Phantom Graph (x^2-1)^2: Complex Roots and Parametric Curves

This example demonstrates the 'phantom graph' phenomenon for the function y=(x^2-1)^2, revealing invisible imaginary trajectories in the real plane by incorporating complex roots and parametric curves. Core mathematical concepts include complex numbers, parametric equations, function dilation and translation, and properties of quadratic functions. Construction highlights: using GeoGebra's ComplexRoot command to compute complex roots of the polynomial, and CurveCartesian to build parametric curves, involving objects such as c, g, a, f, z_1 through z_4, A1-A4, text1, d, b, e, h, i, j, k, with sliders or dynamic parameters controlling the curve shape. Suitable for high school to university mathematics education, helping students understand the deep connection between complex numbers and function graphs, and exploring transformations of quadratic functions and circles.