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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
functions?

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

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
value.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Functions

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