Thread: transformation questions really confusing me!

Ok suppose you have f(x)= sin 2x and and g(x)= sin(2x-30)
What would be the transformation? presumably you would all agree it is a shift of 30 in the x direction?

Now if you have f(x)= 2^(4x)
and g(x)= 2^(4x-3)

Why isn't this simply a translation of 3 in the x direction? It is infact a translation of 3/4
( or a stretch..but forget that for the moment)

Help

Ok suppose you have f(x)= sin 2x and and g(x)= sin(2x-30)
What would be the transformation? presumably you would all agree it is a shift of 30 in the x direction?

Not really. Let's denote the graph of some function f(x) by G(f). Then we have the following rule.

G(g) is G(f) shifted n units to the right iff g(x) = f(x - n)

In the first example, g(x) = sin(2(x - 15)) = f(x - 15). Therefore, G(g) is G(f) shifted 15 degrees to the right.

In the second example, g(x) = 2^(4x - 3) = 2^(4(x - 3/4)) = f(x - 3/4), so G(g) is G(f) shifted by 3/4. Note that f(x - 3) = 2^(4(x - 3)) = 2^(4x - 12), which is not the same as g(x).

The general rule is, when you consider the composition f(h(x)), you take the expression for f(x) and replace x with h(x).

ok..but still not totally clear... Lets look at this example

sinx to sin(3x+45)

I understand this to be a translation -15 x direction followed by a stretch SF 1/3 in x direction.

But i am told that it could equally be a translation of -45 followed by a stretch which makes no sense at all?

sinx to sin(3x+45)

I understand this to be a translation -15 x direction followed by a stretch SF 1/3 in x direction.

After some head scratching, I have to say it's the other way around: a stretch by 1/3 followed by a shift by -15. This transforms the graph of sin(x) into the graph of sin(3x + 45).

Let f(x) = sin(x) and h(x) = sin(3x + 45). There are two ways to get h(x) from f(x). For the first, let g(x) = sin(3x) = f(3x). Then h(x) = g(x + 15). The transition from f to g is a stretch by 1/3, and the one from g to h is a shift by -15.

For the second way, let g(x) = f(x + 45). Then h(x) = g(3x). This is a shift by -45 followed by a stretch by 1/3.

Many thanks! All cleared up for me now.
Just need to remember to use the function notation each time and you can't go wrong!