# CUBIC FUNCTIONS

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:

If y = f(x) + c and c > 0, the graph undergoes a vertical shift c units up along the y-axis.

If y = f(x) + c and c < 0, the graph undergoes a vertical shift c units down along the y-axis.

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:

If y = f(x + d) and d > 0, the graph undergoes a horizontal shift d units to the left.

If y = f(x + d) and d < 0, the graph undergoes a horizontal shift d units to the right.

## SUMMARY

Any function of the form is called a cubic function.

Consider the 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.

Return