How to randomly generate continuous functions

My objective is to randomly generate good looking continuous functions, good looking meaning that functions which can be recovered from their plots.

Essentially I want to generate a random time series data for 1 second with 1024 samples per second. If I randomly choose 1024 values, then the plot looks very noisy and nothing meaningful can be extracted out of it. In the end I have attached plots of two sinusoids, one with a frequency of 3Hz and another with a frequency of 100Hz. I consider 3Hz cosine as a good function because I can extract back the timeseries by looking at the plot. But the 100 Hz sinusoid is bad for me as I cant recover the timeseries from the plot. So in the above mentioned meaning of goodness of a timeseries, I want to randomly generate good looking continuos functions/timeseries.

The method I am thinking of using is as follows (python language):

(1) Choose 32 points in x-axis between 0 to 1 using x=linspace(0,1,32).

(2) For each of these 32 points choose a random value using y=np.random.rand(32).

(3) Then I need an interpolation or curve fitting method which takes as input (x,y) and outputs a continuos function which would look something like func=curve_fit(x,y)

(4) I can obtain the time seires by sampling from the func function

Following are the questions that I have:

1) What is the best curve-fitting or interpolation method that I can use. They should also be available in python.

2) Is there a better method to generate good looking functions, without using curve fitting or interpolation.