A quadratic function is a polynomial function of degree 2. Its general form is f of x equals a x squared plus b x plus c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, which is a U-shaped curve.
The opening direction of a parabola depends on the coefficient a. When a is greater than zero, the parabola opens upward like a U shape. When a is less than zero, the parabola opens downward like an upside-down U. The vertex is the highest or lowest point of the parabola, located at x equals negative b over 2a.
Every parabola has an axis of symmetry, which is a vertical line passing through the vertex. The equation of this line is x equals negative b over 2a. The parabola is symmetric about this line. For intercepts, the y-intercept occurs where the parabola crosses the y-axis at point zero comma c. The x-intercepts are found by solving the quadratic equation, and their existence depends on the discriminant b squared minus 4ac.
Quadratic functions have either a maximum or minimum value at their vertex. When the coefficient a is positive, the parabola opens upward and has a minimum value at the vertex. When a is negative, the parabola opens downward and has a maximum value at the vertex. The value at the vertex can be calculated using the formula 4ac minus b squared, all divided by 4a. The domain of any quadratic function is all real numbers, while the range depends on the vertex and opening direction.
To summarize the key properties of quadratic functions: They have the standard form f of x equals a x squared plus b x plus c. Their graphs are parabolas with characteristic U-shaped curves. The opening direction depends on the coefficient a, and the vertex provides either the maximum or minimum value. The axis of symmetry always passes through the vertex, making the parabola perfectly symmetric.