Parametric Graphs Formula

Q: How do you use the Parametric Graphs formula?

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.

Q: What do the symbols mean in the Parametric Graphs formula?

Arrows on the curve indicate direction of increasing $t$. Cusps occur where $dx/dt = 0$ and $dy/dt = 0$ simultaneously.

Parametric graphs are plotting and analyzing curves defined by parametric equations x = f(t), y = g(t), including eliminating the parameter, determining.

The Formula

Tangent slope:

\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

Second derivative:

\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}

When to use: 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.

Quick Example

Given

x = t^2

y = t^3

:
- Eliminate

t

t = x^{1/2}

, so

y = x^{3/2}

(but this misses the portion where

t < 0

).
- At

t = 1

: slope

= \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t}\Big|_{t=1} = \frac{3}{2}

Notation

Arrows on the curve indicate direction of increasing

t

. Cusps occur where

dx/dt = 0

and

dy/dt = 0

simultaneously.

What This Formula Means

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.

Formal View

\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

;

\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\!\left(\frac{dy}{dx}\right)}{dx/dt}

; arc length

= \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^{\!2} + \left(\frac{dy}{dt}\right)^{\!2}}\,dt

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.

Common Mistakes

Connecting points by $x$ order - join them in increasing- $t$ order, since the path can reverse.
Taking $\frac{dy}{dx}$ as $\frac{dx/dt}{dy/dt}$ - it is $\frac{dy/dt}{dx/dt}$ , the $y$ -rate over the $x$ -rate.
Ignoring direction arrows after eliminating $t$ - the Cartesian shape hides which way the curve is traced.

Why This Formula Matters

Sketching with direction arrows and computing $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ is how you read velocity and turning points off a path; cusps appear exactly where both $\frac{dx}{dt}$ and $\frac{dy}{dt}$ vanish. This turns an abstract pair of functions into a traceable, analyzable trajectory. Recognizing it by "Am I sketching or analyzing the actual traced path of $(x(t),y(t))$ , including its direction or tangent?" — rather than by familiar numbers — is what lets a student tell it apart from parametric equations (the definition) and cartesian curve sketching and polar graphs in a mixed problem set.

Frequently Asked Questions

What is the Parametric Graphs formula?

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.

How do you use the Parametric Graphs formula?

What do the symbols mean in the Parametric Graphs formula?

Arrows on the curve indicate direction of increasing $t$ . Cusps occur where $dx/dt = 0$ and $dy/dt = 0$ simultaneously.

Why is the Parametric Graphs formula important in Math?

What do students get wrong about Parametric Graphs?

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.

What should I learn before the Parametric Graphs formula?

Before studying the Parametric Graphs formula, you should understand: parametric equations, trigonometric functions.

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Related Concepts

Parametric Equations Trigonometric Functions Polar Graphs

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Parametric Equations Formula Trigonometric Functions Formula Polar Graphs Formula