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.