Quadratic function
A quadratic function is a polynomial function of degree two, defined by the equation f(x) = ax^2 + bx + c, where a, b, and c are real constants and a \neq 0.[1] The graph of such a function forms a parabola, a U-shaped curve that opens upward if a > 0 (indicating a minimum vertex) or downward if a < 0 (indicating a maximum vertex).[2] Quadratic functions can be expressed in multiple forms, including the general form f(x) = ax^2 + bx + c, the vertex form f(x) = a(x - h)^2 + k (where (h, k) is the vertex), and the factored form f(x) = a(x - r)(x - s) (where r and s are the roots).[2] The vertex of the parabola is located at x = -\frac{b}{2a}, and the axis of symmetry is the vertical line passing through this point.[1] The roots, or x-intercepts, are found by solving ax^2 + bx + c = 0 using the quadratic formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where the discriminant b^2 - 4ac determines the number and nature of real roots: positive for two distinct real roots, zero for one real root, and negative for no real roots (complex conjugates).[2] Historically, quadratic equations trace back to ancient Babylonian mathematics around 2000 BCE, where geometric methods solved problems equivalent to quadratics, later systematized by Persian mathematician Al-Khwarizmi in the 9th century CE through completing the square.[3] In modern applications, quadratic functions model real-world phenomena such as projectile motion (e.g., the parabolic trajectory of a thrown ball), optimization problems (e.g., maximizing area or revenue), and physical structures like satellite dish designs.[1][4]Etymology and History
Origin of the Term
The term "quadratic" originates from the Latin adjective quadratus, meaning "squared" or "made square," which directly alludes to the second-degree term in the polynomial expression, involving the square of the variable.[5] This linguistic root emphasizes the mathematical essence of squaring as the defining operation, distinguishing it from higher-degree polynomials.[6] In English mathematical literature, "quadratic" first appeared in a mathematical context during the 1660s, initially describing forms or equations centered on squares without higher powers.[5] By the 1680s, the specific phrase "quadratic equation" had emerged to denote algebraic expressions of this form.[5] As the modern concept of a function gained prominence in the 18th and 19th centuries, the terminology extended to "quadratic function," applying the same etymological foundation to the broader polynomial entity.[7] A related geometric term, "quadrate," shares this etymology from Latin quadratus (past participle of quadrare, "to square"), historically referring to a square shape or the act of forming a square in figures and constructions.[8]Historical Development
The earliest known solutions to quadratic equations date back to the ancient Babylonians around 2000–1800 BCE, who recorded algorithmic methods on clay tablets for problems involving areas and lengths, effectively using techniques akin to completing the square to find positive real roots.[9] These methods addressed practical issues like land measurement but were presented verbally and geometrically without algebraic notation.[10] In ancient Greece, around 300 BCE, Euclid advanced the geometric approach to quadratic equations in his Elements, particularly in Book II, Proposition 14, where he demonstrated how to construct a square equal to a given rectilinear figure, equivalent to solving for roots through mean proportionals without conceptualizing equations as such.[9] This geometric framework influenced subsequent European mathematics, emphasizing constructions with ruler and compass.[11] Following the Greek period, significant advancements occurred during the Indian and Islamic Golden Ages. In 628 CE, the Indian mathematician Brahmagupta provided the first explicit formula for solving quadratic equations in his treatise Brāhmasphuṭasiddhānta.[9] Later, around 820 CE, the Persian scholar Muhammad ibn Musa al-Khwarizmi systematized the solutions to all types of quadratic equations using geometric methods, particularly completing the square, in his book Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (The Compendious Book on Calculation by Completion and Balancing), from which the word "algebra" derives.[9][3] During the Renaissance in the 16th century, Italian mathematicians like Niccolò Tartaglia and Gerolamo Cardano contributed to the algebraic treatment of quadratic equations as part of broader advancements in solving polynomial equations. Cardano's Ars Magna (1545) provided a systematic account of quadratic solutions, including cases with irrational roots, building on earlier verbal methods and integrating them into a more formal algebraic structure.[10][12] In the 17th and 18th centuries, the concept of quadratic functions emerged through the formalization of analytic geometry and function theory. René Descartes, in La Géométrie (1637), linked algebraic quadratic equations to their geometric representations as parabolas via coordinate systems, enabling the graphing of curves defined by second-degree polynomials.[3] Later, Leonhard Euler advanced the understanding by introducing modern function notation f(x) in 1734 and exploring quadratic expressions within infinite series and analysis, solidifying their role as functions rather than isolated equations.[11] The 19th and 20th centuries saw extensions of quadratic functions to multivariate settings and deeper integration with calculus. Carl Friedrich Gauss's Disquisitiones Arithmeticae (1801) developed the theory of binary quadratic forms, generalizing quadratics to two variables in number theory contexts like representing integers.[13] In calculus, Brook Taylor's series (1715), further generalized in the 19th century to multivariable functions by figures like Joseph-Louis Lagrange, employed quadratic terms as second-order approximations for local behavior of smooth functions.[10] These developments facilitated applications in optimization and physics, where multivariate quadratics model phenomena like energy potentials.[14]Definition and Terminology
Coefficients and General Form
A quadratic function is a polynomial function of degree two, expressed in its general form asf(x) = ax^2 + bx + c,
where a, b, and c are real numbers, and a \neq 0 to ensure the degree is exactly two.[15][6] This form represents the standard polynomial expansion for univariate quadratic functions. The coefficient a plays a crucial role in determining the direction and width of the parabola that graphs the function: if a > 0, the parabola opens upward, indicating a minimum point, while if a < 0, it opens downward, indicating a maximum; additionally, the magnitude of |a| affects the width, with larger values producing a narrower parabola and smaller values (closer to zero but non-zero) resulting in a wider one.[6][16] The coefficient b provides a linear shift, influencing the horizontal position of the parabola's vertex, while c serves as the vertical intercept, giving the value of f(0).[15][16] For example, in the function f(x) = 2x^2 - 3x + 1, the coefficient a = 2 (positive, so the parabola opens upward and is relatively narrow), b = -3 (shifts the vertex rightward), and c = 1 (y-intercept at (0, 1)).[6]