A quartic function is a polynomial function of degree four, meaning its highest exponent on the variable is 4. They are useful in modeling systems with multiple peaks and valleys, and they appear in physics, engineering, and curve fitting when simpler polynomials aren’t flexible enough.
In standard form, a quartic function looks like the following:
f(x) = ax⁴ + bx³ + cx² + dx + e
where a ≠ 0, and a, b, c, d, and e are constants.
Quartic functions are a type of polynomial that can model more complex relationships than quadratics or cubics. Depending on the coefficients, their graphs can have a wide range of shapes. The leading term (ax⁴) dominates the behavior of the function as x becomes very large or very small.
Learn more in-depth about Polynomials Explained: Definition, Degree & Factoring.
Key Features of a Quartic Function
- The graph is typically U-shaped or W-shaped, but may also appear as a flattened or stretched version depending on the coefficients.
- It can have 0 to 4 real roots (x-intercepts).
- It may have 1 to 3 turning points, which are local maxima or local minima.
- If a > 0, the ends of the graph rise upward. If a < 0, the ends fall downward.
- It is continuous and smooth, with no breaks or sharp corners.
Graph Example
Consider the quartic function:
f(x) = x⁴ − 4x³ + 3x² + 2x − 5
This function has:
- A leading coefficient of 1, so the graph opens upward.
- Degree 4, so the ends of the graph both go to positive infinity.
- A complex shape with potential for multiple turning points and up to four real roots.
Below is a graph of this quartic function.
