Graph one period of the following function. Show all work . y=2sinx+2cosx
To graph one period of the function y = 2sin(x) + 2cos(x), we need to determine the amplitude, period, phase shift, and vertical shift of the function.
1. Amplitude: The amplitude of a sinusoidal function is the absolute value of the coefficient of the trigonometric functions. In this case, the coefficient of sin(x) is 2, and the coefficient of cos(x) is also 2. Hence, the amplitude is ∣2∣ = 2.
2. Period: The period of a sinusoidal function is the distance between two consecutive peaks or troughs. Since sin(x) and cos(x) both have a period of 2π, the period of the given function is also 2π.
3. Phase Shift: The phase shift of a sinusoidal function is the horizontal shift of the graph. In this case, there is no phase shift since there are no additional terms inside the sin or cos functions.
4. Vertical Shift: The vertical shift of a sinusoidal function is the vertical translation of the graph. The constant term, in this case, is 0, so there is no vertical shift.
Combining these values, we can construct the graph of the function.
Start by drawing the x and y axes.
The graph of y = 2sin(x) will oscillate between two peaks and two troughs within the period of 2π, with a maximum value of 2 and a minimum value of -2. At x = 0, the graph will intersect the x-axis, and at x = π/2 and 3π/2, it will intersect the y-axis.
Similarly, the graph of y = 2cos(x) will have the same period of 2π and maximum and minimum values of 2 and -2, respectively. However, it will intersect the x-axis at x = π/2 and 3π/2 and intersect the y-axis at x = 0.
To graph the function y = 2sin(x) + 2cos(x), we can add the corresponding y-values of sin(x) and cos(x) at each x-value.
Here is a step-by-step summary of the graph construction for one period:
Step 1: Mark the x and y axes.
Step 2: Label key points on the x-axis: 0, π/2, π, 3π/2, 2π to represent one period.
Step 3: Label key points on the y-axis: -2, 0, 2 to represent the range of the function.
Step 4: Plot the points (0, 2), (π/2, 2), (π, 0), (3π/2, -2), (2π, -2) for the function y = 2sin(x).
Step 5: Plot the points (0, 2), (π/2, 0), (π, -2), (3π/2, 0), (2π, 2) for the function y = 2cos(x).
Step 6: Add the corresponding y-values for each x-value for y = 2sin(x) + 2cos(x).
Step 7: Connect the points with a smooth curve to complete the graph.
Note: Since there is no phase shift and vertical shift, the graph will pass through the origin at (0, 0). The amplitude of 2 means the graph will extend 2 units above and below the x-axis.
This process will give you the graph of one period of the function y = 2sin(x) + 2cos(x).
To graph one period of the function y = 2sin(x) + 2cos(x), we need to plot the function for values of x ranging from 0 to 2π (one complete cycle of the sine and cosine functions).
First, let's analyze the behavior of the function. It is a combination of sine and cosine functions, and they both have a period of 2π. Therefore, the combined function will also have a period of 2π.
To graph the function, follow these steps:
Step 1: Determine the amplitude:
The amplitude represents the maximum displacement from the central line. Here, the amplitudes of both the sine and cosine functions are 2. Therefore, the combination will also have an amplitude of 2.
Step 2: Plot the central line:
The central line is the equilibrium position around which the function oscillates. Since the sine and cosine functions have average values of 0, the central line will be y = 0.
Step 3: Identify key points within one period:
Since the period is 2π, we can identify some key points by evaluating the function at critical values of x.
- Start with x = 0:
Substituting x = 0 into the function, we get:
y = 2sin(0) + 2cos(0)
y = 0 + 2
y = 2
So, (0, 2) is one key point.
- Next, consider x = π/2:
y = 2sin(π/2) + 2cos(π/2)
y = 2 + 0
y = 2
So, (π/2, 2) is another key point.
- Lastly, use x = π:
y = 2sin(π) + 2cos(π)
y = 0 - 2
y = -2
So, (π, -2) is the third key point.
Step 4: Plot the points and complete the graph:
Now, using the amplitude and key points, we can plot the graph. The graph will show a combination of sine and cosine functions, resulting in a waveform that oscillates above and below the central line.
Connecting the key points, the graph will look like an oscillating wave with a peak at (0, 2), a peak at (π/2, 2), and a trough at (π, -2). The graph will repeat itself every 2π units in the x-axis.
Here is a visual representation of the graph:
```
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2 | +
| + +
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---------|-------------
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-2 |
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```
Remember, the graph will continue to repeat itself in both the positive and negative x-directions.
That's how you graph one period of the function y = 2sin(x) + 2cos(x).