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Transformation of graphs , y= f(2x)

What is the reasoning behind the x being stretch by a factor of 1/2 rather than a factor of 2 ?, since it is 2x , which is rather counter intuitive.

(edited 6 years ago)

Reply 1

Notnek

Original post by Bilbao

What is the reasoning behind the x being stretch by a factor of 1/2 rather than a factor of 2 ?, since it is 2x , which is rather counter intuitive.

An example to get you thinking:

f(x) = 2x

f(2x) = g(x) = 2(2x)

f(1) = 2
g(0.5) = 2

f(2) = 4
g(1) = 4

So to get to the same output in g as you do in f, you need to halve the input.

Reply 2

uponthyhorse

I struggled with this for a bit but recently got it

Say you have the function y = f(x). It goes through the point (2, 4). You also have y = f(2x). Where does the point (2, 4) move to on the new graph? Well you know if you put in 2 into the function you will receive 4 as a value of y. So now you can "solve" for the new X coordinate: 2x = 2, so x = 2/2 = 1. From this it's very intuitive to see why you stretch by 1/2. Think of it as solving an equation.

Reply 3

RDKGames

Study Forum Helper

Original post by Bilbao

What is the reasoning behind the x being stretch by a factor of 1/2 rather than a factor of 2 ?, since it is 2x , which is rather counter intuitive.

It is counter intuitive, yes, but then again there are plenty of other things in maths which are counter intuitive.

Take a function

y=f(x)

which takes on a certain

y

value when

x=X

. Then introduce a transformation

y=f(2x)

. We want to retain the exact same

y

value as the previous one, so we need to set

2x=X

which hence means that

x=\frac{X}{2}

. Hence the new input into your transformed function is HALF the original one, and this applies to all

x

hence we say

f(x) \mapsto f(2x)

is a stretch by a factor 1/2 parallel to the x-axis.

Same thing works when you transform

y=f(x)

2y=f(x)

. You half every

y

value, but more often you would rewrite the second eq. by saying that

y=\frac{1}{2}f(x)

and hence it is much more obvious to say that

y

halves, and so

y \mapsto 2y

is a transformation of a stretch factor 1/2 parallel to the y-axis.

(edited 6 years ago)

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