Graphs Of Factorised Cubic Functions
Learn to sketch factorised cubic functions for Queensland Year 11 Mathematical Methods (QCAA). When a cubic is a product of linear factors, its graph follows from those factors.
You will learn to plot x-intercepts from the zeros, decide where the curve crosses or touches the x-axis from a repeated factor, mark the y-intercept, and read the behaviour for large positive and negative x from the sign of a.
Every question with a fully worked solution.
- Graphs Of Factorised Cubic Functions - Video - Graphs of factorised cubic functions Watch
Theory
In Year 11 Mathematical Methods (QCAA, Unit 1), a factorised cubic \(y=a(x-x_1)(x-x_2)(x-x_3)\) can be sketched straight from its factors: the x-intercepts are the zeros, a squared factor makes the graph touch the axis while a single factor makes it cross, and the sign of \(a\) fixes the end behaviour. This page shows how to find the intercepts, read touch-versus-cross and sketch the cubic without calculus.
A factorised cubic is written \(y=a(x-x_1)(x-x_2)(x-x_3)\). The numbers \(x_1,x_2,x_3\) are the zeros, so the curve cuts the \(x\)-axis at \(x=x_1,\,x_2,\,x_3\). The y-intercept is found by putting \(x=0\).
The power of a factor is its multiplicity. A single (odd) factor makes the graph cross the axis; a squared (even) factor makes it just touch the axis and turn back. A cube factor \((x-h)^3\) flattens as it crosses.
The leading coefficient \(a\) sets the end behaviour. When \(a>0\) the cubic falls on the left and rises on the right; when \(a<0\) it does the opposite.
For \(y=a(x-x_1)(x-x_2)(x-x_3)\):
How to sketch a factorised cubic
- Zeros: set each factor to \(0\) to get the \(x\)-intercepts.
- Multiplicity: mark each intercept as a cross (single or cube factor) or a touch (squared factor).
- y-intercept: substitute \(x=0\) into the factor form.
- Ends: use the sign of \(a\) — \(a>0\) rises to the right, \(a<0\) falls — then join the points into a smooth curve.
x-intercepts — set each factor to \(0\):
| \((x+2)(x-1)(x-3)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(-2,\ 1,\ 3\) |
Each is a single factor, so the curve crosses at all three.
y-intercept — substitute \(x=0\):
| \(y\) | \(=\) | \((0+2)(0-1)(0-3)\) |
| \(=\) | \((2)(-1)(-3)\) | |
| \(=\) | \(6\) |
End behaviour — the leading term is \(x^3\), so \(a=1>0\):
| \(\text{left}\) | \(\to\) | \(-\infty\) |
| \(\text{right}\) | \(\to\) | \(+\infty\) |
Intercepts \((-2,0),(1,0),(3,0)\) and \((0,6)\); rises to the right.
x-intercepts:
| \(-(x+1)(x-2)^2\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(-1\ \text{(single)},\ 2\ \text{(squared)}\) |
At \(x=-1\) the graph crosses; at \(x=2\) it touches and turns back.
y-intercept — put \(x=0\):
| \(y\) | \(=\) | \(-(0+1)(0-2)^2\) |
| \(=\) | \(-(1)(4)\) | |
| \(=\) | \(-4\) |
End behaviour — leading term \(-x^3\), so \(a=-1<0\):
| \(\text{left}\) | \(\to\) | \(+\infty\) |
| \(\text{right}\) | \(\to\) | \(-\infty\) |
Crosses at \((-1,0)\), touches at \((2,0)\), \(y\)-intercept \(-4\); falls to the right.
x-intercepts — the factor \(x\) gives a zero at the origin:
| \(2x(x+2)(x-3)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(-2,\ 0,\ 3\) |
All three factors are single, so the curve crosses at each.
y-intercept — substitute \(x=0\):
| \(y\) | \(=\) | \(2(0)(0+2)(0-3)\) |
| \(=\) | \(0\) |
The curve passes through the origin \((0,0)\).
End behaviour — leading term \(2x^3\), so \(a=2>0\):
| \(\text{left}\) | \(\to\) | \(-\infty\) |
| \(\text{right}\) | \(\to\) | \(+\infty\) |
Intercepts \((-2,0),(0,0),(3,0)\); rises to the right.
Factors — a cross is a single factor, a touch is a squared factor:
| \(y\) | \(=\) | \(a(x+3)(x-1)^2\) |
Find \(a\) — substitute the point \((0,-3)\):
| \(-3\) | \(=\) | \(a(0+3)(0-1)^2\) |
| \(-3\) | \(=\) | \(a(3)(1)\) |
| \(-3\) | \(=\) | \(3a\) |
| \(a\) | \(=\) | \(-1\) |
Write the rule with \(a=-1\):
| \(y\) | \(=\) | \(-(x+3)(x-1)^2\) |
Rule: \(y=-(x+3)(x-1)^2\).
Common pitfalls
Frequently asked questions
How do you find the x-intercepts of a factorised cubic?
Set each factor equal to \(0\) and solve. For \(y=a(x-x_1)(x-x_2)(x-x_3)\) the intercepts are \(x=x_1,x_2,x_3\).
What does a squared factor do to a cubic graph?
A squared factor \((x-h)^2\) makes the graph touch the \(x\)-axis at \(x=h\) and turn back, rather than crossing through it.
How do you find the y-intercept of a factorised cubic?
Substitute \(x=0\) into the factor form: \(y=a(0-x_1)(0-x_2)(0-x_3)\).
How does the sign of a change the graph?
If \(a>0\) the cubic rises to the right (and falls to the left); if \(a<0\) it falls to the right (and rises to the left).
Do I need calculus to sketch a factorised cubic?
No. In Year 11 you sketch from the intercepts, the multiplicity of each factor, the \(y\)-intercept and the sign of \(a\) — turning-point coordinates are not required.