A sine function has the following key features:
Frequency = 1/4π
Amplitude = 2
Midline: y = 2
y-intercept: (0, 2)
The function is not a reflection of its parent function over the x-axis.
Use the sine tool to graph the function. The first point must be on the mid-line and the second point must be a maximum or minimum value on the graph closest to the first point.
To graph the given sine function with the provided key features, you can follow these steps:
Step 1: Determine the period of the sine function.
The period of a sine function is the distance between two successive peaks (or troughs) of the graph. The frequency is given as 1/4π, which means the period is 2π divided by the frequency: 2π / (1/4π) = 8. Therefore, the period of the sine function is 8.
Step 2: Determine the amplitude.
The amplitude is given as 2, which represents half the vertical distance between the peak and trough of the graph. This means the maximum value of the graph is 2 units above the midline, and the minimum value is 2 units below the midline.
Step 3: Identify the midline.
The midline of the sine function is given as y = 2, which means the graph is shifted vertically 2 units above the x-axis. This line acts as a reference point for the function.
Step 4: Determine the y-intercept.
The y-intercept is given as (0, 2), which means the graph intersects the y-axis at the point (0, 2).
Step 5: Plot the first two key points.
Start by plotting the first point on the midline, which is (0, 2) as given. Then plot the second point, which will be a maximum or minimum value closest to the first point. Since the graph is not reflected over the x-axis, the second point will be maximum.
Step 6: Use the period to find other key points.
Based on the period of 8, divide the x-axis into equal intervals of 8. For example, you can choose points such as (8, 4), (16, 2), (24, 0), (32, 2), and so on, to complete a full cycle of the graph. You can plot these points by adding or subtracting the period from the x-coordinate of the second point.
Step 7: Sketch the graph.
Connect the plotted points smoothly with a periodic shape that starts at the first point on the midline and reaches the second point, repeating the same pattern for each cycle. The graph should alternate between reaching the maximum and minimum values, remaining within the amplitude range, and crossing the midline.
By following these steps, you can accurately graph the given sine function with the provided key features.
To graph the given sine function, you will need to follow these steps:
1. Start by plotting the y-intercept, which is (0, 2). This point represents the point on the midline.
2. Next, determine the frequency of the function, which is 1/4π. The general formula for the sine function is y = A*sin(Bx) + C, where A represents the amplitude, B represents the frequency, and C represents the vertical shift or midline.
Here, the frequency is 1/4π, which means that B = 1/4π.
3. The amplitude is given as 2, so the equation becomes y = 2*sin(1/4π*x) + 2.
4. Find the point closest to the first point, which should be a maximum or minimum value depending on the starting point. To find this point, calculate the phase shift. The general formula for the phase shift is φ = -C/B.
Here, the midline is y = 2, so the phase shift becomes φ = -2/(1/4π) = -8/π.
5. To find the next point, add or subtract the period value from the x-coordinate of the first point. The period of a sine function is T = 2π/B.
Here, the period is 2π/(1/4π) = 8.
6. Plot the second point on the graph using the calculated x-coordinate and the corresponding y-coordinate obtained by plugging the x-value into the function.
7. Repeat steps 5 and 6 for additional points if necessary, depending on the desired accuracy of the graph.
Following these steps, you can continue plotting points until you have enough to sketch the graph accurately.
Frequency = 1/4π ==> period = 4π ... y = sin(1/2 x)
Amplitude = 2 ... y = 2sin(1/2 x)
Midline: y = 2 ... y = 2 + 2sin(1/2 x)
y-intercept: (0, 2) ... works for me.