Cubic graphs and cubic inequalities
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 4 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- A curve with two bends
- The ends decide the shape
- Sketching a factorised cubic
Read the transcript of the teaching part
Last time
Before we start: last time, in The remainder and factor theorems and cubic equations, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
A curve with two bends
A quadratic has one bend. A cubic can have two: it rises, turns, falls, turns again, and rises. You do not need calculus to sketch one. You need the roots, the y intercept, and the direction of the two ends.
The ends decide the shape
A cubic with a positive x cubed term starts low on the left and ends high on the right. With a negative x cubed term, it starts high on the left and ends low on the right. In both cases the ends go in opposite directions, so a cubic always crosses the x axis at least once.
Sketching a factorised cubic
One: the roots. Set each bracket to zero. Mark them on the x axis. Two: the y intercept. Put x equals zero in the whole expression. Three: the ends. Look at the sign when you multiply out the x terms. Four: a repeated factor means the curve touches the axis there, and does not cross. Five: draw a smooth curve through the roots. Label every intercept.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Cubic inequalities solved graphically.
The whole chapter: Equations, Inequalities and Graphs.
Chapter 4: Equations, inequalities and graphs
- Cubic graphs and cubic inequalities 5 min
- Modulus equations and inequalities 5 min
Also for O Level Additional Maths 4037
Which topics come up most in O Level Additional Maths 4037 past papers, counted from every paper since 2004.