How do you transform y=1/x to y= (3x-2)/(x+1) ?

The answer says it's a reflection in the x-axis, stretching by a scale factor of 5 and translate it by (-1.3) but how do I do it?

1 Answer
Aug 5, 2018

That's an interesting question! I've never done something like it before, but I've figured out how to do it.

A typical transformation of a function #y=f(x)# takes the form

#y=af[b(x-c)] + d#

where

  • #a# stretches/squishes the function up and down,
  • #b# stretches/squishes the function left to right,
  • #c# shifts the function left to right, and
  • #d# shifts the function up and down.

In this form, a change in any of #a, b, c, d# corresponds directly to a change in the function by that same amount. (For example, increasing #c# by 1 always means the function shifts to the right by 1.)

Thus, if we can write a given "final transformation" in this form, we can extract the stretches and shifts from it directly. That's what we'll try to do.

For the function #y = 1/x#, the generic transformation form is

#y=a/(b(x-c)) + d#

Can we write #y=(3x-2)/(x+1)# in this form?

Yes we can!

The first (and hardest) thing to do is to express the numerator is a way that uses #x+1# in place of #x#. To do this, add and subtract 1 to #x#, like this:

#y=(3(x+color(blue)1-color(red)1)-2)/(x+1)#

#color(white)y=(3(x+color(blue)1)-color(red)3-2)/(x+1)#

#color(white)y=(3(x+1)-5)/(x+1)#

Why did we do that? Because, when we split the function into two fractions, like this:

#y=(3(x+1))/(x+1) -(5)/(x+1)#

the first fraction has a cancellation we can do:

#y=(3cancel((x+1)))/cancel(x+1) -(5)/(x+1)#

#color(white)y=3 - 5/(x+1)#

And now, if we reorder the tems, we get

#y=(-5)/(x+1) + 3#

And hey, look—this is the form we were aiming for! This form tells us:

#a=–5#
#b=1#
#c=–1#
#d=3#

In other words, to translate #y=1/x# into #y=(3x-2)/(x+1),# we stretch the function up/down by –5 (i.e. reflect it about the #x#-axis and then stretch that by 5), shift it left by 1 (i.e. right by –1), and shift it up by 3.

(Since #b=1#, there is no left-to-right stretch or squish.)

These instructions can be expressed the way your answer does it:

  • reflect around the #x#-axis,
  • stretch (vertically) with a scale factor of 5, and
  • translate by #(–1, 3)#.

Hope this helps!