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Graphing Two Functions with One Equation?

Date: 10/21/2003 at 22:02:55
From: Vivian
Subject: Is it possible to use ONE equation to graph two functions?

Hi, Dr. Math. 

I am just wondering if it is possible to use one equation to graph two 

For example, is there one equation that gives the same graph as the 
two equations

  1. y = sqrt(x) - 4
  2. y = sqrt(x) - 8

Is there something to do with the negative/positive value of the 
"vertical stretch factor" and the "horizontal stretch factor"? 

Date: 10/21/2003 at 22:42:13
From: Doctor Peterson
Subject: Re: Is it possible to use ONE equation to graph two functions

Hi, Vivian.

Interesting question!

Obviously you can't have an equation of the form

  y = f(x)

represent two functions of x, since y can only have one value for each 
x.  But an equation like

  f(x,y) = 0

can easily have two values of y for each x.  For example, the equation 
of a circle represents two functions, each one being a semi-circle.

In fact, let's suppose we have any two functions, f and g.  You want 
to make an equation such that (x,y) satisfies it whenever either

  y = f(x)  or  y = g(x)

Can we do that?  I've never tried before, but I have an idea.  In 
fact, two ideas.  First, we can use the zero-product property.  That is, 

  (y - f(x))(y - g(x)) = 0

will be true under exactly the right conditions.  That's a first 
answer to your question.

Second, we can use the fact that

  |y| = h(x)

is true when y is either h(x) or -h(x).  So if we take the average of 
our two functions, [f(x) + g(x)]/2, then we have to add either the 
positive or negative of the same quantity to get either function, and 
this equation represents the union of both functions:

  |y - [f(x) + g(x)]/2| = [f(x) - g(x)]/2

Think about that and you should see why it works.  It's not nearly as 
nice as the first way, but maybe that makes it more fun!

Can you apply this to a specific pair of functions like yours and get 
an equation to graph?  Try simplifying each equation; one way in the 
second case is to square the equation to avoid having an absolute 

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Functions

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