1. What are cubic equations?
Algebraic equations in which the highest power of the variable is 3 are called cubic equations.
For example:- y = x³ + 5x - 3, 2x³ + 3 = 0, y = 7x³ - x are all cubic equations. The simplest example of a cubic equation is y = x³.
2. What are cubic curves and their characteristics?
The graphs produced by cubic equations are called cubic curves. Cubic curves have two main shapes depending on the coefficient of x³.
Important points to remember for cubic curves:-
- The basic shape of the graph is determined by the highest power of the variable.
- Sign of the highest power (x³) determines the orientation of the graph.
- Putting x = 0 in the equation gives you the y-intercept.
- A cubic graph may have one, two, or three x-intercepts. When the equation is in the factored form (for example y = (x + a)(x + b)(x + c)), you can let y = 0 to get the x-intercepts.
- The points where the curve turns are the points where the gradients of the curve are zero. [to learn that please refer to the gradient of curves]
Example 1:
The graph of y = x³ - 4x - 4 is shown below.
The root of the equation x³ - 4x - 4 = 0 is p.
a) Calculate y when x = 2.
b) Show that 2 < p < 3
c) Find the smaller interval that contains the value of p.
Example 2:
(x + b)(x - 8)(x + 5) = ax³ + cx² + dx - 80
Find the values of a, b, c and d.
