Where does y = (x − 1)(x − 2)(x − 3) cross the y-axis?

(0, −6). (−1)(−2)(−3) = −6.

Where does y = x(x + 3)(x − 5) cross the x-axis?

0, −3 and 5. Each factor is zero in turn.

A cubic has a positive x³ term. How does its graph run?

from bottom left to top right. Large positive x gives large positive y.

For which values of x is (x + 1)(x − 2)(x − 4) negative?

x < −1 or 2 < x < 4. The graph is below the axis there.

What does the graph of y = (x − 2)²(x + 1) do at x = 2?

touches the x-axis. A squared factor means the curve touches and turns back.

Graphs of cubics, and cubic inequalities · Cambridge IGCSE Additional Mathematics

In plain words

A cubic written as three brackets tells you almost everything about its graph: where it crosses the x-axis, where it crosses the y-axis, and which way it runs. Once it is sketched, cubic inequalities can simply be read off.

6 things to know

y = (x − a)(x − b)(x − c) crosses the x-axis at x = a, x = b and x = c.
It crosses the y-axis where x = 0: multiply the three constants together.
If the x³ term is positive, the graph comes up from the bottom left and goes off to the top right. If it is negative, it runs from top left to bottom right.
A squared bracket, (x − a)², means the graph touches the x-axis at a without crossing it.
For y = |cubic|, reflect the parts below the x-axis in the x-axis.
To solve a cubic inequality, sketch the graph. f(x) > 0 where the graph is above the x-axis, and f(x) < 0 where it is below.

Worked example

Sketch y = (x + 1)(x − 2)(x − 4), and solve (x + 1)(x − 2)(x − 4) > 0.

It crosses the x-axis at −1, 2 and 4. At x = 0, y = (1)(−2)(−4) = 8.
The x³ term is positive: the curve rises from the bottom left, through −1, over a hump, down through 2, and up through 4.
The graph is above the axis between −1 and 2, and to the right of 4.
−1 < x < 2 or x > 4.

Tips and tricks

Label all four intercepts on a sketch: three on the x-axis and one on the y-axis.
Check a region with one test value. At x = 0 the value is 8, which is positive, so the stretch containing 0 is above the axis.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Graphs of cubics, and cubic inequalities: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

Where does y = (x − 1)(x − 2)(x − 3) cross the y-axis?
- (0, 6)
- (0, −6) (the answer)
- (0, 1)
- (0, 0)
(−1)(−2)(−3) = −6.
Where does y = x(x + 3)(x − 5) cross the x-axis?
- 3 and −5
- 0, 3 and −5
- 0, −3 and 5 (the answer)
- −3 and 5
Each factor is zero in turn.
A cubic has a positive x³ term. How does its graph run?
- from top left to bottom right
- from bottom left to top right (the answer)
- it is U-shaped
- it is a straight line
Large positive x gives large positive y.
For which values of x is (x + 1)(x − 2)(x − 4) negative?
- −1 < x < 2 or x > 4
- x < −1 or 2 < x < 4 (the answer)
- x > 4 only
- x < −1 only
The graph is below the axis there.
What does the graph of y = (x − 2)²(x + 1) do at x = 2?
- crosses the x-axis
- touches the x-axis (the answer)
- crosses the y-axis
- has an asymptote
A squared factor means the curve touches and turns back.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

2 questions, 6 marks. Write your answers on paper, then check them.

Graphs of cubics, and cubic inequalities

Cambridge IGCSE Additional Mathematics 0606 · 6 marks · papermunch.org

Name ______________________________ Date ______________

State the coordinates of the points where y = (x + 2)(x − 1)(x − 3) meets the axes.[3]
Show answerHide answer
(−2, 0), (1, 0) and (3, 0) on the x-axis, and (0, 6) on the y-axis.
I got this one
Solve (x + 2)(x − 1)(x − 3) ≤ 0.[3]
Show answerHide answer
x ≤ −2 or 1 ≤ x ≤ 3. The graph is on or below the axis there.
I got this one

Answers: Graphs of cubics, and cubic inequalities

1. (−2, 0), (1, 0) and (3, 0) on the x-axis, and (0, 6) on the y-axis.
2. x ≤ −2 or 1 ≤ x ≤ 3. The graph is on or below the axis there.