The main difference between the two is that cosine wave leads the sine wave by an amount of 90 degrees. Image Courtesy: reddit. Add new comment Your name. Plain text. This question is for testing whether or not you are a human visitor and to prevent automated spam submissions. Since A is negative, the graph of the cosine function has been reflected about the x -axis. We can use the transformations of sine and cosine functions in numerous applications.
As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function. A point rotates around a circle of radius 3 centered at the origin.
Sketch a graph of the y -coordinate of the point as a function of the angle of rotation. The constant 3 causes a vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph in Figure Because the outputs of the graph will now oscillate between —3 and 3, the amplitude of the sine wave is 3.
Sketch a graph of this function. A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P , as shown in Figure Sketch a graph of the height above the ground of the point P as the circle is rotated; then find a function that gives the height in terms of the angle of rotation. Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in Figure Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other.
A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection. Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example. Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4.
Putting these transformations together, we find that. A weight is attached to a spring that is then hung from a board, as shown in Figure As the spring oscillates up and down, the position y of the weight relative to the board ranges from —1 in. Assume the position of y is given as a sinusoidal function of x. Sketch a graph of the function, and then find a cosine function that gives the position y in terms of x. The London Eye is a huge Ferris wheel with a diameter of meters feet.
It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. With a diameter of m, the wheel has a radius of The height will oscillate with amplitude Passengers board 2 m above ground level, so the center of the wheel must be located The midline of the oscillation will be at The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.
Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.
For each function, state the amplitude, period, and midline. State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.
Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure Explain why the graph appears as it does.
Did the graph appear as predicted in the previous exercise? A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The wheel completes 1 full revolution in 10 minutes. Find the amplitude, midline, and period of h t. Find a formula for the height function h t. How high off the ground is a person after 5 minutes?
Skip to main content. Module 2: Periodic Functions. Search for:. Figure 2. The sine function. Figure 3. Next, so the period is. There is no added constant inside the parentheses, so and the phase shift is. Inspecting the graph, we can determine that the period is the midline is and the amplitude is 3. Determine the formula for the cosine function in Figure. To determine the equation, we need to identify each value in the general form of a sinusoidal function.
When the graph has an extreme point, Since the cosine function has an extreme point for let us write our equation in terms of a cosine function. We can see that the graph rises and falls an equal distance above and below This value, which is the midline, is in the equation, so.
The greatest distance above and below the midline is the amplitude. The maxima are 0. So Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so Also, the graph is reflected about the x -axis so that. The graph is not horizontally stretched or compressed, so and the graph is not shifted horizontally, so. Determine the formula for the sine function in Figure.
Determine the equation for the sinusoidal function in Figure. With the highest value at 1 and the lowest value at the midline will be halfway between at So. The distance from the midline to the highest or lowest value gives an amplitude of. The period of the graph is 6, which can be measured from the peak at to the next peak at or from the distance between the lowest points. Therefore, Using the positive value for we find that.
So far, our equation is either or For the shape and shift, we have more than one option. We could write this as any one of the following:. While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes. Again, these functions are equivalent, so both yield the same graph. Write a formula for the function graphed in Figure. Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs.
Now we can use the same information to create graphs from equations. Given the function sketch its graph. Sketch a graph of. The quarter points include the minimum at and the maximum at A local minimum will occur 2 units below the midline, at and a local maximum will occur at 2 units above the midline, at Figure shows the graph of the function. Sketch a graph of Determine the midline, amplitude, period, and phase shift.
Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph. Draw a graph of Determine the midline, amplitude, period, and phase shift. Given determine the amplitude, period, phase shift, and horizontal shift. Then graph the function. Begin by comparing the equation to the general form and use the steps outlined in Figure. Since is negative, the graph of the cosine function has been reflected about the x -axis. Figure shows one cycle of the graph of the function.
We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function.
A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the y -coordinate of the point as a function of the angle of rotation. Recall that, for a point on a circle of radius r , the y -coordinate of the point is so in this case, we get the equation The constant 3 causes a vertical stretch of the y -values of the function by a factor of 3, which we can see in the graph in Figure.
Notice that the period of the function is still as we travel around the circle, we return to the point for Because the outputs of the graph will now oscillate between and the amplitude of the sine wave is. What is the amplitude of the function Sketch a graph of this function. A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P , as shown in Figure.
Sketch a graph of the height above the ground of the point as the circle is rotated; then find a function that gives the height in terms of the angle of rotation. Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in Figure.
Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection. Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.
Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that. A weight is attached to a spring that is then hung from a board, as shown in Figure. As the spring oscillates up and down, the position of the weight relative to the board ranges from in. Assume the position of is given as a sinusoidal function of Sketch a graph of the function, and then find a cosine function that gives the position in terms of.
The London Eye is a huge Ferris wheel with a diameter of meters feet. It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. With a diameter of m, the wheel has a radius of The height will oscillate with amplitude Passengers board 2 m above ground level, so the center of the wheel must be located m above ground level.
The midline of the oscillation will be at The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes. Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.
Access these online resources for additional instruction and practice with graphs of sine and cosine functions. The sine and cosine functions have the property that for a certain This means that the function values repeat for every units on the x -axis. How does the graph of compare with the graph of Explain how you could horizontally translate the graph of to obtain. For the equation what constants affect the range of the function and how do they affect the range?
The absolute value of the constant amplitude increases the total range and the constant vertical shift shifts the graph vertically. How does the range of a translated sine function relate to the equation. How can the unit circle be used to construct the graph of. At the point where the terminal side of intersects the unit circle, you can determine that the equals the y -coordinate of the point.
For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y -values and their corresponding x -values on one period for Round answers to two decimal places if necessary.
For the following exercises, graph one full period of each function, starting at For each function, state the amplitude, period, and midline. State the maximum and minimum y -values and their corresponding x -values on one period for State the phase shift and vertical translation, if applicable.
Round answers to two decimal places if necessary. Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure.
Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure.
For the following exercises, let. On solve. On Find all values of. On the maximum value s of the function occur s at what x -value s? On the minimum value s of the function occur s at what x -value s? On solve the equation. On find the x -intercepts of.
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