Yes, there is only one true linear line. All the linear lines come from the origin function y = x. The reason why they look different is only because it is translated to other positions and or scaled. We will prove this statement in this post for all real functions!
So what is the translation of an function?
Well to put it in simplest terms, it is the process of modifying a function such that it translates (moves) to a different location on the Cartesian plane.
If you change the x value like this y = (x - z) then you will move the functions z units to the right.
If you change it like this y = (x + z) then you move it z units to the left.
If you scale a linear function then you basically changed its slope.
Here is an example of y = zx. This means you scaled the function up to a factor of z.
On the other hand you can scale it down like this : y = (1/z)* x. This would mean that you scaled the line down to one z th.
Let us think intuitively why this theorem must be true by studying the base formula of a linear line:
y= mx + c
Where:
- y is the output value
M the slope of the line
c is a constant showing where the line intercepts the y axis.
If you think of the origin function y = x then you can see the slope is equal to 1 and the constant c = 0. That would mean that any other linear line function is just a origin line that has been translated and the slope was just adjusted. So we can easily see that all lines come from the same place and it’s face values translated the line to different areas on the cartesian plane and we also scaled the function.
Let’s start proving it.
So this is a simple prove let’s assume the following:
Let f be a linear function such that all of f is continuous and differentiable.
f(x) = ax + b
Where a and b is any real number.
We want to prove that you can translate the function f such that f = x.
We can translate functions by plugging in an x value that can change the whole equation.
We will plug into x :
And now our function has been translated so it can fit neatly in the y= x position this can only happen if all the linear lines are the same….
Thus we have proven there is only one true function!
This was a micro proof post. Once again all images was made by myself!
As always thanks for reading!
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