Given vertical translation

**a**(+ to shift up, - to shift down), vertical scaling

**b**(larger to grow, smaller to shrink), horizontal translation

**c**(+ to shift right, - to shift left), and horizontal scaling

**d**(larger to grow, smaller to shrink), the function transformation would be:

(f((x-

**c**)/**d**)+**a**)/**b**You can pick and choose values. If you don't have

**a**or

**c**, replace with 0. If you don't have

**b**or

**d**, replace with 1.

Assume our function f was the sine function.

**sin(x):**

Say we wanted to shift the whole function to the right by +1, i.e. up the x-axis. c=1 and our transformation would be

**sin(x - [+1]):**

Note how the graph used to cross the x-axis at 0 (red), but now crosses it at 1 (green).

Say we now wanted to stretch it longer, i.e. horizontally scale it. Maybe stretch it twice as long. If we choose a horizontal scaling factor of d=2, our new function would be

**sin((x-1)/2):**

Now perhaps we want to shift it upwards, e.g. a=1. Our new function is

**sin((x-1)/2)+1:**

And finally, we want to stretch our function in the vertical axis, perhaps also by double, so b=2.

**(sin((x-1)/2)+1)*2:**

Before, it ranged from a minimum y of 0 to a maximum y of 2, now the maximum y is 4.

## 2 comments:

I'm going to be nitpicky on your last two formulas of sin(x). To shift it upwards, the function should be sin((x-1)/2)+1.

Meaning a should be applied after the sin function. Same with your last equation. It should be (sin((x-1)/2)+1)*2

:)

No, that's not nitpicky, that's a typo on my part. :) I had it correct in the top formula, just not the application. Thanks for the correction!

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