**Contents:**

## What is a Quartic Function?

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) = a_{4} x^{4} + a_{3} x^{3} + a_{2} x^{2} +a_{1} x + a_{0}.

Where:

- a
_{4}is a nonzero constant. - a
_{3}, a_{2}, a_{1}and a_{0}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 a_{0} 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.

Three basic shapes are possible. For a > 0:

## 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.

## Quartic Curve Examples

A quartic curve is any curve given by a fourth degree polynomial. It can be defined by the following equation

Ax^{4} + By^{4} + Cx^{3}y + Dx^{2}y^{2} + Exy^{3} + Fx^{3} + Gy^{3} + Hx^{2}y + Ixy^{2} + Jx^{2} + Ky^{2} + Lxy + Mx + Ny + P = 0.

Examples of quartic curves:

- Ampersand curve,
- Bean curve,
- Bow curve,
- Bullet nose curve,
- Butterfly curve,
- Cardioid curve (polar function),
- Cassini ovals,
- Cruciform curve,
- Deltoid curve,
- Devil’s curve,
- Eight curve,
- Fish curve,
- Hippopede,
- Kampyle of Eudoxus,
- Klein quartic,
- Piriform curve,
- Spiric section
- Trifolium curve (three leaf clover, Kepler’s folium).

## References

Davidson, Jon. Fourth Degree Polynomials. Retrieved from https://www.sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html on May 16, 2019.