quartic function

(noun)

Any polynomial function whose greatest exponent is of power four.

Related Terms

Examples of quartic function in the following topics:

  • Zeros of Polynomial Functions with Real Coefficients

    • A root, or zero, of a polynomial function is a value that can be plugged into the function and yield $0$.
    • The zero of a function, $f(x)$, refers to the value or values of $x$ that will result in the function equaling zero, $f(x)=0$.
    • These are often called the roots of the function.
    • For example, if given the function:
    • More complicated equations also exist for the higher functions, such as cubic and quartic functions, though their expressions are beyond the scope of this atom.

  • Other Equations in Quadratic Form

    • For example, if a quartic equation is biquadratic—that is, it includes no terms of an odd-degree— there is a quick way to find the zeroes of the quartic function by reducing it into a quadratic form.  
    • Consider a quadratic function with no odd-degree terms which has the form:
    • It is important to realize that the same kind of substitution can be done for any equation in quadratic form, not just quartics.

  • Exercises

    • First we could begin by dropping the quartic term in the square root in comparison to the quadratic.
    • When we showed that $\gamma$ was the full-width at half-max of the function $\rho^2 = \left( (\omega _0^2 - \omega^2)^2 + \gamma^2 \omega^2\right)^{-1}$ , we dropped certain terms by assuming that $\gamma$$\rho^2 (\omega_0 + \gamma/2)$ was small.

  • B.13 Chapter 13

    • There are also simply solvable quartics, but this is beyond the scope of the question.

  • Functional Groups

    • Functional groups are atoms or small groups of atoms (two to four) that exhibit a characteristic reactivity when treated with certain reagents.
    • A particular functional group will almost always display its characteristic chemical behavior when it is present in a compound.
    • Because of their importance in understanding organic chemistry, functional groups have characteristic names that often carry over in the naming of individual compounds incorporating specific groups.
    • In the following table the atoms of each functional group are colored red and the characteristic IUPAC nomenclature suffix that denotes some (but not all) functional groups is also colored.

  • Inverse Trigonometric Functions

    • Each trigonometric function has an inverse function that can be graphed.
    • To use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse.
    • However, the sine, cosine, and tangent functions are not one-to-one functions.
    • As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one.
    • The arcsine function is a reflection of the sine function about the line $y = x$.

  • Introduction to Rational Functions

    • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
    • A rational function is any function which can be written as the ratio of two polynomial functions.
    • Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
    • Note that every polynomial function is a rational function with $Q(x) = 1$.
    • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.

  • Functional Structure

    • An organization with a functional structure is divided based on functional areas, such as IT, finance, or marketing.
    • Functional departments arguably permit greater operational efficiency because employees with shared skills and knowledge are grouped together by functions performed.
    • Functional structures may also be susceptible to tunnel vision, with each function perceiving the organization only from within the frame of its own operation.
    • This organizational chart shows a broad functional structure at FedEx.
    • Each different functions (e.g., HR, finance, marketing) is managed from the top down via functional heads (the CFO, the CIO, various VPs, etc.).

  • Functional Groups

    • Functional groups also play an important part in organic compound nomenclature; combining the names of the functional groups with the names of the parent alkanes provides a way to distinguish compounds.
    • Functionalization refers to the addition of functional groups to a compound by chemical synthesis.
    • In materials science, functionalization is employed to achieve desired surface properties; functional groups can also be used to covalently link functional molecules to the surfaces of chemical devices.
    • Alcohols are a common functional group (-OH).
    • Define the term "functional group" as it applies to organic molecules

  • Expressing Functions as Power Functions

    • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
    • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
    • Polynomials are made of power functions.
    • Functions of the form $f(x) = x^3$, $f(x) = x^{1.2}$, $f(x) = x^{-4}$ are all power functions.
    • Describe the relationship between the power functions and infinitely differentiable functions