4.2PureStretch

Quartic Graphs

A quartic is a degree-4 polynomial, $y = a x^{4} + \dots$ . Its graph is a “W” or “M” style curve with up to four roots and three turning points. The same factorising-and-sketching ideas as cubics apply.

25 min Video by Zeeshan Zamurred Graphs and Transformations

Edexcel AS Level Maths: 4.2 Quartic GraphsWatch the full walkthrough before the notes below.

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What you'll be able to do

Recognise the shape of a quartic from the sign of a
Find roots from a factorised quartic
Handle repeated roots on a quartic
Sketch a quartic with its key features

Overall shape

A positive quartic looks like a wide “W” — both ends point upwards. A negative quartic is flipped, like an “M”, with both ends pointing down. This is because the $x^{4}$ term dominates for large $∣ x ∣$ .

a > 0 : both ends up (W) a < 0 : both ends down (M)

Even-degree polynomials have both ends pointing the same way.

Roots from factors

As with cubics, each linear factor gives a crossing point and each squared factor gives a touch. A quartic can have up to four real roots.

1Set each factor to zero.

Answerroots at

x = - 3, - 1, 1, 2

Repeated roots

A factor such as $(x - 1)^{2}$ makes the curve touch the axis at $x = 1$ , reducing the number of crossings. The $y$ -intercept again comes from setting $x = 0$ .

Tip — Both ends of a quartic point the same way — unlike a cubic, where they point opposite ways.

Formula recap

y = a x^{4} + b x^{3} + c x^{2} + d x + e

General quartic.

a > 0 \Rightarrow W shape

Both ends up.

a < 0 \Rightarrow M shape

Both ends down.

Common mistakes to avoid

Drawing a quartic with ends pointing in opposite directions (like a cubic).

A quartic is even-degree: both ends point the same way.

Assuming a quartic always has four roots.

It can have 4, 3, 2, 1 or 0 real roots depending on the factors.

Key takeaways

Positive quartic = W (ends up); negative = M (ends down).
Each factor gives a root; squared factors give touches.
Up to four real roots and three turning points.

Test yourself

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