Transformations of Functions Help

[BakerDad12]

Hi guys, I have two questions I am struggling on. They are transforming coordinates on a graph. The questions are attached below and they are 11) c and 22) d.

Thank you!

[Jefferson]

Hi,

Transformation works by translating/shifting the graph left, right, up or down.
For example, if the original function is
y = x²,
Then y = (x+2)² would shift it 2 units to the left (always the opposite of the sign).

I noticed that for the questions you're struggling with, the term x has a coefficient. To find out how transformation works on your graph, you must first isolate the x by factorising the coefficient out.

i.e. for:
y = 2x - 6
the -6 applies to 2x rather than just x, so by factorising:
y = 2(x-3)
Hence, it is clear that the function above is y = 2x shifted 3 units to the right.

tldr;
Isolate the variable before determining transformation.


(Sorry it's like midnight atm so my choice of words may be iffy)

[BakerDad12]

Hi, I understand that but I still am getting it wrong. I don't know why...

[RuiAce]

Take 22d): \( \boxed{y = -3 f(-3(x-3))+1 } \). Suppose that \( (x,y)\) lies on \(y = f(x)\).

First deal with the inner: Replacing \(x\) with \(x-3\) translates to the right by 3, so we now have the point \( (x+3, y)\)

Replacing \(x-3\) with \(-3(x-3)\) reflects about the \(y\)-axis, and introduces a dilation factor of \( \frac13\), so we now have the point \( \left( -\frac13x + 3, y\right)\).

Now deal with the outer: Replacing \( f(-3(x-3)\) with -3f(-3(x-3))\) reflects about the \(x\)-axis, and introduces a dilation factor of \(3\), so we now have the point \( \left( -\frac13 x+3, -3y\right) \)

Replacing \( -3f(-3(x-3)) \)  with  \( -3f(-3(x-3)) \)  translates upwards by 1, so we now have the point \( \left( - \frac13x+3, -3y+1\right) \).

So our final point is
\[ \left( - \frac13x+3, -3y+1\right). \]

If you need help with the other one, please post any relevant progress.

[BakerDad12]

Hi Rui, thanks for the reply. I bloody knew it! The answers say -1/3(x+3), -3y -1. I got the correct answer (by your standards) and I thought the answers were wrong but even my teacher said they were right!! Far out...

Just a quick note, the answers ARE wrong, aren't they?? We have a test on this topic coming up soon...