Year 11 / Binomial expansion and cubic functions
Follow a cubic to its ends.
A cubic has degree three. With positive leading coefficient, its left end falls and its right end rises. Negative leading coefficient reverses this. Translation form a(x − h)³ + k places the central point of the basic cubic at (h, k).
A factor (x − r) gives a zero at r. A repeated squared factor touches the axis there instead of crossing. A graph can reveal useful features, but an exact root requires algebraic evidence or an explicitly approximate numerical solution.
INTERACTIVE MODEL
Try it. Watch it change.
y = 1(x − 0)³ + 0
The graph is calculated from the displayed rule. Drag the highlighted point or use the sliders.
Explore: Move the central translation point and reverse the leading coefficient. Compare the ends of the graph.
Factorised form
Describe y = (x + 1)(x − 2)².
- Zeros are x = −1 and x = 2.
- The graph crosses at −1 and touches at the repeated zero 2.
- Its leading coefficient is positive, so it falls left and rises right.
Assumed knowledge
Quadratic factors and multiplying brackets.
Learning checkpoints & source
YOUR LEARNING CHECKPOINT- Connect factors to zeros and multiplicity.
- Read translations and end behaviour of cubic graphs.
READY TO TRY IT?