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Does the Order Matter When Transforming a Function?

Date: 08/31/2007 at 23:09:21
From: Julie
Subject: order for translations of graphs

If a function has a lot of translations, such as a reflection, a 
stretch, a horizontal shift, and a vertical shift, how do you decide 
in which order to perform the shifts?  -2(x-5)^3+3

I get doing what is inside parenthesis first, but then do you do the 
stretch, and then the reflection?  Do you reflect first, and then 
stretch?  Previous teachers haven't said that the order matters, but 
this one insists that it is very important, and we have to list the 
order in which translations are performed.

I guess my thoughts are, I would shift right first, then reflect (do 
you always reflect before a stretch?) then stretch by 2, then shift 
up 3.

Date: 09/01/2007 at 20:57:50
From: Doctor Peterson
Subject: Re: order for translations of graphs

Hi, Julie.

Yes, order matters in many cases.  I'm always very careful with this,
taking the transformations one at a time to make sure I don't confuse

I start with the "basic" function, and do one thing at a time:

      x^3     the basic cubic
    (x-5)^3   replacing x with x-5 shifts right by 5
   2(x-5)^3   multiplying by 2 stretches vertically by a factor of 2
  -2(x-5)^3   multiplying by -1 reflects in the x-axis
  -2(x-5)^3+3 adding 3 shifts up by 3

Here the shift right could actually be done at any point, and the 
stretch and reflection could be interchanged, but the vertical shift
must be done after them (or it would involve a different quantity). 
Your order is fine.

See this page for a further discussion, including coverage of the 
tricky case with ax+b inside parentheses, and the fact that some
transformations can be interchanged and others can't (without
modifying one of them):

  Order of Transformations of a Function 

I'd recommend that you try different orders for yourself and see what
happens, taking one step at a time as I did.  This will give you 
valuable experience with these details, so you'll know them from
personal experience.  For example, if you first shift up by 3 and then
stretch vertically by 2, you get

  x^3 + 3
  2(x^3 + 3) = 2x^3 + 6

which is different.

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

- Doctor Peterson, The Math Forum 

Date: 09/01/2007 at 22:47:10
From: Julie
Subject: Thank you (order for translations of graphs)

Dr. Peterson - Thank you so much for your time and very clear 
explanation.  I like your suggestion of trying the translations in
different orders to see how the graphs look.  I appreciate the link to
more info.  Julie
Associated Topics:
High School Functions

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