graph y=sin2x, and y=sin1/2x . How does the k value affect the graphs? Describe the transformation in words. What key feature of the graph is changed?
new link
https://www.wolframalpha.com/input/?i=graph+y+%3D+sinx%2C+y+%3D+sin%282x%29%2C+0+%3C+x+%3C+2%CF%80
I wanted the domain from 0 to 2π
and I wanted you to compare the number of complete cycles of the sine curve
for y = sin kx , from 0 to 2π
sin(kx) has a period of 2π/k
To understand the effect of the k value on the graphs y = sin(2x) and y = sin(1/2x), let's begin by discussing the general form of the sine function, which is y = sin(kx). Here, the k value represents the coefficient of x, which affects the period and the frequency of the graph.
For y = sin(2x), the k value is 2. This means the coefficient of x is 2, implying that the graph will have a shorter period compared to the standard sine graph. The period of the standard sine graph is 2π, but when k is 2, the period becomes 2π/2 = π. Consequently, the graph will oscillate twice as fast as the standard sine graph. It appears more compressed horizontally, with each complete oscillation occurring over a smaller interval.
On the other hand, for y = sin(1/2x), the k value is 1/2. This implies that the coefficient of x is 1/2, resulting in a longer period compared to the standard sine graph. The period of the standard sine graph is 2π, but when k is 1/2, the period becomes 2π/(1/2) = 4π. Hence, the graph oscillates half as fast as the standard sine graph. It appears stretched horizontally, with each complete oscillation happening over a larger interval.
In summary, the k value affects the period and frequency of the sine function graph. When k is greater than 1, like in y = sin(2x), the graph is compressed horizontally, oscillating at a faster rate. When k is less than 1, like in y = sin(1/2x), the graph is stretched horizontally, oscillating at a slower rate. The key feature of the graph that is changed is the time taken for one complete oscillation.
I suggest you go to Wolfram and graph the following:
y = sinx
y = sin(2x)
y = sin(3x)
y = sin(1/2x)
y = (1/4x)
etc
keep the y = sinx with each case, as a reference
e.g.
https://www.wolframalpha.com/input/?i=graph+y+%3D+sinx%2C+y+%3D+sin%282x%29