Terminology: Quadratic Equation vs. Quadratic Function
Date: 04/07/2004 at 13:36:10 From: Rachel Subject: Terminology: quadratic equation vs. quadratic function As I read various algebra books for high school kids, I find what appears to be an inconsistent use of the words "quadratic equation", and I wanted to be sure I use it correctly myself. Is it correct to call y = ax^2 + bx + c a "quadratic equation"? It is an "equation" in the sense that it sets two expressions equal to each other, however frequently textbooks seem to call this a "quadratic function" (since it is a function) and reserve the phrase "quadratic equation" for ax^2 + bx + c = 0 only. Are there times when it is more appropriate to refer to y = ax^2 + bx + c as an equation, as opposed to a function? Are the ordered pairs that fall on the graph of the parabola considered "solutions" to the equation or to the function? Can a function have "solutions"? Is the word "equation" appropriate any time 2 expressions are set equal to each other? And is the word "quadratic" appropriate any time the highest degree of a variable in the equation is 2? If so, could the equation of a circle also be considered a "quadratic equation in 2 variables"? Yet a circle could not be called a quadratic function since it's not a function. What about x = ay^2 + by + c on the x-y plane. This is not a function. But can it be called a quadratic equation? Anything you could say to help clarify this would be much appreciated.
Date: 04/07/2004 at 16:02:31 From: Doctor Peterson Subject: Re: Terminology: quadratic equation vs. quadratic function Hi, Rachel. You've raised some good questions. Almost all of them can be answered "yes". It turns out that the usages you're asking about are consistent with the definitions, but not always with conventions that kids see, which they can think are more absolute than they really are. Let's look at the definitions. An _equation_ is any statement that two expressions are equal. That includes y = ax^2 + bx + c x = ay^2 + by + c ax^2 + bx + c = 0 x^2 + y^2 = r^2 Some equations involve only one variable; those can be solved to find the value(s) of the variable that make them true. Others involve more than one variable; we don't usually say that a pair (x,y) for which the equation is true is a "solution" of the equation, but in some cases it makes sense to speak that way; we do say we "solve" the equation for one variable in terms of, or as a function of, the other. A _function_ is a relation between two variables such that for any value of the independent variable (in the domain) there is exactly one value of the dependent variable. This can be expressed as an equation like y = ax^2 + bx + c which tells how to find y given a value of x; we say we have expressed y as a function of x. But this also represents a function: x = ay^2 + by + c The only difference is that we have gone against the usual convention that x is independent; we have instead written x as a function of y. Note that the equation is not the function; it only represents or defines it. When we want to emphasize the function itself, we can name it, and define for example f(x) = ax^2 + bx + c Then we say that the function is f (that is, the relationship named "f"--NOT the equation we wrote that defines it). This page discusses the distinction: Are All Functions Equations? http://mathforum.org/library/drmath/view/53273.html Also, for the fact that functions do not have solutions (or roots), while equations do, see: What Are the Definitions of Zero and Root? http://mathforum.org/library/drmath/view/64502.html Finally, something is called _quadratic_ if it involves a second- degree polynomial. So all the equations I listed are quadratic; some are univariate (one variable), others bivariate (two variables). At high school level, texts (or students) tend to equate "quadratic equation" with the univariate kind, and in particular with one in standard form, since that is what they spend inordinate amounts of time learning to solve. For an example of bivariate quadratics, a.k.a. conics, see Quadratic Curve http://mathworld.wolfram.com/QuadraticCurve.html Compare that with Quadratic Equation http://mathworld.wolfram.com/QuadraticEquation.html Quadratic Surface http://mathworld.wolfram.com/QuadraticSurface.html In summary, there are different kinds of "quadratic equations"; as with most definitions in English or in math, the term depends somewhat on the context. When the book is talking about solving univariate equations, they can say "a quadratic equation has the form ax^2 + bx + c = 0" as if that were the only way the phrase is used, because for the moment it is; but later when they want to graph quadratic functions or quadratic curves, they can use the word a little differently. So you should try to be consistent within your context, and using words according to their definitions; but don't worry too much about being completely uniform across the board. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 04/07/2004 at 17:02:02 From: Rachel Subject: Thank you (Terminology: quadratic equation vs. quadratic function) Thank you Dr. Peterson! I appreciate the thorough response, the quick turn-around, and the fact that you are there to answer these questions. You have cleared things up for me. I can now use these terms and feel confident I am using them correctly.
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