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Graphing an Equation


Date: 11/08/97 at 22:27:10
From: Perry Loh
Subject: Graphing an equation

I was wondering if you could explain the following question to me:

   y = a+b(x)+c(x^2)+...

I don't know how to work with this equation or what kind of graph this 
equation produces. All help would be great. Thanks.


Date: 11/09/97 at 09:25:47
From: Doctor Chita
Subject: Re: Graphing an equation

Hi Perry:

This does look mysterious, doesn't it! Actually, the equation is a 
general one that represents any polynomial function. A polynomial is 
an expression containing many terms, usually separated by + signs. 
A function is a special set of ordered pairs (x, y).

In this equation, you have two variables, and y is a function of a 
polynomial in terms of the variable x. x is the independent variable, 
and y is the dependent variable.

The coefficients of x are real numbers, represented as a, b, c, etc.

The exponents of x in every term are non-negative integers. The first 
term in this equation can be written as a*x^0 where x^0 = 1. That's 
why you don't see an "x" term.

The three dots after the third term is called an ellipsis. It means 
that the polynomial can go on indefinitely as the exponents increase.

For example, y = 3 + 2x + 5x^3 - 4 x^5 is a polynomial function. 
Here, the exponents of x are 0 (implied), 1, 3, and 5. The "visible" 
coefficients of x are 3, 2, 5, and -4. That is, a = 3, b = 2, c = 0, 
d = 5, 3 = 0, and f = -4.

Once you identify the exponents and coefficients, you can graph the 
function on a coordinate plane. 

Here's a graph of this function:

          

The shape of a graph becomes very interesting, depending on the number 
of terms in the expression and the constants and exponents. If you 
have a graphing calculator or a computer, try graphing different 
functions. Change the constants and/or exponents and see how each 
change affects the corresponding graph.

Polynomial functions are very interesting, especially around the 
origin. That's where most of the "real" action occurs. Check it out.

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations
High School Polynomials

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