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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) = a4 x4 + a3 x3 + a2 x2 +a1 x + a0.
Where:
- 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.
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
Ax4 + By4 + Cx3y + Dx2y2 + Exy3 + Fx3 + Gy3 + Hx2y + Ixy2 + Jx2 + Ky2 + 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.