How Do Function Transformations Work?

[WannabeFeynman]

Hello all, I need some help to clear my doubts.

Why does a horizontal translation (f(x + c)) move to the left if c is positive?

Can someone graphically explain what effect a stretch and compression (vertical and horizontal) has on the original parent function?

Similar to the first question, why does f(ax) actually stretch by a factor of 1/a instead of a?

Thanks, I might have more questions later.

 

[.Scott]

Is you graph y1=f(x) and y2=f(x+C) and then compare them, whatever "y1" you see at any "x1" will be seen to the left (assuming C>0) at x2="x1-C" for y2 because y2 = f(x2+C) = f(x1-C+C) = f(x1) = y1.
Similarly:
f(Ax) will horizontally thin the graph by a factor of "A" (or stretch it by 1/A)
Af(x) will vertically stretch the graph by a factor of "A".

 

[Seydlitz]

Take a look at this simple example, suppose $f(x)=x$, a simple straight line through the origin. If $x=0$, then $f(x)=0$ as well. But if you take $f(x+a)$, the origin will be at $-a$. If you draw the line, the origin will move to the left because the origin is now in negative part of the x-axis.

 

[Mark44]

Take a look at this simple example, suppose $f(x)=x$, a simple straight line through the origin. If $x=0$, then $f(x)=0$ as well. But if you take $f(x+a)$, the origin will be at $-a$. If you draw the line, the origin will move to the left because the origin is now in negative part of the x-axis.

The origin doesn't move around, but the x-intercept does.

 

[CellCoree]

Hello there,

I am happy to help clarify your doubts about function transformations.

Firstly, let's talk about horizontal translations. In a function f(x + c), the value of c represents the amount of translation in the horizontal direction. When c is positive, it means that the function is being shifted to the right by c units. This is because in the original function, the x values are being increased by c, causing the function to shift to the right.

To better understand the effects of stretch and compression, it is helpful to think about the original parent function as a rubber band. When we stretch the rubber band vertically, it becomes longer and thinner, resulting in a vertical stretch of the function. On the other hand, when we compress the rubber band vertically, it becomes shorter and wider, resulting in a vertical compression of the function. Similarly, when we stretch the rubber band horizontally, it becomes longer and thinner, resulting in a horizontal stretch of the function. And when we compress the rubber band horizontally, it becomes shorter and wider, resulting in a horizontal compression of the function.

Now, to answer your question about why f(ax) stretches by a factor of 1/a instead of a, it is important to understand that in a function, the x values represent the inputs and the y values represent the outputs. So, when we multiply the x values by a, we are essentially changing the inputs of the function. This results in a change in the scale of the function, which is why it stretches or compresses. However, the y values remain the same, hence the factor of 1/a.

I hope this explanation helps to clear your doubts. Please feel free to ask more questions if you have any. I am always happy to help with any scientific concepts. Keep learning and exploring!