Graphing_hand.jpgcubic function is a polynomial of degree three.

e.g. y = x3 + 3x2 − 2x + 5

Cubic graphs can be drawn by finding the x and y intercepts.
Because cubic graphs do not have axes of symmetry the turning points have to be found using calculus.

Sketching Cubics

Method 1: Factorisation.

If the equation is in the form y = (x − a)(x − b)(x − c) the following method should be used:

Step 1:  Find the x-intercepts by putting y = 0.
Step 2:  Find the y-intercept by putting x = 0.
Step 3:
Plot the points above to sketch the cubic curve.

e.g. Sketch the graph of y = (x − 2)(x + 3)(x − 1)

Step 1:

Find the x-intercepts by putting y = 0.

0 = (x − 2)(x + 3)(x − 1)

x = 2 or -3 or 1

Step 2:

Find the y-intercepts by putting x = 0.

y = (0 − 2)(0 + 3)(0 − 1)

y = -2 x 3 x -1

y = 6

Step 3:

Plot the points and sketch the curve.

Y11_Cubic_Graphs_01.gif

Note: Functions with a repeated factor have a graph which just touches the x-axis. e.g. y = (x − 2)2(x + 1)

Y11_Cubic_Graphs_02.gif

 

 

 

Method 2: Transformation

The graph of the basic cubic y = x3 is shown in the diagram.         Y11_Cubic_Graphs_03.gif

 

This basic cubic is moved or transformed as follows:

 

y = ax3 The a has the effect of changing the basic cubic in the y- direction.

It affects the steepness of the graph.

  • If a is large, the cubic is steeper.
  • If a is small, the cubic is flatter.
  • If a is negative, the cubic is inverted.
Y11_Cubic_Graphs_04.gif

y = x3 + k

The k has the effect of moving the cubic up or down the y-axis by k units.

 

Y11_Cubic_Graphs_05.gif

y = (x − h)3

The h has the effect of moving the basic cubic along the x-axis by h units.

Y11_Cubic_Graphs_06.gif