Any function of the form

is referred to as a **cubic function**. We shall also refer to this
function as the "parent" and the following graph is a sketch of
the parent graph.

We also want to consider factors that may alter the graph. Let's begin by considering the functions

and their graphs.

The parent graph is shown in red and the variations of this graph appear
as follows: the function y = f(x) + 2 appears in green; the graph of y =
f(x) + 5 appears in blue; the graph of the function y = f(x) - 1 appears
in gold; the graph of y = f(x) - 3 appears in purple.

It is now easy to generalize:

It is also necessary to evaluate the functions at specific values and examine their graphs. Let's investigate the changes to the graph for the following values: (x + 1), (x + 3), (x - 2), and (x - 4). Given these values, our new functions would be

Now, let's examine the graphs and make our observations.

As before, our parent graph is in red, y = f(x + 1) is shown in green,
y = f(x + 3) is shown in blue, y = f(x - 2) is shown in gold, and y = f(x
- 4) is shown in purple.

Let's make our observations:

is called a cubic function.

1) If c > 0, the graph shifts c units up; if c < 0, the graph shifts
c units down.

2) If d > 0, the graph shifts d units to the left; if d < 0, the graph
shifts d units to the right.