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 Condition:
a ≠ 0 Vertex Form:
y = a(x +
b2a)² + c −
b²4a Axis of Symmetry:
x = −
b2a Coefficient (a):
determines the opening direction and width. Coefficient (b):
helps determine the vertex and axis of symmetry. Coefficient (c):
determines the y-intercept.
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.
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.
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. Angles are measured in radians. 2π rad = 360°. The ω slider ranges from −2 to 2; 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.
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.
0.25 (a < 1)4 (a > 1)
−44
0.001 (x > 0)256
Use this line to see the mirror symmetry of the two inverse functions.