Quadratic function in the context of "Ulam spiral"

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship ${\displaystyle xy=1.}$ In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others.

In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings:

• A fixed and well-defined number or other non-changing mathematical object, or the symbol denoting it. The terms mathematical constant or physical constant are sometimes used to distinguish this meaning.
• A function whose value remains unchanged (i.e., a constant function). Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question.

For example, a general quadratic function is commonly written as:

In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as$${\displaystyle ax^{2}+bx+c=0\,,}$$where the variable ⁠${\displaystyle x}$⁠ represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.

The values of ⁠${\displaystyle x}$⁠ that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation $${\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}$$where r and s are the solutions for ⁠${\displaystyle x}$⁠.

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup ${\displaystyle \cup }$ (or a straight line like a linear function), while a concave function's graph is shaped like a cap ${\displaystyle \cap }$.

A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function ${\displaystyle f(x)=cx}$ (where ${\displaystyle c}$ is a real number), a quadratic function ${\displaystyle cx^{2}}$ (${\displaystyle c}$ as a nonnegative real number) and an exponential function ${\displaystyle ce^{x}}$ (${\displaystyle c}$ as a nonnegative real number).