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Cubic Functions

A cubic function is one in the form f ( x ) = a x 3 + b x 2 + c x + d .

The "basic" cubic function, f ( x ) = x 3 , is graphed below.


The function of the coefficient a in the general equation is to make the graph "wider" or "skinnier", or to reflect it (if negative):

The constant d in the equation is the y -intercept of the graph.

graphs of y=x^3, y=x^3-2, and y=x^3+3

The effects of b and c on the graph are more complicated. However, if you can factor the right side of the equation, you can find one or more x -intercepts , and use these to sketch the graph. (Some cubics, however, cannot be factored.)

graph of y=(x+3)(x+1)(x-2)

A cubic function may have one, two or three x -intercepts, corresponding to the real roots of the related cubic equation.