Quadratic functions

Let us first see how quadratic functions are defined. A function is a relation between two or more variables which is one to one and onto. If the independent variable is single with degree 2 and the dependent variable is of degree 1, then the function is said to be quadratic. That is, the output of the function is related to the square of the input.

The general form of a quadratic function is y = f(x) = ax2 + bx + c, where a, b and c are all constants. Obviously, the constant a cannot be 0, in which case the degree of the variable will be no longer 2 and hence the function will not qualify as quadratic function.

Quadratic-functions are the most prominent functions in algebra as many practical situations can be cited as quadratic functions examples. Thus the knowledge of solving quadratic functions is extremely important. The solving can mean in two types of solving. One is to find y for a given value of x. But this is very simple task to do and has no great practical implications.

On the other hand, for a given value of y, to solve forx is extremely important. This is what generally referred as solving quadratic functions. The very first exercise is to solve for solve for x,when y = 0. The values of x under such conditions are called as the zeroes of the function or the x-intercepts of the function. In case of linear functions y-intercepts are more prominent but in case of functions of quadratic form, the x-intercepts are more prominent.

In certain cases, you may find the solutions to be imaginary and at a later stage we will explain the implication of such cases.

A quadratic function can be transformed to a quadratic equation for a given value of y. That is, the constant term is adjusted accordingly and is equated to 0. There are many methods of solving for the variable. The easiest method, if possible, is to factor the trinomial as the product to two binomials and apply the zero product property. In cases where factorization is not possible, one can use the quadratic formula and find the solutions.

We cannot predict for sure, the shape of a graph for a given function except in two cases. In case of a linear function, the graph is a straight line. The shapes of graphs of quadratic functions are invariably vertical parabolas. The points of intersections of the parabolas with x-axis are the zeroes of the function. If there are no such intercepts, the zeroes of the functions are imaginary.

The Art of Simplifying Rational Expressions

The concept of simplifying rational expressions can be understood only after the studying the basics. Once the basics are clear, it becomes very easy to solve these. The concept of solving algebraic equations can also be learnt from these basics.

Once this is understood simplifying algebraic expressions can be very simple. Basic algebraic expressions will contain the arithmetic operations of addition, subtraction, multiplication and division. They also contain constants and variables. The value of constants does not change throughout and the value of the variables can change. This is the basic difference between the two.

To simplify rational expressions one must understand what the definition of these expressions is. These can be written with the fractions. The denominator in the fraction must not be zero; otherwise it is difficult to define the fraction.

The degree of the expression plays a very important role in deciding the method to solve the expression. The question how to simplify rational expressions can be understood only if the basics are understood. It is important to learn the concept of polynomials to understand this concept. Ratio of the polynomials can be treated as an expression representing the concept. The concept of ratio is very clear and simple to understand.

There can be various terms in the polynomial. These terms can be rational or need not be rational in nature. But this will be true only for the real numbers. The complex numbers do not come into the picture here. The complex numbers consist of a real part and an imaginary part in them.

So, this cannot become a part of the polynomials that appear in the ratio. The polynomial that appears in the denominator must not be reducible to zero otherwise the whole expression is valid.This concept is also useful in geometry. The expression can be represented graphically. Simplifying these fractions can be learnt from the simplification of simpler fractions. The process in simplification is very simple.

The common terms that are present in both the numerator and the denominator are cancelled, so that the common terms do not appear twice. This is the basic idea of simplification. The same process is followed in simplifying these expressions as well. The polynomials must be reduced into simpler terms. If there are any common terms in the numerator and the denominator they must be cancelled. This makes the term simpler and easy to understand.