Question

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?

Answer 1

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!