Quadratic Function
A quadratic function is a type of function in mathematics that involves a variable raised to the power of 2 (also known as squared)
A quadratic function is a type of function in mathematics that involves a variable raised to the power of 2 (also known as squared). It can be written in the general form as:
f(x) = ax^2 + bx + c
where a, b, and c are constants. The variable x represents the input values, and f(x) represents the output values of the function.
The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the value of the coefficient a. If a > 0, the parabola opens upwards and if a < 0, the parabola opens downwards. The coefficient a determines the overall shape and the steepness of the parabola. The coefficient b affects the horizontal shift of the parabola, and the constant term c represents the vertical shift. To analyze a quadratic function, we can consider a few key features: 1. Vertex: The vertex of a parabola is the point where it reaches either its highest or lowest point. The x-coordinate of the vertex can be found using the formula x = -b/2a, and the corresponding y-coordinate can be obtained by substituting this x-value into the function. 2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry can be found using the formula x = -b/2a. 3. Discriminant: The discriminant, denoted as ∆, is used to determine the nature of the roots of a quadratic equation. It is calculated using the formula ∆ = b^2 - 4ac. If ∆ > 0, the equation has two distinct real roots, if ∆ = 0, the equation has one real root (known as a “double root”), and if ∆ < 0, the equation has no real roots (only complex roots). 4. x-intercepts and y-intercept: The x-intercepts (or zeros) of a quadratic function are the points where the function crosses the x-axis. They can be found by setting f(x) = 0 and solving the quadratic equation. The y-intercept is the point where the parabola crosses the y-axis, and it can be found by substituting x = 0 into the function. These are the basic concepts and features associated with quadratic functions. Understanding these key points can help solve problems, graph the function, and analyze its behavior.
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