves so that

x = \cos(2t)

and

y = \sin(3t)

for

t \in [0, 2\pi]

. How many times does the curve cross itself in one full period? Justify by finding the period.

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Related Concepts

Function Trigonometric Functions Parametric Graphs Polar Coordinates

Background Knowledge

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

function definitiontrigonometric functions