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The graph of a degree 1 polynomial (or linear function) f(x) = a0 + a1x, where a1 ≠ 0, is an oblique line with y-intercept a0 and slope a1 . The graph of a degree 2 polynomial. f(x) = a0 + a1x + a2x2, where a2 ≠ 0. is a parabola . The graph of a degree 3 polynomial. f(x) = a0 + a1x + a2x2 + a3x3, where a3 ≠ 0.
Graph polynomial. In mathematics, a graph polynomial is a graph invariant whose value is a polynomial. Invariants of this type are studied in algebraic graph theory. [1] Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. The chromatic polynomial, a polynomial whose values at integer ...
The polynomial is the Tutte polynomial of the bull graph. The red line shows the intersection with the plane , which is essentially equivalent to the chromatic polynomial. The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role ...
The Chebyshev polynomials form a complete orthogonal system. The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives.
Sextic equation. Graph of a sextic function, with 6 real roots (crossings of the x axis) and 5 critical points. Depending on the number and vertical locations of minima and maxima, the sextic could have 6, 4, 2, or no real roots. The number of complex roots equals 6 minus the number of real roots. In algebra, a sextic (or hexic) polynomial is a ...
Chromatic polynomial. All non-isomorphic graphs on 3 vertices and their chromatic polynomials, clockwise from the top. The independent 3-set: k3. An edge and a single vertex: k2(k – 1). The 3-path: k(k – 1)2. The 3-clique: k(k – 1) (k – 2). The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of ...
The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the y -axis. If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros of the corresponding quadratic function. The bivariate case in terms of variables x ...
Characteristic polynomial. In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector ...