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.

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: Sketch $(x(t),y(t))$ point by point as $t$ rises, marking direction and slope $\frac{dy/dt}{dx/dt}$ .

Common stuck point: The procedure for parametric graphs is the easy part; the trap is connecting points by $x$ order. Asking "Am I sketching or analyzing the actual traced path of $(x(t),y(t))$ , including its direction or tangent?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I sketching or analyzing the actual traced path of $(x(t),y(t))$ , including its direction or tangent?

Worked Examples

Example 1

easy

Sketch the direction of motion for the parametric curve

x = t

y = t^2

t

increases from

-2

2

Answer

\text{Parabola } y = x^2 \text{, traced left to right as } t \text{ increases}

First step

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)

Full solution

2
The rectangular equation is $y = x^2$ (a parabola opening upward).
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 $(2, 4)$ .

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.

Example 3

medium

For

x = 3\cos t

y = 2\sin t

, sketch direction and starting point.

Example 4

medium

For

x = \cos t

y = \sin t

t \in [-\pi/2, \pi/2]

, what arc of the unit circle is traced?

Example 5

medium

For

x = 1 + 2\cos t

y = -3 + 2\sin t

, describe the Cartesian curve.

Example 6

hard

For the cycloid

x = t - \sin t

y = 1 - \cos t

, find the tangent slope at

t = \pi/2

Example 7

hard

For

x = 3 + \cos t

y = -2 + \cos t

, eliminate the parameter and describe the graph.

Example 8

challenge

Find the concavity of

x = t^2

y = t^3

t = 1

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

Example 3

easy

For

x = t

y = 2t

, list the points at

t = 0, 1, 2

Example 4

easy

t

increases for

x = t

y = t

, in which direction does the curve move?

Example 5

easy

Eliminate the parameter to graph

x = t

y = -t + 3

Example 6

easy

For

x = t^2

, what is the smallest possible

x

-value?

Example 7

easy

Plot the curve

x = \cos t

y = \sin t

t = \pi/2

Example 8

easy

For

x = 2t

y = 3

, what does the graph look like?

Example 9

easy

Find where

x = t - 1

y = t + 1

crosses the

y

-axis.

Example 10

easy

For

x = t

y = t^2

, is the graph symmetric about the

y

-axis?

Example 11

medium

Sketch direction: for

x = \cos t

y = \sin t

, does the point move clockwise or counterclockwise as

t

increases from

0

Example 12

medium

Eliminate the parameter and describe:

x = t + 1

y = (t+1)^2 - 3

Example 13

medium

For

x = 2t

y = t^2

, find the Cartesian equation and the vertex.

Example 14

medium

A particle follows

x = 3 - t

y = 2t - 1

. Find the slope of its straight-line path.

Example 15

medium

For

x = t^2

y = t^3

, where is the curve's cusp?

Example 16

medium

For

x = \sin t

y = \sin t

, describe the graph including any restriction.

Example 17

medium

Find the tangent slope of

x = \cos t

y = \sin t

t = \pi/4

Example 18

medium

For

x = t^2 - 4

y = t

, find the

x

-intercepts of the graph.

Example 19

medium

For

x = 3\cos t

y = 3\sin t

, find the point at

t = \pi

Example 20

challenge

The curve

x = t^2

y = t^3 - 3t

crosses itself. Find the point of self-intersection.

Example 21

challenge

For

x = t - \sin t

y = 1 - \cos t

(a cycloid), find the points where the tangent is horizontal on

[0, 2\pi]

Example 22

challenge

For

x = 2\cos t

y = 3\sin t

, eliminate the parameter and name the curve and its intercepts.

Example 23

easy

List the points on

x = t

y = t + 2

t = -1, 0, 1, 2

Example 24

easy

At what parameter value does

x = t^2

y = t - 1

reach the

y

-axis?

Example 25

easy

t = \pi

, find the point on

x = \cos t

y = \sin t

Example 26

medium

For

x = \sin t

y = \cos t

t = 0

, find the starting point and initial direction.

Example 27

medium

Find the slope of the tangent to

x = t^2 - 1

y = t^3 - t

t = 1

Example 28

medium

Find the tangent slope of

x = 2\cos t

y = 3\sin t

t = \pi/4

Example 29

medium

For

x = t + 1

y = (t+1)^2 - 4

, find vertex of the Cartesian graph.

Example 30

medium

For

x = t^2

y = t^3

, find where the tangent is vertical.

Example 31

medium

For

x = 2t - 1

y = 4 - t

, find the

x

-intercept.

Example 32

hard

For

x = t^2 - 2

y = t^3 - 4t

, find all parameter values where the curve crosses the

x

-axis.

Example 33

hard

For

x = 2\cos t

y = \sin(2t)

, find the parameter values where the tangent is horizontal on

[0, 2\pi)

Example 34

hard

For

x = t^2

y = t^3 - t

, find the slope of the tangent at

t = -1

Example 35

challenge

For

x = t^3 - 4t

y = t^2 - 4

, find the point of self-intersection.

Example 36

challenge

For

x = e^t \cos t

y = e^t \sin t

(logarithmic spiral), find the tangent slope at

t = 0

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Related Concepts

Parametric Equations Trigonometric Functions Polar Graphs

Background Knowledge

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

parametric equationstrigonometric functions