Cubics

Understanding Cubics

A cubic function is a function of the form f(x) = ax^3 + bx^2 + cx + d where a ≠ 0.
There are various forms for cubic functions including the general form, the factored form, and the vertex form.

They have the unusual property that they can possess either one or three real roots.
They can change direction once or twice, which means they can have between zero and two turning points.
The ends of the cubics graph, called the ‘end behavior’, point in same direction if the leading coefficient (a) is positive, and in opposite directions if a is negative.
Cubics can intersect the x-axis up to three times, but must intersect at least once, implying they have either 1 or 3 real roots.

To solve cubic equations, one method is by factorization. If you can write the cubic as a product of three linear factors ((x - a)(x - b)(x - c)), then the roots of the cubic equation are a, b, c.
Another way involves using the cubic formula, which is complex and rarely used.
Synthetic division is another technique used to solve cubic equations.

To sketch a cubic, one should identify the y-intercept (the value of d), the x-intercepts (the roots), turning points, and end behavior.
At the roots of the cubic, the graph intersects the x-axis.
Draw a smooth curve through these points making sure to match the end behaviors at the ends.

Cubic equations appear in a variety of natural phenomena including fluid dynamics and the shape of certain curves in nature.
They are also used in physics and engineering, for example in calculating the stress on a beam.