The cubic function belongs to the family of polynomial functions. The image below summarises the different shapes of cubics that you might encounter. The black circle is indicating the point of inflection on each photo. It is interesting to see that the cubics in photo A have turning points, called a local maximum or minimum, while the other cubics do not. While we don't need to use the terms written on the photos right now, they will reappear when we study Calculus later on.
Just as with linear and quadratic functions, cubics can be represented in different forms. There are three main forms:
General Form: $y=ax^3+bx^2+cx+d$y=ax3+bx2+cx+d
Translated form: $y=k\left(xb\right)^3+c$y=k(x−b)3+c
Factored form ($3$3 linear factors): $y=k\left(xa\right)\left(xb\right)\left(xc\right)$y=k(x−a)(x−b)(x−c)
In this course we will focus on graphing polynomials in factored form.
Depending on the cubic involved, the function could be a onetoone relation or a manytoone relation. (Recall that a onetoone relation is a relation where no two values of $x$x are mapped to the same value of $y$y, and a manytoone relation is a relation where two or more values of $x$x are mapped to the same value of $y$y).
$y=k\left(xa\right)\left(xb\right)\left(xc\right)$y=k(x−a)(x−b)(x−c)
Just like factored form for quadratics, this form allows us to read off the $x$xintercepts directly. Try changing the values in the applet below, try cases with all 3 values different and other cases where some or all of the values match.

The $y$yintercept can be found by making $x=0$x=0. The $x$xintercepts for the factored form can be determined directly from the equation as described above or solved in a similar way to a quadratic equation using the null factor law.
Consider the function $f(x)=2(x1)(x3)(x+2).$f(x)=2(x−1)(x−3)(x+2).
a) Find the $x$xintercepts of the function.
b) Sketch the graph of $y=f(x).$y=f(x).
c) Sketch the graph of $y=f(x).$y=f(x).
a) Think: The $x$xintercepts occur at $y=0$y=0. The equation is in factored form though so we can use the null factor law to read off the $x$xintercepts.
Do: Set the value of $y$y to $0$0 and read off the $y$yintercepts.
$2(x1)(x3)(x+2)$2(x−1)(x−3)(x+2)  $=$=  $0$0 
$(x1)(x3)(x+2)$(x−1)(x−3)(x+2)  $=$=  $0$0 
$x$x  $=$=  $2,1,3$−2,1,3 using the null factor law 
b)
Think: The curve will pass through the $x$xaxis at the each of the above intercepts since the power of each factor is $1$1. The $y$yintercept of the curve can be found by substituting $x=0$x=0 into the function.
Do: Substitute $x=0$x=0 into the equation:
$y$y  $=$=  $2(01)(03)(0+2)$2(0−1)(0−3)(0+2) 
$=$=  $12$12 
Now that we have all the intercepts we can graph the function:
c) To graph the absolute value function, we need to reflect any parts of the above curve that are below the $x$xaxis by the $x$xaxis, so that the whole function lies above the $x$xaxis. The parts of the curve that we need to reflect are highlighted below:
So by reflecting these parts about the xaxis we get the following graph of the function $y=f(x)$y=f(x):
Consider the graph of the function.
For what values of $x$x is the cubic concave up?
For what values of $x$x is the cubic concave down?
State the coordinates of the point of inflection in the form $\left(a,b\right)$(a,b).
Consider the given graph of a cubic function.
Determine whether the cubic is positive or negative.
negative
positive
negative
positive
State the coordinates of the $y$yintercept.
Which of the following could be the equation of the function?
$y=2x^3+3$y=−2x3+3
$y=2x^3+3$y=2x3+3
$y=2x^33$y=2x3−3
$y=2x^33$y=−2x3−3
$y=2x^3+3$y=−2x3+3
$y=2x^3+3$y=2x3+3
$y=2x^33$y=2x3−3
$y=2x^33$y=−2x3−3
Consider the curve $y=\left(x+3\right)\left(x+2\right)\left(x2\right)$y=(x+3)(x+2)(x−2).
Find the $x$xvalue of the $x$xintercept(s).
Find the $y$yvalue of the $y$yintercept(s).
Sketch a graph of the curve.
Which of the following functions has a curve that passes through the origin?
$y=\left(x2\right)^2\left(x+3\right)$y=(x−2)2(x+3)
$y=\left(x+1\right)^3$y=(x+1)3
$y=\left(x4\right)\left(x+7\right)\left(x5\right)$y=(x−4)(x+7)(x−5)
$y=x\left(x6\right)\left(x+8\right)$y=x(x−6)(x+8)
$y=\left(x2\right)^2\left(x+3\right)$y=(x−2)2(x+3)
$y=\left(x+1\right)^3$y=(x+1)3
$y=\left(x4\right)\left(x+7\right)\left(x5\right)$y=(x−4)(x+7)(x−5)
$y=x\left(x6\right)\left(x+8\right)$y=x(x−6)(x+8)
Sketch the graphs of cubic polynomials and their moduli, when given in factorised form y = k(x – a)(x – b)(x – c).