Show a graph of 3 sin(x)/2 for -2π < x < 2π
Understand the Problem
The question is asking for a graph of the function (3 sin(x)/2) over the interval -2π to 2π. This involves plotting the sine function scaled by a factor of 3/2 within the specified x-range.
Answer
The graph of \( y = \frac{3}{2} \sin(x) \) oscillates from \( -\frac{3}{2} \) to \( \frac{3}{2} \) over the interval \( -2\pi \) to \( 2\pi \).
Answer for screen readers
The graph of the function ( y = \frac{3}{2} \sin(x) ) oscillates between ( -\frac{3}{2} ) and ( \frac{3}{2} ) over the interval ( -2\pi ) to ( 2\pi ).
Steps to Solve
- Identify the function to plot
We are given the function ( y = \frac{3}{2} \sin(x) ).
- Determine the x-values to use
We will plot the function over the interval from ( -2\pi ) to ( 2\pi ). We can select several points within this range to create a smooth graph. For example, choose points: $$ x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi $$
- Calculate the corresponding y-values
Using the function ( y = \frac{3}{2} \sin(x) ), we will calculate the y-values for the selected x-values.
For each of the x-values:
- For ( x = -2\pi ), ( y = \frac{3}{2} \sin(-2\pi) = 0 )
- For ( x = -\frac{3\pi}{2} ), ( y = \frac{3}{2} \sin(-\frac{3\pi}{2}) = -\frac{3}{2} )
- For ( x = -\pi ), ( y = \frac{3}{2} \sin(-\pi) = 0 )
- For ( x = -\frac{\pi}{2} ), ( y = \frac{3}{2} \sin(-\frac{\pi}{2}) = \frac{3}{2} )
- For ( x = 0 ), ( y = \frac{3}{2} \sin(0) = 0 )
- For ( x = \frac{\pi}{2} ), ( y = \frac{3}{2} \sin(\frac{\pi}{2}) = \frac{3}{2} )
- For ( x = \pi ), ( y = \frac{3}{2} \sin(\pi) = 0 )
- For ( x = \frac{3\pi}{2} ), ( y = \frac{3}{2} \sin(\frac{3\pi}{2}) = -\frac{3}{2} )
- For ( x = 2\pi ), ( y = \frac{3}{2} \sin(2\pi) = 0 )
- Draw the graph
Plot the points calculated according to the sine function behavior. The graph will oscillate between ( -\frac{3}{2} ) and ( \frac{3}{2} ), creating a wave-like pattern.
- Label the graph
Make sure to label the x-axis from ( -2\pi ) to ( 2\pi ) and y-axis from ( -\frac{3}{2} ) to ( \frac{3}{2} ) to help readers understand the graph.
The graph of the function ( y = \frac{3}{2} \sin(x) ) oscillates between ( -\frac{3}{2} ) and ( \frac{3}{2} ) over the interval ( -2\pi ) to ( 2\pi ).
More Information
This function is a sine wave that is vertically stretched by a factor of ( \frac{3}{2} ). The sine function has a regular wave pattern, oscillating between its maximum and minimum values, which in this case are ( \frac{3}{2} ) and ( -\frac{3}{2} ).
Tips
- Forgetting to correctly apply the scaling factor to the sine function.
- Mislabeling the axes or not including key points in the graphing, which can lead to confusion.
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