Learning Outcomes • To recognise a cubic function: y = f (x) = ax3 + bx2 + cx + d.
• To be able to plot a graph of a cubic function and recognise the various shapes of cubic graphs.
• To be able to fi nd properties of cubic functions (crossing axes). • To work with intersecting cubic functions. • To use cubic functions to solve real-life problems. • To understand how to transform functions.
18
18.1 What is a cubic function?
A cubic function is a relationship between two variables x and y which can be written in the form y = f (x) = ax 3 + bx 2 + cx + d, where a, b, c, d are constant real numbers, a ≠ 0.
a is called the coeffi cient of x 3. b is called the coeffi cient of x 2. c is called the coeffi cient of x. d is called the constant term.
EXAMPLE 1
(a) If y = 2x 3 − 5x 2 + 7x − 1, fi nd y when x = 2. (b) If f (x) = −3x 3 + 5x 2 − x − 3, fi nd f (−3).
Solution
(a) x = 2: y = 2(2)3 − 5(2)2 + 7(2) − 1 = 9 (b) f (−3) = −3(−3)3 + 5(−3)2 − (−3) − 3 = 126
