Hi guys!!!

Please help.

**Question:**

The graph with equation y = f(x) undergoes a series of transformations so that the equation of the image is y = −f(5x − 7). State the transformations in the correct order.

Can someone explain how to do this using the pre-image method?

I am so confused with the fractions and all.

You can use the pre-image method but it's certainly not necessary.

\(y=f\left(x\right)\) maps to \(y=-f\left(5x-7\right)\) which is equivalent to \(y=-f\left(5\left(x-\frac{7}{5}\right)\right)\)

From here, you should be able to determine it by inspection. The sequence of transformations are as follows:

1. Dilation of factor \(\frac{1}{5}\) from the \(y\) axis

2. Reflection in the \(x\) axis

3. Translation of \(\frac{7}{5}\) units in the positive direction of the \(x\) axis (right)

Note: The order step 2 can be wherever (since it's the only transformation that affects the images position on the \(y\) axis) but the translation must be after the dilation. This is because if you were to translate it, then dilate it you would also have to dilate the translation. This is why you should make sure you consider the order! It is very important.

If you didn't understand how I did the above by inspection, I used the fact that \(f(x)\) maps to \(f\left(\frac{x}{a}\right)\) after a dilation of factor \(a\) from the \(y\) axis and \(f\left(x-h\right)\) is a translation of \(h\) units right and \(-f\left(x\right)\) flips the graph over the \(x\) axis. I suggest you get use to this way of reading transformations as well, I have attatched the diagram which can be accessed in the methods 11 textbook in the spoiler below.

If you wanted to go through the transformations using the pre-image method it would follow as such

Let the transformation map \(\left(x,y\right)\mapsto \left(x',y'\right)\)

\(y=f\left(x\right)\) maps to \(y'=-f\left(5x'-7\right)\). Rearranging the equation gives

\(-y'=f\left(5x'-7\right)\). We choose to write \(y=-y'\) and \(x=5x'-7\), now solve for \(x'\) and \(y'\) and we get

\(\left(x,y\right)\mapsto \left(\frac{x}{5}+\frac{7}{5},-y\right)\) which leads to the sequence of transformations.

I hope this helps