for y=sin nx
'n' is called the frequency.
what effect does 'n' have on the graph ??
for a graph from 0 to 2pi, there will be n complete periods of the sine curve.
The period of each sine curve will be 2pi/n
e.g. y = sin 4x will have 4 complete sine curves from 0º to 360º, and each sine curve will have a period of 90º
or
y = sin 4x will have 4 complete sine curves from 0 to 2pi, and each sine curve will have a period of pi/2
The parameter 'n' in the function y = sin(nx) is called the frequency, and it directly affects the graph of the function.
The frequency determines the number of cycles or oscillations in the graph within a given interval. A higher frequency of 'n' will result in a greater number of cycles within the same interval, while a lower frequency will result in fewer cycles within the same interval.
Specifically, when 'n' is increased, the graph becomes more compressed horizontally, meaning the distance between consecutive peaks and troughs decreases. On the other hand, when 'n' is decreased, the graph becomes more stretched out horizontally, increasing the distance between peaks and troughs.
In summary, increasing the value of 'n' increases the number of oscillations within a given interval and compresses the graph horizontally, while decreasing 'n' does the opposite, resulting in fewer oscillations and stretching out the graph horizontally.
The parameter 'n' in the sine function y = sin(nx) determines the frequency of the graph. The frequency represents how many complete cycles of the sine wave occur within a given interval.
To understand the effect of 'n' on the graph, it is helpful to compare different values. Let's consider two cases: n = 1 and n = 2.
1. When n = 1:
y = sin(x)
In this case, 'n' is equal to 1. The graph of y = sin(x) completes one full cycle between 0 and 2π (or 0 and 360 degrees). It starts at zero, rises to a maximum value of 1, returns to zero, falls to a minimum value of -1, and then completes the cycle by returning to zero.
2. When n = 2:
y = sin(2x)
Here, 'n' is equal to 2. The graph of y = sin(2x) completes two full cycles between 0 and 2π (or 0 and 360 degrees). It starts at zero, rises to a maximum value of 1, returns to zero, falls to a minimum value of -1, rises again to zero, and goes through a second cycle with the same pattern.
Therefore, as 'n' increases, the frequency of the graph increases, resulting in more cycles occurring within a given interval. This causes the graph to become more "compressed" or "squished" horizontally, as the peaks and troughs appear closer together. On the other hand, if 'n' decreases, the frequency decreases, causing the graph to become more "spread out" or "stretched" horizontally, as the peaks and troughs become more spaced out.