Interactive Math Lab

Math Graphic Lab

Explore mathematics through interactive graphics.

Choose an experiment

Select a module to begin exploring.

01 · Radial Line ExplorerL = |Atⁿ + B|

Explore how radial lines create geometric patterns.

02 · Taiji Spiral Explorerr = Atⁿ + B

Explore how mathematical parameters shape spirals.

03 · Linear Function Explorery = kx + b

Explore how slope and y-intercept change a straight line.

04 · Quadratic Function Explorery = ax² + bx + c

Explore how the coefficients change a parabola.

05 · Sine Function Explorer (sin)y = A sin(ωx + φ) + D

Explore amplitude, angular frequency, and phase shift in a sine wave.

06 · Cosine Function Explorer (cos)y = A cos(ωx + φ) + D

Explore how the parameters change a cosine wave.

07 · Sine & Cosine Comparison (sin & cos)y = A sin(ωx + φ) + D; y = A cos(ωx + φ) + D

Compare sine and cosine on the same coordinate plane.

08 · Unit Circlecos²θ + sin²θ = 1

Connect a point on the unit circle to the sine and cosine curves.

09 · Exponential & Logarithmic Functionsy = aˣ; y = logₐ x

Compare inverse functions and explore their symmetry about y = x.

10 · Cubic Power Functiony = x³

Explore the cubic power function and its moving point.

11 · Negative Quadratic Powery = x⁻²

Explore the reciprocal-square curve on both sides of zero.

12 · Negative Cubic Powery = x⁻³

Explore the signed reciprocal-cubic curve.

13 · Square Root Functiony = $x^{\frac{1}{2}}$

See how the square-root function grows from the origin.

14 · Cube Root Functiony = $x^{\frac{1}{3}}$

Explore cube roots for positive and negative inputs.

15 · Negative Half Powery = $x^{- \frac{1}{2}}$

Explore the reciprocal square-root function.

16 · Negative One-Third Powery = $x^{- \frac{1}{3}}$

Explore the reciprocal cube-root function.

17 · General Power Explorery = xᵃ

Switch among 15 common integer and fractional exponents.

18 · Quartic Function Explorery = x⁴

Explore the quartic function and compare its shape with a quadratic curve.

19 · Quadratic & Quartic Comparisony = x² and y = x⁴

Compare how the quadratic and quartic curves change around the origin.

1. Radial Line Explorer

Cartesian coordinates · Non-negative radial lengths

Length: L = |Atⁿ + B|
Endpoint: x = L cos(t), y = L sin(t)
The absolute value keeps every line length non-negative.

Red lines: radial segments. Green dots: endpoints. Equal x/y scale preserves the true geometry.

2. Taiji Spiral Explorer

Polar coordinates · Signed radius · Endpoint trajectory

Radius: r = Atⁿ + B
Cartesian conversion: x = r cos(t), y = r sin(t)
The radius may be negative. The angle t is measured in radians.

Blue curve: endpoint trajectory. Concentric circles indicate radial distance; rays indicate angle.

3. Linear Function Explorer

Cartesian coordinates · Slope and y-intercept

Function: y = kx + b
Slope (k): determines the direction and steepness of the line.
Y-intercept (b): determines where the line crosses the y-axis.

4. Quadratic Function Explorer

Cartesian coordinates · Parabolas and their properties

Function y = ax² + bx + c, a ≠ 0

Vertex V(−b2a, c − b²4a)

Vertex form y = a(x + b2a)² + c − b²4a

Axis of symmetry x = −b2a

Parabola Vertex and axis y-intercept Real x-intercepts

5. Sine Function Explorer (sin)

Trigonometric functions · Radians on the horizontal axis

Function: y = A sin(ωx + φ) + D
A: amplitude factor · ω: angular-frequency factor · φ: phase offset · D: vertical shift
Angles are measured in radians. 2π rad = 360°. The ω slider ranges from −2 to 2; at ω = 0 the function is constant.
Five curve reference points: A = start (rising midline crossing) · B = crest · C = descending midline crossing · D = trough · E = end of one period. All five points follow the curve; the red draggable point is separate.

6. Cosine Function Explorer (cos)

Trigonometric functions · Radians on the horizontal axis

Function: y = A cos(ωx + φ) + D
A: amplitude factor · ω: angular-frequency factor · φ: phase offset · D: vertical shift
Angles are measured in radians. 2π rad = 360°. The ω slider ranges from −2 to 2; at ω = 0 the function is constant.
Five curve reference points: A = crest · B = descending midline crossing · C = trough · D = rising midline crossing · E = end of one period. The points follow the curve; the red draggable point is separate.

7. Sine & Cosine Comparison (sin & cos)

Compare both functions with the same parameters

Sine: y = A sin(ωx + φ) + D
Cosine: y = A cos(ωx + φ) + D
Both curves share the same A, ω, φ, and D so you can compare their shapes directly.
Curve key: Red = sine · Blue = cosine.
Drag either colored point to change the shared x-coordinate; both points move together. Angles are measured in radians. 2π rad = 360°. At ω = 0 both functions are constant.

8. Unit Circle & Sine/Cosine Explorer

One rotating point · Two coordinates · Two trigonometric curves

Unit circle: x² + y² = 1
Horizontal coordinate: cos θ · Vertical coordinate: sin θ
Move θ from 0° to 720° and watch the same point generate two full turns. 2π rad = 360°.

Periodicity of Sine and Cosine: sin(θ + 360°k) = sin θ; cos(θ + 360°k) = cos θ, where k ∈ ℤ. Each complete turn repeats the same sine and cosine values.

Black point group angle (θ)

0° (0)720° (4π)

Red point angle (α · sine)

0° (0)720° (4π)

Green point angle (β · cosine)

0° (0)720° (4π)

9. Exponential & Logarithmic Function Explorer

Inverse functions · Shared base · Reflection across y = x

What to look for
• The exponential graph always passes through (0, 1); the logarithmic graph passes through (1, 0).
• When a > 1, both functions increase. When 0 < a < 1, both decrease.
• The exponential function has domain all real numbers and range y > 0. The logarithmic function has domain x > 0 and range all real numbers.

Base (a)

0.25 (a < 1)4 (a > 1)

Exponential input (x)

−44

Logarithm input (x)

0.001 (x > 0)256

Show y = x reference line

Use this line to see the mirror symmetry of the two inverse functions.

Math Graphic Lab

Choose an experiment

01 · Radial Line ExplorerL = |Atⁿ + B|

02 · Taiji Spiral Explorerr = Atⁿ + B

03 · Linear Function Explorery = kx + b

04 · Quadratic Function Explorery = ax² + bx + c

05 · Sine Function Explorer (sin)y = A sin(ωx + φ) + D

06 · Cosine Function Explorer (cos)y = A cos(ωx + φ) + D

07 · Sine & Cosine Comparison (sin & cos)y = A sin(ωx + φ) + D; y = A cos(ωx + φ) + D

08 · Unit Circlecos²θ + sin²θ = 1

09 · Exponential & Logarithmic Functionsy = aˣ; y = logₐ x

10 · Cubic Power Functiony = x³

11 · Negative Quadratic Powery = x⁻²

12 · Negative Cubic Powery = x⁻³

13 · Square Root Functiony = x12

14 · Cube Root Functiony = x13

15 · Negative Half Powery = x−12

16 · Negative One-Third Powery = x−13

17 · General Power Explorery = xᵃ

18 · Quartic Function Explorery = x⁴

19 · Quadratic & Quartic Comparisony = x² and y = x⁴