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Consider the general quadratic function g(x)=ax2+bx+c. The first applet below illustrates how changes in the coefficients a, b, and c will affect the graph. The red graph is f(x)=x2, and the green graph is g(x)=ax2+bx+c (initially, both graphs coincide). Use the sliders to explore the effect of changing the values of a, b, and c.
MultiGraph applet written by David Eck (http://math.hws.edu/javamath/index.html)
Notice that changing the coefficient a appears to cause some kind of vertical stretch (and reflect it vertically
if a is negative),
and changing c
causes a vertical shift. However, the role of b is unclear - changes in b seem to combine both a horizontal and vertical shift.
The graph of a quadratic function can be better understood by completing the square
and rewriting g(x) in the form g(x)=a(x-h)2+k. The next applet
below shows how changes in a, h, and k will affect the graph. Again, the red graph is f(x)=x2, and the green graph is g(x)=a(x-h)2+k (initially, both graphs coincide). Use the sliders to explore the effect
of changing the values of a, h, and k.
This time you can see that changing a will stretch the graph vertically (and reflect it vertically if a is negative), changing h causes a horizontal shift,
and changing k
causes a vertical shift.
NEXT: Discussion of cubic
polynomial functions