This series of explorations examines the patterns established between critical points (inflection points and relative extrema) of a function when one of its coefficients is modified. For example, in any cubic function of the form Y = ax3 + bx2 + cx + d, the inflection point will trace the red function above when you vary the coefficient b, and keep the values a, c, and d constant. In other words, the red curve is the set of the inflection points for:
Y = ax3 + -5x2 + cx + d,
Y = ax3 +-2.3x2 + cx + d,
Y = ax3 + ex2 + cx + d,
Y = ax3 + 100000x2 + cx + d,
etc., where a, c, and d, are constant values.
Upon further examination, you can prove that the red curve formed by the inflection point is a quadratic function.
These investigations will examine the locus of critical points that is produced when each of the coefficients of a function is varied. In this case, you might wonder what type of locus occurs when you modify each of the other coefficients, a, c, and d. In addition, these explorations will examine the locus of critical points in other polynomial functions, such as a quadratic and quartic, and other functions that have critical points, such as asymptotic functions. Click on the various critical points on the navigation bar at the left to begin exploring!
This exploration was inspired by the vertex trace of a parabola I found at ExploreMath.com
Last revised: July 24, 2000