Graphing Cubic Polynomials

1. Identify various types of cubic curves (Refer to Excel Applet for Cubic Curves) Sketch simple cubic curves Form equation of cubic polynomial from sketch Graphing Cubic Polynomials

2. Graphing Cubic Polynomials The real roots of the polynomial equation P(x) = 0 are given by the values of the intercepts of the function y = P(x) with the x-axis. Nature of roots: 3 real and distinct  x = x1, x = x2 and x= x3 are the solutions. 2 real and equal and 1 real and distinct 1 real and 2 complex roots

3. 10 5 0 -4 -2 0 2 4 -5 -10 Definition of Cubic Function A cubic function is a polynomial function of the form ax3 + bx2 + cx + d, where a, b, c and d are constants and a cannot be 0. Example 1: y = x3

4. 10 5 0 -4 -2 0 2 4 -5 -10 Use the excel applet to investigate Example 2: y = x3 – 5x2 + 2x + 8

5. 10 5 0 -2 -1 0 1 2 3 4 -5 -10 Use the excel applet to investigate Example 3: y = x3 – x2 - x +1

6. Graphing Cubic Polynomials How to graph a cubic function? Example : y = x3 – 2x2 –x + 2 (Note: a > 0) Step 1: Check if y can be factorise into 3 linear factors y = (x + 1)(x -2)(x -1) (Sometimes, you may get 1 linear factor and a quadratic factor that cannot be factorised When this happens – use the quadratic formula to solve for x. If it cannot be solved, then there will be 2 complex roots and 1 real root) Step 2: Set y = 0, x = -1, x = 2, x = 1

7. y = x3 – 2x2 –x + 2 Graphing Cubic Polynomials Step 3: Finding the y-intercept. When x = 0, y = 2.  (0, 2)