﻿ Families of Curves

# Families of Curves

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 The graph tool can graph a family of curves by changing a constant.  Just push the button     For example, if you have entered the function y = ax2.       I have set a to be 1 and the delta to be 0.25.  The graph shows a family of curves, centered around a = 1 with a varying from -0.25 to 2.25.  In other words we have graphed:   y = -0.25x^2        y = 0x^2        y = 0.25x^2        y = 0.5x^2        y = 0.75x^2        y = 1x^2        y = 1.25x^2        y = 1.5x^2        y = 1.75x^2        y = 2x^2        and        y = 2.25x^2   all on the same set of axes.   This graph also shows one of the graph tool 4's new features where graphs away from the "main" value of a have reduced transparency.  This can more clearly demonstrate some concepts.  This, in turn, brings us to the other window in the constant controllers.     Two of controls in this window affect families of curves.  The  slider changes the number of curves in the family, or how many steps away from the "main" value the graph tool moves.  By moving the slider all the way to the right, we greatly increase the number of curves in the family.  Underneath the slider, the graph tool shows the range of values that a will take based on your current settings.  In our case a is varying from -0.25 to 2.25.     You can also turn off the transparency effect using the indicated button.