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The graph of a real single-variable quadratic function is a parabola. If a quadratic function is equated with zero, then the result is a quadratic equation . The solutions of a quadratic equation are the zeros (or roots ) of the corresponding quadratic function, of which there can be two, one, or zero.
Therefore, the graph of the function f(x − h) = (x − h) 2 is a parabola shifted to the right by h whose vertex is at (h, 0), as shown in the top figure. In contrast, the graph of the function f(x) + k = x 2 + k is a parabola shifted upward by k whose vertex is at (0, k), as shown in the center figure.
The function f(x) = ax 2 + bx + c is a quadratic function. [16] The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of a, b, and c. If a > 0, the parabola has a minimum point and opens upward.
Marc van Kreveld suggested the algorithmic problem of computing shortest paths between vertices in a line arrangement, where the paths are restricted to follow the edges of the arrangement, more quickly than the quadratic time that it would take to apply a shortest path algorithm to the whole arrangement graph. [42]
The Paley graph of order 9 is a locally linear graph, a rook's graph, and the graph of the 3-3 duoprism. The Paley graph of order 13 has book thickness 4 and queue number 3. [9] The Paley graph of order 17 is the unique largest graph G such that neither G nor its complement contains a complete 4-vertex subgraph. [10]
In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y = a x k {\displaystyle y=ax^{k}} – appear as straight lines in a log–log graph, with the exponent corresponding to ...
Graph of the logistic map (the relationship between + and ). The graph has the shape of a parabola, and the vertex of the parabola changes as the parameter r changes. As mentioned above, the logistic map can be used as a model to consider the fluctuation of population size.
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
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