## Quadratic Function

### A quadratic function is a type of polynomial function with a degree of 2

A quadratic function is a type of polynomial function with a degree of 2. It is called “quadratic” because the highest power of the variable is squared. The general form of a quadratic function is:

f(x) = ax^2 + bx + c

Where ‘a’, ‘b’, and ‘c’ are constants and ‘x’ represents the variable. The coefficient ‘a’ cannot be zero, as it determines the shape and direction of the parabolic graph.

Quadratic functions graphically form a U-shaped curve known as a parabola. The vertex of the parabola is a point (h, k), where ‘h’ represents the x-coordinate and ‘k’ represents the y-coordinate. The value of ‘a’ determines whether the parabola opens upward (if ‘a’ is positive) or downward (if ‘a’ is negative).

Some important features and concepts related to quadratic functions include:

1. Vertex: The vertex of a quadratic function is the highest or lowest point on the parabola. It is obtained by finding the x-coordinate using the formula ‘h = -b/2a’ and then substituting this value into the function to get the y-coordinate.

2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation can be determined using the formula ‘x = -b/2a’.

3. Discriminant: The discriminant is a value calculated from the coefficients of the quadratic function, given by ‘b^2 – 4ac’. It can be used to determine the nature of the solutions of the quadratic equation or the x-intercepts of the graph. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is one real solution (where the parabola touches the x-axis); and if it is negative, there are no real solutions (the parabola does not intersect the x-axis).

4. X-intercepts: The x-intercepts of a quadratic function are the points where the graph intersects the x-axis. They can be found by solving the quadratic equation ‘ax^2 + bx + c = 0’.

5. Maximum or Minimum: Depending on the coefficient ‘a’, a quadratic function either has a maximum point (if ‘a’ is negative) or a minimum point (if ‘a’ is positive), which is the vertex of the parabola.

Quadratic functions have various applications in real life, such as modeling the path of a thrown object or optimizing certain processes.

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