Determining if 3 curves have the same period

[negation]

Suppose there are 3 different curves.

How do I determine if they have the same period? I am aware of how to calculate period but how do we determine if they have the same period graphically?

1) what is the idea behind x(t) = x(t+T)?

2) must all 3 curves cuts a particular horizontal at the same time?

 

[negation]

I managed to understand it.
Thread can be closed.

 

[tycoon515]

The period of a function is the minimum width of an interval such that the value of the function at one of the interval's endpoints is equal to the value at the other endpoint, and the entire range of values that the dependent variable can take on is contained within the interval.

To determine whether three curves have the same period, start by marking a point of your choice on one of the curves. Then find and mark the next point to the right on the same curve such that a horizontal line would connect both points, and the entire range of values that the dependent variable can take on is covered between the two points. Repeat this process for the other two curves, and compare the horizontal distances between the two points marked on each of the curves. These distances measure the period of each of the curves.

A less hardline approach is to observe whether the dependent variable seems to cover its entire range of values and get back to the same value more frequently (i.e. in lesser-width intervals) on one of the curves. This works very well if the vertical boundaries (i.e. ranges) of the three curves are pretty close to one another because then you can easily see if one reaches its crest or trough sooner relative to when the others crest or trough with each successive cycle. Even if their periods are nearly identical, just follow all three curves out really far, and eventually you'll notice one peaking/troughing sooner relative to the others.

1) The idea behind x(t) = x(t+T) is that a dependent variable (i.e. horizontal position of a point in an oscillating spring-mass system) takes on some value at time t and then takes on that value again T amount of time later, where T is the period of the curve, and the dependent variable takes on all of the values it can from time t to time t+T. You can actually generalize this equation in the following way: x(t) = x(t+n*T) for integer n. This says that the dependent variable takes on the same value at time t that it does at any other time that is an integer multiple of T chronologically distant from t (i.e. t-2T, t-T, t, t+T, t+2T, etc...).

2) If all three curves have the same period, then they all cut out horizontals of a particular width, but not necessarily a particular horizontal. If they do cut out a particular horizontal, then all three curves must intersect at both endpoints of the horizontal.