Mastering the Basics of Quadratic Functions: Understanding the Vertex, Axis of Symmetry, Roots, and Discriminant

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 can be written in the form:

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

Where “a”, “b”, and “c” are constants, and “x” represents the independent variable.

The graph of a quadratic function is a U-shaped curve called a parabola. The vertex of the parabola is the minimum or maximum point, depending on whether the coefficient “a” is positive or negative.

Here are some key terms related to quadratic functions:

1. Vertex: The vertex is the point on the parabola where the curve changes direction. It can be found using the formula:

x-coordinate of vertex = -b / (2a)
y-coordinate of vertex = f(x-coordinate of vertex)

2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetric halves. It can be found using the formula:

x-coordinate of axis of symmetry = -b / (2a)

3. Roots/Zeros: The roots (also called zeros or x-intercepts) of a quadratic function are the values of “x” where the graph intersects the x-axis. They correspond to the solutions of the equation f(x) = 0. The roots can be found using several methods, such as factoring, completing the square, or using the quadratic formula:

x = (-b ± sqrt(b^2 – 4ac)) / (2a)

4. Discriminant: The discriminant of a quadratic equation is the value inside the square root in the quadratic formula, b^2 – 4ac. It determines the nature of the roots:

– If the discriminant is positive, there are two distinct real roots.
– If the discriminant is zero, there is exactly one real root (the parabola touches the x-axis at one point).
– If the discriminant is negative, there are no real roots (the parabola does not intersect the x-axis).

Quadratic functions have many applications in various fields, such as physics, engineering, and economics. They can be used to model real-life situations, such as projectile motion or revenue optimization.

When working with quadratic functions, it is important to understand the properties of the parabola, find the vertex and axis of symmetry, determine the roots, and analyze the discriminant to gain insights into the behavior of the function.

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