C1 Transformation help

I just want to find out examples of graph transformations for stretches of graphs. I wrote down in my notes a sort of way to answer similar transformation questions. I wrote "y=af(x) stretches y=f(x) by a scale factor of a parallel to the y-axis" and "y=f(ax) stretches y=f(x) by a scale factor of 1/a to the x axis"

But when doing some past papers I wasn't able to indicate which one it would be. What graphs would follow the formula y=af(x) and y=f(x)

Reply 1

c223

Original post by Fruitbasket786

Your notes are absolutely correct :smile:

For example, take the graph y=x^2 (squared).
To stretch this graph parallel to the y axis, you want to stretch the y coordinates of the graph, which is after you've squared the x values. Therefore to stretch with scale factor 4 would be y=4(x^2) as you take an x value, square it (which gives you the corresponding y value) and then multiply this by 4 to get the y stretch.
To stretch this graph parallel to the x axis, you want to stretch the x coordinates, which is before you've squared them to get the corresponding y values. Therefore to stretch with scale factor 4 would be y=1/4x^2 (no brackets needed due to BIDMAS rules). Sorry if this isn't too clear, it's hard to type value like that, it reads a quarter x squared.

Take another graph: y=7x-22.
To stretch in the y direction, s.f. 5 for example -> y=5(7x-22) as you need to stretch the y values which are the x values after they are put into the function.
To strecth in the x direction, s.f. 5 for example -> y=(5*7)x-22, therefore y=35x-22 as you want to stretch the x values.

Hope that made some form of sense and I didn't just confuse you more! The more practice you do of these, the easier it will become. Just think that stretching parallel to the y axis means the y values are altered and parallel to the x axis means the x values are altered. Y values are found from the 'output' of the function whereas x values are the 'input'.

Reply 2

Fruitbasket786

I kind of see what you're trying to say, but I thinking about what the letter "a" could be when it comes to graphs and my equation.

Reply 3

c223

Original post by Fruitbasket786

I kind of see what you're trying to say, but I thinking about what the letter "a" could be when it comes to graphs and my equation.

The letter a is the scale factor, therefore substitute a in for where I've used 4 and 5. It's likely in the exam you'll be given actual numbers (at least this tends to happen on my exam board, I do OCR).

Reply 4

Fruitbasket786

I do OCR aswell, but I'm really confused with the the difference in the placement of a which is the scale factor. I just don't see the difference in the general translation formula with y=af(x) y=f(ax). Would y=af(x) be something like y=8x^3 and y=f(ax) be something like y=2x.

Reply 5

c223

Original post by Fruitbasket786

y=a(fx) means that you put the scale factor, a, outside of a bracket with the main function inside, e.g. y=4x^2 + 19x + 4 => y=a(4x^2 + 19x + 4). Think of it as being y=a(f(x)) where a is a constant which multiplies the whole function.
y=f(ax) means that you put the scale factor, a, before the x value only, e.g. y=24x - 19 => y=24ax - 19. You can see the -19 remains unaffected.

The example you gave with y=8x^3 would be y=af(x) since x^3 is the function, then you multiply the whole function by 8 to get y=8x^3.
The example with y=2x could either be a stretch scale factor 2 in the y direction or a stretch scale factor 1/2 in the x direction, here they are equivalent. Something that is y=f(ax) could follow the pattern y=ax^2 + ax + c, where the c is not affected by the a.

C1 Transformation help

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