Mastering Quadratic Functions: Coefficients, Graphing, Vertex, X-intercepts, and Solutions

quadratic function

A quadratic function is a type of function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0

A quadratic function is a type of function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola.

The coefficient a determines whether the parabola opens upwards (when a > 0) or downwards (when a < 0). The coefficient b affects the position of the vertex, while the constant term c determines the y-intercept. The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. This form is obtained by completing the square. From this form, we can easily determine the vertex of the parabola. To find the x-intercepts of a quadratic function, we set f(x) = 0 and solve for x. The solutions can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). This formula gives us the values of x where the graph of the quadratic function crosses or touches the x-axis. The axis of symmetry of a quadratic function is a vertical line passing through the vertex. It is given by the equation x = -b / (2a). The discriminant of a quadratic function is the value inside the square root in the quadratic formula, i.e., b^2 - 4ac. It provides information about the nature of the solutions. If the discriminant is positive, there are two distinct real solutions. If the discriminant is zero, there is one real solution. If the discriminant is negative, there are no real solutions, and the graph of the quadratic function does not intersect the x-axis. Overall, understanding quadratic functions involves analyzing their coefficients, graphing them, finding the vertex, x-intercepts, axis of symmetry, and using the quadratic formula to solve for solutions.

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