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A quadratic function is a polynomial function of degree 2. They have the expression (standar form):

The graph of a quadratic function is a parabola.

Some parabolas cut the x-axis in two points. We call this points the roots (or zeros) of the polynomial.

Polynomials Functions. Quadratic functions: a quadratic function with two roots | matematicasVisuales

We can obtain these roots solving the quadratic equation

The solutions of a quadratic equation are given by:

The discriminant is defined as:

If the discriminant is greater than 0 the quadratic equation has two roots, x1, x2. Then we can write a quadratic function in the factored form:

Some parabolas only cut (or touch) the x-axis in one point.

Polynomials Functions. Quadratic functions: A quadratic function with only one root | matematicasVisuales

In this case the discriminant is equal to zero and the solution of the quadratic equation is:

Then we say that this root is a double root. In this case, the quadratic function in the factored form is:

Some parabolas do not cut the x-axis. In this case, the discriminant is less than 0 and the quadratic equation has no real solution.

Polynomials Functions. Quadratic functions: a quadratic function without real roots | matematicasVisuales

When a is a positive number the parabola opens upward but if a is a negative number the parabola opens downward. Here we can see one example with two real roots, one with only one root and one with no real roots:

Polynomials Functions. Quadratic functions: a parabola with two real roots | matematicasVisuales
Polynomials Functions. Quadratic functions: a parabola with only one real root | matematicasVisuales
Polynomials Functions. Quadratic functions: a parabola with no real roots | matematicasVisuales

Every parabola has a maximum or a minimum (it is a maximum if a is a negative number and it is a minimum if a is a positive number). This point is called the vertex. The vertical line through the vertex is the axis of symmetry of the parabola. The equation of the axis is:

Polynomials Functions. Quadratic functions: axis of a parabola | matematicasVisuales

And the vertex of the parabola is the point with coordinates:

Polynomials Functions. Quadratic functions: the vertex of a parabola | matematicasVisuales

Quadratic Functions with real or complex coefficients always have two roots (real or complex) (Fundamental Theorem of Algebra):

Complex Polynomials Functions. Quadratic functions have two roots | matematicasVisuales

REFERENCES

Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.

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