Consider functions that have the property that f(x+6) = f(x) for all values of x. Find two functions that have this property and describe the geometric symmetry of all such functions. What word describes functions that have this property?

Question

adamk · Answer

@Loser66 But then f(x+6) would equal 12, which is not f(x).

adamk · Answer

I don't think I'm totally understanding.

satellite73 · Answer

i bet this is for a trig class
am i right?

satellite73 · Answer

guess i'll never know, but that is the usual introduction to periodic functions

loser66 · Answer

To me, 
1) a constant function gives us f(x+6) =f(x) for all x
2) a periodic one, any of them, you can use cos or sin to find it out. As long as you get the period = 6, you are ok.

adamk · Answer

@satellite73 Nope, it's for an AP Calc AB class.

adamk · Answer

@Loser66 What are you saying?

mathmale · Answer

Surely sounds as tho' your function f(x) is periodic with period 6.

Let's look at the sine and cosine functions:

The general form is y=a sin (bx + c), where a is the amplitude, b is the frequency and c represents a phase shift (not applicable to the problem at hand).  The period of this function, or of the cosine function similarly written, is 2pi/b, where b is defined above.

We are told that the period is 6.  Since the formula for period is 2pi/b, we can equate 6 to 2pi/b and solve for b.

This discussion would be identical for y=a cos (bx + c).

Can you now answer the original questions?   "Find two functions that have this property and describe the geometric symmetry of all such functions. What word describes functions that have this property?"

adamk · Answer

@mathmale So it's the same function but just with a vertical shift up 6?

phi · Answer

A periodic function  repeats its "basic shape" 
For example, a sin wave    f(x) = sin(x)
repeats over and over again
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