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Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis—where y = 0). The case shown has two critical points. Here the function is f(x) = (x3 + 3x2 − 6x − 8)/4.
In mathematics, a cubic function is a function of the form that is, a polynomial function of degree three. In many texts, the coefficientsa, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers.
Setting f(x) = 0 produces a cubic equation of the form
whose solutions are called roots of the function. The derivative of a cubic function is a quadratic function.
A cubic function with real coefficients has either one or three real roots (which may not be distinct);[1] all odd-degree polynomials with real coefficients have at least one real root.
The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic function is monotonic. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. Up to an affine transformation, there are only three possible graphs for cubic functions.
Cubic functions are fundamental for cubic interpolation.
^Bostock, Linda; Chandler, Suzanne; Chandler, F. S. (1979). Pure Mathematics 2. Nelson Thornes. p. 462. ISBN 978-0-85950-097-5. Thus a cubic equation has either three real roots... or one real root...
cubicfunction is a function of the form f ( x ) = a x 3 + b x 2 + c x + d , {\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,} that is, a polynomial function of...
called roots of the cubicfunction defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic equation are real...
In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite...
of a quartic function is a cubicfunction. Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic...
Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which...
vertex Cubic crystal system, a crystal system where the unit cell is in the shape of a cube Cubicfunction, a polynomial function of degree three Cubic equation...
continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a cubicfunction with a local...
2016 1709 update. CUBIC is a less aggressive and more systematic derivative of BIC TCP, in which the window size is a cubicfunction of time since the...
In mathematics, a concave function is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex...
straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubicfunction: Third degree polynomial. Quartic function: Fourth degree polynomial...
Cubic zirconia (abbreviated CZ) is the cubic crystalline form of zirconium dioxide (ZrO2). The synthesized material is hard and usually colorless, but...
volumes Conversion of units § Volume Cube (arithmetic) Cube root Cubic equation Cubicfunction NIST Guide to the SI - B.8 Factors for Units Listed Alphabetically...
ex-tangential quadrilateral. Critical points of a cubicfunction and inflection points of a quartic function are found by solving a quadratic equation. Babylonian...
quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions...
a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies...
field. Because they have an odd degree, septic functions appear similar to quintic and cubicfunctions when graphed, except they may possess additional...
consider the algebraic function determined by the equation y 3 − x y + 1 = 0. {\displaystyle y^{3}-xy+1=0.\,} Using the cubic formula, we get y = − 2...
18.[citation needed] CUBIC is a less aggressive and more systematic derivative of BIC, in which the window is a cubicfunction of time since the last...
range from −25 °C to 40 °C is approximated by the following empirical cubicfunction: L water ( T ) ≈ ( 2500.8 − 2.36 T + 0.0016 T 2 − 0.00006 T 3 ) J/g...
smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable...
value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function. For both real and complex...
basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous...
{1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}} is the probability density function for the normal distribution with expected value 0 and standard deviation...
b_{2}=-k_{2}(x_{2}-x_{1})+(y_{2}-y_{1})=-1.6875.} In the figure, the spline function consisting of the two cubic polynomials q 1 ( x ) {\displaystyle q_{1}(x)} and q 2 (...
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