Quadratic functions, with their unique parabolic graphs, are fundamental in mathematics. They are defined by the equation f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. This text delves into their various representations, including standard, factored, and vertex forms, and discusses their defining features, such as the degree, leading term, and vertex. Techniques for solving quadratic equations, like the quadratic formula, factoring, and completing the square, are also covered, along with insights into their inverse functions.
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Polynomial functions of degree two, represented by the equation \(f(x) = ax^2 + bx + c\)
Vertex
The point where the function attains its maximum or minimum value, located on the parabola's axis of symmetry
Y-intercept
The point at which the parabola crosses the y-axis, with coordinates \((0, c)\)
Standard Form
\(y = ax^2 + bx + c\), useful for analyzing the general shape and position of the parabola
Factored Form
\(y = a(x - r)(x - s)\), valuable for finding the x-intercepts
Vertex Form
\(y = a(x - h)^2 + k\), facilitates graphing and understanding the function's maximum or minimum value
Quadratic functions are defined by their degree, which is two, distinguishing them from linear and higher-degree polynomial functions
The leading term, \(ax^2\), must have a non-zero coefficient \(a\) to ensure the function is quadratic
Represents the optimal value of the function, either a maximum or a minimum, and is located at the intersection of the parabola with its axis of symmetry
To identify a quadratic function, look for an expression where the highest power of the variable is two
Quadratic Formula
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), provides a reliable method for calculating the roots of a quadratic equation
Factoring and Completing the Square
Other techniques used to solve quadratic equations
The inverse of a quadratic function is found by interchanging the roles of \(x\) and \(y\) and solving for \(x\)
\(f^{-1}(y) = \frac{-b \pm \sqrt{b^2 - 4ac + 4ay}}{2a}\), exists in the real number domain if and only if \(b^2 - 4ac + 4ay\) is non-negative
Quadratic functions are integral to various scientific and mathematical applications due to their distinctive properties and the parabolic shape of their graphs
The highest power of the variable in a quadratic function is always two
The graph of a quadratic function features a unique vertex and axis of symmetry
The quadratic formula can be used to determine the roots of a quadratic function, and the function can be represented in different forms to facilitate analysis and problem-solving
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Quadratic function definition
Polynomial of degree two, form: f(x) = ax^2 + bx + c, a ≠ 0.
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Vertex of a parabola
Point of max/min value, on axis of symmetry x = -b/(2a).
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Y-intercept of a quadratic
Point where parabola crosses y-axis, coordinates (0, c).
