1. Cubic curve sketching General shape: a > 0 x y x y a < 0 General form Characteristics: usually 2 humps, may or may not be asymmetrical Important point to note: If there are 3 factors, then there will be 3 x-intercepts Just like in quadratic curve, when there are 2 factors, there are 2 x-intercepts.

2. Getting the factors from the curve: quadratic x y 3 y=f(x) In order to get the x-intercepts, we are actually solving for x  x = -1 and x = 3 means (x +1)(x – 3) = 0 We can get the equation of the curve from its x-intercepts!!

3. Getting the factors from the curve: cubic In order to get the x-intercepts, we are actually solving for x  x = -1, -3 and x = 4 means (x +1)(x + 3)(x – 4) = 0 Again, we can get the equation of the curve from its x-intercepts!! x y 4 -3

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

5. 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 = x 1, x = x 2 and x= x 3 are the solutions. 2 real and equal and 1 real and distinct 1 real and 2 complex roots Graphing Cubic Polynomials

6. Definition of Cubic Function Example 1: y = x 3 -10 -5 0 5 10 -4-2024 A cubic function is a polynomial function of the form ax 3 + bx 2 + cx + d, where a, b, c and d are constants and a cannot be 0.

7. Use the excel applet to investigate Example 2: y = x 3 – 5x 2 + 2x + 8 -10 -5 0 5 10 -4-2024

8. Use the excel applet to investigate Example 3: y = x 3 – x 2 - x +1 -10 -5 0 5 10 -201234

9. Graphing Cubic Polynomials How to graph a cubic function? Example : y = x 3 – 2x 2 –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

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