Cubic Graphs

A cubic's factorised form is a sketch waiting to happen — every root tells you exactly where the curve meets the x-axis, and whether it crosses through or just touches.

The shape of a cubic

A cubic function has the form $y = ax^3+bx^2+cx+d$, with $a \neq 0$. Its overall shape is fixed entirely by the sign of $a$: if $a>0$, the curve rises from bottom-left to top-right; if $a<0$, it falls from top-left to bottom-right. Between those two ends, the curve can wiggle — rising, turning, falling, turning again — but the far-left and far-right behaviour never changes.

Reading roots from factorised form

When a cubic is written as a product of linear factors, each factor gives a root — a point where the curve meets the $x$-axis — and how the factor appears tells you exactly how the curve behaves there:

Single root $(x-p)$: the curve crosses the axis at $x=p$, passing from one side to the other.
Repeated root $(x-p)^2$: the curve touches the axis at $x=p$ and turns back — the same behaviour as a repeated root of a quadratic.
Triple root $(x-p)^3$: the curve crosses the axis at $x=p$, but flattens out as it does — an inflection through the axis rather than a clean crossing.

Worked example 1

Sketching from three distinct roots

Sketch the graph of $y = (x-1)(x+2)(x-3)$, marking all axis intercepts.

MethodRoots, y-intercept, then join respecting the end behaviour

1

The leading term, from multiplying the three $x$'s together, is $+x^3$, so $a>0$: bottom-left to top-right.

Positive leading coefficient.

2

Three single roots at $x=1,\,x=-2,\,x=3$ — the curve crosses the axis at each.

Each factor appears once.

3

$y$-intercept: substitute $x=0$: $y = (-1)(2)(-3) = 6$.

The value where the curve crosses the y-axis.

4

Order the roots on the axis: $-2, 1, 3$. Starting from bottom-left, the curve rises, crosses at $-2$, rises to a local max, falls, crosses at $1$, dips to a local min, then rises again, crossing at $3$ and continuing up.

Three crossings force exactly this rise-fall-rise pattern.

▪ sketched

Examiner's eye

A cubic with three distinct real roots always has exactly this "wiggle" shape — one local maximum and one local minimum between the outer roots. You don't need calculus to know a turning point exists between $-2$ and $1$, and another between $1$ and $3$; the crossings alone guarantee it.

Worked example 2

A repeated root · touching, not crossing

Sketch the graph of $y = (x-2)^2(x+1)$, marking all axis intercepts.

MethodIdentify the repeated root's touching behaviour

1

Leading term is $+x^3$ (positive), so bottom-left to top-right.

a > 0.

2

$x=2$ is a repeated root (power 2): the curve touches the axis here and turns back. $x=-1$ is a single root: the curve crosses here.

Different factor powers give different behaviour.

3

$y$-intercept: at $x=0$, $y=(-2)^2(1) = 4$.

Substitute x=0.

4

From bottom-left, the curve rises and crosses at $x=-1$, continues up to a local max, then descends to touch the axis at $x=2$ (without crossing), and rises away again.

Consistent with one crossing and one touch.

▪ sketched

The key insight

At a repeated root, the curve behaves locally like a parabola touching the axis — it doesn't pass through to the other side. This is identical to the repeated-root behaviour you've already seen for quadratics; a cubic with a squared factor is really "a quadratic touch, plus one more crossing" layered together.

Worked example 3

Factorising first, using the factor theorem

Given that $x=2$ is a root of $y = x^3 - 3x^2 - 4x + 12$, factorise fully and sketch the curve.

Not every cubic arrives pre-factorised — often you need the factor theorem first, exactly as in an earlier lesson, before any sketching can begin.

MethodDivide out the known factor, factorise the remainder, then sketch

1

Since $x=2$ is a root, $(x-2)$ is a factor. Divide: $x^3-3x^2-4x+12 = (x-2)(x^2-x-6)$.

Polynomial division or inspection.

2

Factorise the quadratic: $x^2-x-6 = (x-3)(x+2)$.

Standard quadratic factorising.

3

Fully factorised: $y=(x-2)(x-3)(x+2)$ — three distinct roots at $x=2,3,-2$.

Combine both factorisations.

4

Leading coefficient is positive, $y$-intercept at $x=0$ is $(-2)(-3)(2)=12$ (matching the original constant term). Sketch: rise, cross at $-2$, local max, cross at $2$, local min, cross at $3$, rise.

Same sketching method as example 1, now with three roots in order -2, 2, 3.

▪ solved

Common mistake

Always check the $y$-intercept from the factorised form matches the constant term of the original expanded cubic — here both give $12$. If they disagree, a sign error has crept into the factorisation.

Worked example 4

Finding a cubic's equation from its graph · Challenge

A cubic curve crosses the x-axis at $x=-3$, touches the x-axis at $x=1$, and passes through the point $(0, -18)$. Find the equation of the curve in the form $y=ax^3+bx^2+cx+d$.

The root behaviour (crossing vs touching) tells you the power of each factor directly; only the overall scaling constant $a$ remains unknown, found from the extra point given.

MethodBuild the factorised form from the root behaviour, then find a

1

Crossing at $x=-3$ gives a single factor $(x+3)$. Touching at $x=1$ gives a repeated factor $(x-1)^2$.

Crossing = power 1, touching = power 2.

2

So $y = k(x+3)(x-1)^2$ for some constant $k$ (a cubic needs exactly a single plus a repeated root to reach degree 3).

Degrees check: 1 + 2 = 3.

3

Use the point $(0,-18)$: $-18 = k(3)(-1)^2 = k(3)(1) = 3k$.

Substitute x=0, y=-18.

4

Solve: $k=-6$.

Divide both sides by 3.

5

So $y = -6(x+3)(x-1)^2$. Expanding: $(x-1)^2=x^2-2x+1$, so $(x+3)(x^2-2x+1) = x^3+x^2-5x+3$, giving $y=-6x^3-6x^2+30x-18$.

Multiply out to reach the requested form.

▪ solved

Look deeper — negative k flips the whole picture

Since $k=-6$ is negative, the curve actually falls from top-left to bottom-right, the opposite of what the "positive-looking" factorised form might suggest at a glance. Always determine the sign of the overall scaling constant last, from the given point — never assume it's positive.

Check yourself

Sketch $y=(x+1)(x-2)(x+4)$, marking all intercepts, and state the shape of the curve at each root.
(Hint: find the y-intercept, order the three roots, then apply the sign of the leading coefficient.)

Reveal solution

Method

1

Roots at $x=-4,-1,2$, all single, so the curve crosses at each.

2

y-intercept: $x=0$ gives $y=(1)(-2)(4)=-8$.

3

Leading coefficient positive: rise from bottom-left, cross at $-4$, local max, cross at $-1$, local min, cross at $2$, rise to top-right.

Standard three-crossing wiggle.

▪ solved