A sine function has the following key features:
Period = 4
Amplitude = 4
Midline: y = 1
y-intercept: (0, 1)
The function is not a reflection of its parent function over the x-axis.
Use the Desmos graph tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.
Period = 4 ... y = sin(π/2 x)
Amplitude = 4 ... y = 4sin(π/2 x)
Midline: y = 1 ... y = 1 + 4sin(π/2 x)
y-intercept: (0, 1) ... no change, since sin(0) = 0
y not reflected in x-axis. duh. That would be -sin(x)
Of course, you could change things up a bit and use cosine instead of sine.
Graph both versions and see which one you like.
To graph the sine function with the given key features on Desmos, follow these steps:
1. Open the Desmos graph tool by visiting https://www.desmos.com/calculator.
2. Clear any existing equations by clicking on the trash can icon in the top left corner of the equation bar.
3. Enter the equation for the sine function using the key features. The general form of a sine function is:
y = A * sin(B * (x - C)) + D
- The amplitude, A, represents half the distance between the maximum and minimum values of the function. In this case, the amplitude is 4.
- The period, represented by B, is the distance between two consecutive peaks or troughs. The period is given as 4, so B = 2π/4 = π/2.
- The midline, represented by D, is the horizontal line that intersects the middle of the function. The midline is given as y = 1, so D = 1.
- The phase shift, represented by C, determines the horizontal shift of the function. In this case, there is no phase shift, so C = 0.
Combining these values, the equation of the sine function becomes:
y = 4 * sin((π/2) * (x - 0)) + 1
Enter this equation in the Desmos equation bar and press Enter.
4. To graph the function, Desmos will automatically adjust the x and y-axis limits based on the data entered. However, to ensure that the first point is on the midline and the second point is a maximum or minimum value closest to the first point, we need to manually set the x-axis limits.
- Click on the wrench icon in the top right corner of the graph to open the settings menu.
- In the Axis Bounds section, change the values for the x-axis upper and lower bounds to cover a period of the function. In this case, set the lower bound to -2 and the upper bound to 2.
- Close the settings menu.
5. With the x-axis limits set, the graph will now display the function with the first point on the midline and the second point as a maximum or minimum value closest to the first point.
- The y-intercept is given as (0, 1), which means the function passes through the point (0, 1) where x = 0. You will see this point plotted on the graph.
- The second point depends on whether you start with a maximum or minimum value. Since there is no reflection over the x-axis, we will start with a maximum value. In this case, the first maximum value closest to the first point occurs around x = 0.5. The corresponding y-value will give you the second point on the graph.
The graph should now display the sine function with the given key features. Adjust the x-axis limits or other settings as needed to further explore the function.
To graph the sine function with the given key features, follow these steps:
1. Open the Desmos graphing calculator tool: https://www.desmos.com/calculator
2. Clear any existing equations by clicking on the "x" next to them or using the backspace key.
3. Enter the equation of the sine function with the given key features: y = 4sin((2π/4)x) + 1
4. Click on the wrench icon on the top-right corner of the calculator to open the settings menu.
5. In the settings menu, click on the "Graph Settings" tab.
6. Under "Window Size", you can adjust the graph's x-axis and y-axis ranges. To make the period visible, you can set the x-axis range from -2 to 6. Adjust the y-axis range as needed to include the amplitude and midline.
7. Click on the "+" button on the top-left corner of the calculator to add the first point on the midline. Enter the coordinates (0, 1) and press enter. This point ensures that the graph starts at the midline.
8. Click on the "+" button again to add the second point, which should be a maximum or minimum value closest to the first point. The closest maximum or minimum occurs at x = 2 (half the period). To determine the y-coordinate, substitute x = 2 into the equation: y = 4sin((2π/4) * 2) + 1. Calculate the value and enter it as the y-coordinate.
9. The graph should now display the sine function with the given key features.