Function Graph Explorer

Plot Cartesian, polar, and parametric curves. Trace points along each curve, explore properties, and see step-by-step derivative analysis. Updated in real-time.

Functions

Display:

Properties

Enter a function and click Graph to see properties.

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Quick Reference

Derivative Rules

Power Rule

\frac{d}{dx}[x^n] = nx^{n-1}

Product Rule

\frac{d}{dx}[f \cdot g] = f'g + fg'

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Trig Derivatives

\frac{d}{dx}[\sin x] = \cos x

\frac{d}{dx}[\cos x] = -\sin x

\frac{d}{dx}[\tan x] = \sec^2 x

\frac{d}{dx}[e^x] = e^x

\frac{d}{dx}[\ln x] = \frac{1}{x}

Graph Features

Roots Where

f(x) = 0

. The function crosses the x-axis.

Extrema Where

f'(x) = 0

. Local maxima or minima.

Inflection Where

f''(x) = 0

. Concavity changes.

Increasing: $f'(x) > 0$ | Decreasing: $f'(x) < 0$

Polar Curves

A polar curve gives the distance from the origin as a function of the angle, $r = f(\theta)$ . Convert to Cartesian with $x = r\cos\theta$ and $y = r\sin\theta$ .

Cardioid

r = 1 + \cos\theta

Rose

r = \cos(k\theta)

gives

k

petals for odd

k

2k

for even.

Enclosed area

A = \tfrac{1}{2}\int_0^{2\pi} r^2 \, d\theta

Parametric Curves

A parametric curve traces $(x(t),\, y(t))$ as the parameter $t$ runs from 0 to $2\pi$ . It can loop and cross itself, unlike a function graph.

Circle

x = \cos t,\; y = \sin t

Lissajous

x = \sin(3t),\; y = \sin(2t)

Arc length

L = \int_a^b \sqrt{x'(t)^2 + y'(t)^2} \, dt

Interval Notation

Open interval

(a, b)

Excludes endpoints.

Closed interval

[a, b]

Includes endpoints.

Unbounded

(-\infty, a)

(b, \infty)

Union

A \cup B

Combines intervals.

Example

(-\infty, -1) \cup (2, \infty)

means all

x

less than

-1

or greater than

2

TopicsCalculus Derivatives Functions Graphs Process

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Function Graph Explorer

Functions

Properties

Quick Reference

Derivative Rules

Trig Derivatives

Graph Features

Polar Curves

Parametric Curves

Interval Notation

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