Sketching Trig Functions

[generalkorn12]

Hi,

I'm just wondering what was the exact reason behind Trig graph translations. For example, cos(x+1) - indicates the graph is moved one unit to the left, but why exactly? 

I can understand the reasons for f(x), functions, when you consider in asymptotes and such, but I can't understand the reasons for Trig graphs.

Thanks.

[brightsky]

suppose we want to perform the following transformation (x,y) -> (x-1,y) to the function y = x^2. test out the point (0,0). we want this to move to (-1,0). that is, we want the rule to change so that if we substitute -1 in we'll get the same y-value. intuitively, try y = (x-1)^2. if we substitute -1 in, we won't get y = 0, but we do if we change the rule to y = (x+1)^2.

now for the general case. we have a function y = f(x). we want to apply the transformation (x,y) --> (x-1,y). that is we want to change the equation so that when we substitute x-1, we will get the same y. the only way to do so is to change the equation to y = f(x+1), because then when we substitute x-1, we will get y = f(x), which is the same y-value as the original.

[Special At Specialist]

If you accept that the translation from f(x) to f(x + 1) is a translation of 1 unit to the left of f(x), then you also accept that going from cos(x) to cos(x + 1) is a translation of 1 unit to the left.

If you don't accept that, then I'll show you:
Your x-values are being moved 1 up to what they used to be.
Usually, values of x = 0, x = 1 and x = 2 would produce f(0), f(1) and f(2).
Now to produce f(0), f(1) and f(2), we must use the values x = -1, x = 0 and x = 1.
We have changed from x = 0, 1 and 2 to x = -1, 0 and 1, thus decreasing the x-values by 1, or moving 1 to the left on a cartesian plane.