Identify Characteristics of Cubic Functions

The shape of a cubic

The cubic function belongs to the family of polynomial functions and has the general form given by $y=ax^3+bx^2+cx+d$y=ax3+bx2+cx+d.

Various combinations of choices of the four coefficients $a$a, $b$b, $c$c and $d$d will create variations in the shape and position of the curve. However, the curve will always exhibit certain invariant properties that we will outline here.

Before we look at these properties it might be beneficial to describe the shape using a rope analogy.

Take a piece of rope and lay it on the table. Put two turning points in it so that it looks like photo A in the diagram below. One of the turning points is known as a local maximum and the other is known as a local minimum. Photo A is one of the common shapes of a cubic function.

In between the two turning points (circled) is the centre of the cubic function. A cubic function always has a centre. For $y=ax^3+bx^2+cx+d$y=ax3+bx2+cx+d it is always found at $x=\frac{-b}{3a}$x=−b3a​.

Note also the rounded hill shape as it rises around the turning point and down onto the centre. This hill shape part of the curve is said to be concave down. After it reaches the centre, the curve, just like a glacially hollowed valley, turns around the minimum turning point and begins to rise. It rises slowly at first, but gradually picks up speed upward toward infinity. This valley shape part of the curve is said to be concave up.

All cubic functions undergo a single change in concavity (from concave down to concave up, or from concave up to concave down) and this change always occurs at the cubic's centre.

Now imagine that both ends of the rope are pulled - the left end pulled down and the right end pulled up. At some point in the pulling it will become something like that depicted in photo B. The turning points will have merged together to form a 'flat' region known as a horizontal inflection or, alternatively, a stationary point of inflection. Note that the two concavities on either side of the centre have been maintained.

Photo B is another interesting form of the cubic. Whenever the cubic curve looks like photo B, it has the form $y=a\left(x-h\right)^3+k$y=a(x−h)3+k with the centre point having the coordinates $\left(h,k\right)$(h,k).

Finally imagine pulling on the ends further. You will end up with something like that depicted in photo C.

With a real rope you would be able to pull it into a straight line, but with the cubic curve, a complete straightening is impossible. No matter how hard you 'pull' on the cubic function, there will always be a central point of inflection (no longer horizontal but nevertheless an inflection) with a concave down section on the left of the centre and a concave up section to the right.

The forms of the cubic function's equation

The $x$x-intercepts of the cubic curve $y=ax^3+bx^2+cx+d$y=ax3+bx2+cx+d are in general difficult to find. There are at most three of them.

The equation needed to find these roots, given by $ax^3+bx^2+cx+d=0$ax3+bx2+cx+d=0, can sometimes be solved quite easily provided we can factorise the left hand side.

For example, the function $y=2x^3-6x^2-26x+30$y=2x3−6x2−26x+30 can be factorised to $y=2\left(x+3\right)\left(x-1\right)\left(x-5\right)$y=2(x+3)(x−1)(x−5), and we can immediately write down the $x$x-intercepts as $-3$−3, $1$1 and $5$5. If you think about it, with three intercepts, the cubic must have a local maximum and minimum.

The converse however is not true - a cubic can still have a maximum and minimum even though it only has one intercept. In fact we can show using calculus that the general cubic function $y=ax^3+bx^2+cx+d$y=ax3+bx2+cx+d will have a minimum and maximum turning point whenever $b^2>3ac$b2>3ac.

So for example we know the function $y=2x^3+3x^2-4x+1$y=2x3+3x2−4x+1 has the turning points because $a=2$a=2 and $b=3$b=3 are positive and $c$c is less than zero. This is a nice observation, and you can verify its true using the third applet below.

Very often though, the cubic function can be difficult to factorise. Nevertheless what we do know is that it must always have at least one real root, say $x=r$x=r. This is true because, just like the rope, the cubic moves in opposite directions at the extremities.

With that knowledge we know, at least theoretically, that we can always put $y=ax^3+bx^2+cx+d$y=ax3+bx2+cx+d into the form $y=a\left(x-r\right)\left(x^2+px+q\right)$y=a(x−r)(x2+px+q). The quadratic factor thus provides the key as to whether or not the cubic has more than one real root.

More will be said on roots and intercepts later, but based on this observation, there is a variety of factorised forms of a cubic function including forms like $y=\left(x-2\right)^2\left(x-3\right)$y=(x−2)2(x−3), $y=\left(x+1\right)^3$y=(x+1)3, $y=2\left(x+3\right)\left(x-1\right)\left(x-5\right)$y=2(x+3)(x−1)(x−5), $y=\left(x+1\right)\left(x^2+2x+2\right)$y=(x+1)(x2+2x+2) etc.

Explore

The following applets allow you to experiment with the cubic form.

This first applet allows you to play with the form depicted in photo B above. The form of all cubics having a horizontal inflection at $\left(h,k\right)$(h,k) is given $y=a\left(x-h\right)^3+k$y=a(x−h)3+k. If for example both $h$h and $k$k are zero, then the form reduces to $y=ax^3$y=ax3, which is a cubic with a horizontal inflection at the origin.    

This second applet allows you to play with factorised forms of the cubic, and you will see that when the cubic has three distinct roots, there exists both a maximum and minimum turning point. You might discover that the central point is always exactly half way between those turning points.

The third applet allows you to play with all four constants of the form $y=ax^3+bx^2+cx+d$y=ax3+bx2+cx+d. See what you can discover.

Worked Examples

QUESTION 1

QUESTION 2

QUESTION 3