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Sketch cubic and quartic graphs using roots, repeated factors, intercepts and end behaviour. Recover equations from graph features and practise with full solutions.
Before you startFactorising, quadratic equations and coordinate axes
01 / End behaviour
A cubic has degree 3; a quartic has degree 4. Their leading coefficients are non-zero. Far enough from the origin, the highest-power term determines which way the graph goes.
Cubic, positive leading coefficient:
left end down, right end up.
Quartic, positive leading coefficient:
both ends up.
A negative leading coefficient reverses these directions. This describes the ends, not every slope in the middle. A cubic may have two turning points or none. A quartic need not have a “W” shape.
Try the graph choices and the negative multiplier. Multiplying by −1 reflects the graph in the x-axis and keeps its roots fixed.
The cubic (x + 2)(x − 1)² crosses at −2, touches at 1, and has y-intercept 2. Its left end goes down and right end up.
02 / Roots and contact
A root r gives the point (r, 0). If the factor (x − r) appears m times, r has multiplicity m.
These statements concern a polynomial with all copies of that factor counted. Near the root, the other factors are non-zero, so (x − r)m determines whether the sign changes.
f(x) = (x + 2)(x − 1)²
Root −2: simple → crosses
Root 1: double → touches
The triple-root cubic (x − 1)³ crosses at (1, 0) despite being flat there. A horizontal tangent does not automatically mean a turning point.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
03 / Sketch method
A sketch is not a scale drawing. Unless asked, exact turning-point coordinates are unnecessary here. Do not invent them from a rough picture.
With a factor such as (3 − x), the leading contribution is −x. Count that minus sign before deciding the end behaviour.
y = x³ − 2x² − 8x
Take out the common x.
y = x(x − 4)(x + 2)
Three simple roots: −2, 0, 4.
y-intercept: (0, 0)
The graph crosses at each root.
Signs: −, +, −, +
From left to right across the three roots. Left end down; right end up.
04 / Quartics
A quartic has at most four distinct real roots. Repeated roots count more than once toward its degree, but mark just one point on the axis.
y = (x + 2)²(x − 1)(3 − x)
This is degree 4 with leading coefficient −1. Both ends go down. It touches at −2 and crosses at 1 and 3. Its y-intercept is −12.
Two double roots give two touches; a triple and a simple root give two crossings, one flattened. A fourth-order root gives a single touch. A positive quartic such as x⁴ + 1 has no real roots.
The graph choices above include a quartic with four simple roots, two double roots, and no real roots.
y = (x² − 2)(x² − 5)
Set each quadratic factor equal to zero.
x = −√5, −√2, √2, √5
All four roots are simple. In this order they are about −2.24, −1.41, 1.41, 2.24.
At x = 0: y = (−2)(−5) = 10
Both ends rise, and the graph crosses each root.
Signs: +, −, +, −, +
Use exact surds in labels unless decimals are requested.
05 / Recover an equation
If a cubic has three known simple roots r, s and t, write y = a(x − r)(x − s)(x − t). A point away from the roots determines a.
Use the stated degree and multiplicities. Roots alone do not determine the leading coefficient, and without the degree they do not determine the whole polynomial.
Roots: −3, 1, 2; y-intercept 12
y = a(x + 3)(x − 1)(x − 2)
12 = 6a ⇒ a = 2
y = 2x³ − 14x + 12
The coefficient of x² is zero. Expanding is optional unless the requested form needs it.
Roots −2 and 3, both double
The graph passes through (0, 18)
Degree 4 means these factors account for the whole polynomial.
y = a(x + 2)²(x − 3)²
Insert the non-root point.
18 = a · 4 · 9 ⇒ a = 1/2
Both ends rise because a is positive.
y = ½x⁴ − x³ − 11x²/2
+ 6x + 18
Compare coefficients if the question asks for them.
06 / Fewer real roots
A cubic can have just one distinct real root. For example:
y = (x − 2)(x² + 2x + 5)
x² + 2x + 5 = (x + 1)² + 4 > 0
The only real root is 2. The quadratic factor stays positive, so the cubic has the sign of x − 2. Its y-intercept is −10 and its ends have the usual positive-cubic directions.
For y = (x + 3)(x − 1)(x² + 1), the last factor is always positive. The quartic crosses at −3 and 1, is negative between them, and positive outside. Both ends rise.
A graph window may hide a distant root. Algebra, including a quadratic discriminant when useful, checks whether the sketch accounts for every real root.
07 / Your turn
For each sketch, state the ends, all axis intercepts, and which roots cross or touch. Use the hints only when needed.
y = (2 − x)(x + 1)²
The leading coefficient is −1.
Root −1 is double: touch. Root 2 is simple: cross. The y-intercept is (0, 2). Left end up, right end down. The graph is positive for x < 2 except at −1, and negative for x > 2.
y = 2x³ + 2x² − 12x
Take out 2x, then factorise the quadratic.
y = 2x(x + 3)(x − 2)
Simple roots −3, 0, 2: cross at all three. The y-intercept is the origin. Left end down, right end up. Signs from left to right: −, +, −, +.
y = −(x + 2)³
A triple root has odd multiplicity.
The graph crosses with a horizontal tangent at (−2, 0). Its y-intercept is (0, −8). Left end up, right end down. It has no turning point.
y = (x − 2)²(x + 1)(x − 4)
There are three distinct roots, with multiplicities adding to four.
Both ends up. Cross at −1 and 4; touch at 2. The y-intercept is (0, −16). It is negative between −1 and 4 except at the touching root 2, and positive outside.
A cubic crosses at −2, 1 and 3, and passes through (0, −12). Find its equation in expanded form.
Use a(x + 2)(x − 1)(x − 3).
−12 = 6a ⇒ a = −2
y = −2(x + 2)(x − 1)(x − 3)
y = −2x³ + 4x² + 10x − 12
y = (x − 4)(x² + 4x + 8)
Complete the square in the quadratic factor.
x² + 4x + 8 = (x + 2)² + 4
Only real root: 4, a simple crossing. The y-intercept is (0, −32). Left end down, right end up. Negative to the left of 4; positive to its right.
A quartic touches the axis at −1 and 2, and passes through (0, 8). Find its expanded equation.
Both roots must be double.
y = a(x + 1)²(x − 2)²
8 = 4a ⇒ a = 2
y = 2x⁴ − 4x³ − 6x² + 8x + 8
y = (x² − 3)(x² − 7)
Solve each quadratic factor separately.
Four simple roots, ordered −√7, −√3, √3, √7: cross at each. The y-intercept is (0, 21), and both ends rise. Signs from left to right: +, −, +, −, +.
08 / Recap
Section 1 of 8 · End behaviour