Quadratic Function

A quadratic function is a function of the form:

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

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 vertex of the parabola is the point where the curve changes direction, and it lies on the axis of symmetry of the parabola.

Graphs of Quadratic Functions

The shape of the graph of a quadratic function depends on the sign of the coefficient $a$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

The vertex of the parabola is given by the formula:

$\left(-\frac{b}{2a},\frac{4ac-b^2}{4a}\right)$

If $a$ is positive, the vertex is the minimum point of the parabola, and if $a$ is negative, the vertex is the maximum point of the parabola.

Roots of Quadratic Functions

The roots of a quadratic function are the values of $x$ for which $f(x) = 0$. The roots can be found using the quadratic formula:

$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$

If the discriminant $b^2-4ac$ is positive, the quadratic function has two distinct real roots. If the discriminant is zero, the quadratic function has one real root, which is also the vertex of the parabola. If the discriminant is negative, the quadratic function has no real roots, and the parabola does not intersect the x-axis.

Applications of Quadratic Functions

Quadratic functions are used in a variety of applications, such as physics, finance, and engineering. For example, the height of an object thrown vertically upwards can be modeled by a quadratic function, and the profit of a business can be modeled by a quadratic function.

Conclusion

The quadratic function is a fundamental mathematical concept that is used in many areas of science and engineering. The graph of a quadratic function is a parabola, and the roots of the function can be found using the quadratic formula. Understanding the properties of quadratic functions can help us to analyze and model real-world phenomena.

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