Calculus How To

Quartic Function: Definition, Example

Share on

A quartic function is a fourth-degree polynomial: a function which has, as its highest order term, a variable raised to the fourth power.

It can be written as:

f(x) = a4 x4 + a3 x3 + a2 x2 +a1 x + a0.


  • a4 is a nonzero constant.
  • a3, a2, a1 and a0 are also constants, but they may be equal to zero.

The derivative of every quartic function is a cubic function (a function of the third degree).

The quartic was first solved by mathematician Lodovico Ferrari in 1540.

Graph of a Quartic Function

The graph of a fourth-degree polynomial will often look roughly like an M or a W, depending on whether the highest order term is positive or negative.

  • If the coefficient of the leading term, a, is positive, the function will go to infinity at both sides.
  • If the coefficient a is negative the function will go to minus infinity on both sides.

The term a0 tells us the y-intercept of the function; the place where the function crosses the y-axis. The roots of the function tell us the x-intercepts.

The image below shows the graph of one quartic function. This particular function has a positive leading term, and four real roots.
Quartic Function

Three basic shapes are possible. For a > 0:

Three basic shapes for the quartic function (a>0). For a < 0, the graphs are flipped over the horizontal axis, making mirror images.

Properties of Quartic Polynomials

Fourth degree polynomials all share a number of properties:

  • They have up to four roots,
  • Their derivatives have from 1 to 3 roots,
  • They have no general symmetry,
  • They can have one, two, or no (zero) inflection points,
  • Five points, or five pieces of information, can describe it completely,
  • Every polynomial equation can be solved by radicals.


Davidson, Jon. Fourth Degree Polynomials. Retrieved from on May 16, 2019.

Stephanie Glen. "Quartic Function: Definition, Example" From Calculus for the rest of us!

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!