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Quadratic Functions


Two points with different values of x determine a linear function (a polynomial of degree less or equal to 1)

Polynomial functions and derivative (1): Linear functions
The derivative of a lineal function is a constant function.

Three points not in a line determine a quadratic function, a parabola.

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

The vertex of the parabola is the point with coordinates:

Parabolas are conic sections:

Polynomials Functions. Quadratic functions: parabola as a conic section | 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.
I.M. Gelfand, E.G. Glagoleva, E.E. Shnol, 'Functions and Graphs', Dover Publications, Mineola, N.Y.

MORE LINKS

Powers with natural exponents (and positive rational exponents)
Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)
Polynomial Functions (4): Lagrange interpolating polynomial
We can consider the polynomial function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.
Polynomial functions and derivative (1): Linear functions
The derivative of a lineal function is a constant function.
Polynomial functions and derivative (2): Quadratic functions
The derivative of a quadratic function is a linear function, it is to say, a straight line.
Piecewise Linear Functions. Only one piece
As an introduction to Piecewise Linear Functions we study linear functions restricted to an open interval: their graphs are like segments.
Piecewise Constant Functions
A piecewise function is a function that is defined by several subfunctions. If each piece is a constant function then the piecewise function is called Piecewise constant function or Step function.
Continuous Piecewise Linear Functions
A continuous piecewise linear function is defined by several segments or rays connected, without jumps between them.
Polynomial functions and derivative (3): Cubic functions
The derivative of a cubic function is a quadratic function, a parabola.
Rational Functions (1): Linear rational functions
Rational functions can be writen as the quotient of two polynomials. Linear rational functions are the simplest of this kind of functions.
Rational Functions (2): degree 2 denominator
When the denominator of a rational function has degree 2 the function can have two, one or none real singularities.
Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions)
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.
Polynomial functions and derivative (5): Antidifferentiation
If the derivative of F(x) is f(x), then we say that an indefinite integral of f(x) with respect to x is F(x). We also say that F is an antiderivative or a primitive function of f.
Definite integral
The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the x-axis between two vertical lines.
Monotonic functions are integrable
Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
Indefinite integral
If we consider the lower limit of integration a as fixed and if we can calculate the integral for different values of the upper limit of integration b then we can define a new function: an indefinite integral of f.
Polynomial functions and integral (1): Linear functions
It is easy to calculate the area under a straight line. This is the first example of integration that allows us to understand the idea and to introduce several basic concepts: integral as area, limits of integration, positive and negative areas.
Polynomial functions and integral (2): Quadratic functions
To calculate the area under a parabola is more difficult than to calculate the area under a linear function. We show how to approximate this area using rectangles and that the integral function of a polynomial of degree 2 is a polynomial of degree 3.
Polynomial functions and integral (3): Lagrange polynomials (General polynomial functions)
We can see some basic concepts about integration applied to a general polynomial function. Integral functions of polynomial functions are polynomial functions with one degree more than the original function.
The Fundamental Theorem of Calculus (1)
The Fundamental Theorem of Calculus tell us that every continuous function has an antiderivative and shows how to construct one using the integral.
The Fundamental Theorem of Calculus (2)
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).
Taylor polynomials (1): Exponential function
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Taylor polynomials (2): Sine function
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Taylor polynomials (3): Square root
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (4): Rational function 1
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (5): Rational function 2
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (6): Rational function with two real singularities
This function has two real singularities at -1 and 1. Taylor polynomials approximate the function in an interval centered at the center of the series. Its radius is the distance to the nearest singularity.
Taylor polynomials (7): Rational function without real singularities
This is a continuos function and has no real singularities. However, the Taylor series approximates the function only in an interval. To understand this behavior we should consider a complex function.
Complex Polynomial Functions(1): Powers with natural exponent
Complex power functions with natural exponent have a zero (or root) of multiplicity n in the origin.
Complex Polynomial Functions(2): Polynomial of degree 2
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
Complex Polynomial Functions(3): Polynomial of degree 3
A complex polinomial of degree 3 has three roots or zeros.
Complex Polynomial Functions(4): Polynomial of degree n
Every complex polynomial of degree n has n zeros or roots.