Quadratic Functions


Two points with different values of x determine a linear function (a polynomial of degree less or equal to 1) 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 xaxis in two points. We call this points the roots (or zeros) of the polynomial.
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, x_{1}, x_{2}. Then we can write a quadratic function in the factored form: Some parabolas only cut (or touch) the xaxis in one point. 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 xaxis. In this case, the discriminant is less than 0 and the quadratic equation has no real solution. 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: 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: The vertex of the parabola is the point with coordinates: Parabolas are conic sections: Quadratic Functions with real or complex coefficients always have two roots (real or complex) (Fundamental Theorem of Algebra):
REFERENCES
Michael Spivak, 'Calculus', Third Edition, PublishorPerish, 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.
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The function is not defined for values less than 1. Taylor polynomials about the origin approximates the function between 1 and 1.
The function has a singularity at 1. Taylor polynomials about the origin approximates the function between 1 and 1.
The function has a singularity at 1. Taylor polynomials about the origin approximates the function between 1 and 1.
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