The Cubic FunctionFocused on f(x) = a(x – h)³ + k

Presented by Narendran Sairam9 Yale

Introduction to the cubic function● 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.

Introduction 2

● 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.

● 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.

Forms of cubic functions

Cubic functions can be found in two forms.......f(x)= ax3 + bx2 + cx + d, where a, b, c, and d are constants and a is not equal to 0, or

f(x) = a(x – h)3 + k, where a, h, and k are constants and a is not equal to 0.

The Graph of 'f(x) = a(x – h)³ + k

The values of h and k specify the horizontal and vertical translation, or shift, of the curve. The graph shifts h units right when h is positive and shifts h units left when h is negative. The graph shifts upwards k units when k is positive and shifts k units downward when k is negative. It is easiest to use this method by looking at the point of inflection (a key point) and shifting appropriately. In this case, the origin is the point of inflection of the original function.

Picture of f(x) = a(x – h)³ + k

From this picture we can conclude that (-h,k) is the turning point of the graph. It is also the vertex of the function.