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Dilations of the Graph of y = f(x)

Date: 08/22/2004 at 20:01:07
From: Heather
Subject: Horizontal stretch/shrink vs. vertical stretch/shrink

Why is it that when doing a horizontal shrink or stretch you multiply 
by the reciprocal but when doing a vertical stretch or shrink you 
multiply by just the number?

For example, to stretch y = f(x) vertically by a factor of 2 we just
use y = 2*f(x), but to stretch it horizontally by a factor of 2 we use
y = f(x/2).  Why isn't it y = f(2x)?



Date: 08/22/2004 at 22:21:20
From: Doctor Peterson
Subject: Re: Horizontal stretch/shrink vs. vertical stretch/shrink

Hi, Heather.

There's a different way to express the dilations (stretching and
shrinking) that makes the two look more similar.

As in your example, say we have an equation

  y = f(x)

If you stretch the graph vertically by a factor of k, the new y will
be k times what it was for a given x:

  y = k * f(x)

But you can also think of it as replacing y in the original equation
with y/k:

  y/k = f(x)

That means that y for the new equation has to be k times as great as
in the original equation, in order to be on the graph; dividing it by
k gives you the equivalent "y" in the original equation, which is why
we replace y with y/k to get the new equation.

Now let's stretch horizontally by a factor of k.  That can be done by
replacing x with x/k this time:

  y = f(x/k)

When thought of in that way, the two approaches are the same--only
the variable replaced changes depending on if you dilate vertically or
horizontally.

As an aside, a similar discussion holds for simple translations.  The
usual approach uses the notation:

  y = f(x - h) + k

To move the graph 3 units right and 2 units up, we would have:

  y = f(x - 3) + 2

Again, students wonder why the upward shift of 2 is done with +2 while
the shift to the right is done with -3.  That doesn't seem consistent,
either.

But if we do the same thing and rewrite the equation as

  y - k = f(x - h)

we can see that the two translations are in fact accomplished the same
way.

If you have any further questions, feel free to write back.

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

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