Figure 7 shows that the cosine function is symmetric about the y -axis. Again, we determined that the cosine function is an even function. As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical.
Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are. Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.
Notice in Figure 8 how the period is indirectly related to B. Returning to the general formula for a sinusoidal function, we have analyzed how the variable B relates to the period. A represents the vertical stretch factor, and its absolute value A is the amplitude. The local minima will be the same distance below the midline. Figure 9 compares several sine functions with different amplitudes. The amplitude is A, and the vertical height from the midline is A. In addition, notice in the example that.
Is the function stretched or compressed vertically? The function is stretched. The negative value of A results in a reflection across the x -axis of the sine function , as shown in Figure Now that we understand how A and B relate to the general form equation for the sine and cosine functions, we will explore the variables C and D. Recall the general form:. The greater the value of C , the more the graph is shifted.
While C relates to the horizontal shift, D indicates the vertical shift from the midline in the general formula for a sinusoidal function. Any value of D other than zero shifts the graph up or down. So the phase shift is. We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before C. If the value of C is negative, the shift is to the left. See Figure Using the positive value for B , we find that.
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. 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. On find the x -values at which the function has a maximum or minimum value. Graph on Explain why the graph appears as it does. Graph on Did the graph appear as predicted in the previous exercise? This function also occurs in nature as seen in ocean waves, sound waves and light waves. Even average daily temperatures for each day of the year resemble this function.
The term sinusoid was first used by Scotsman Stuart Kenny in while observing the growth and harvest of soybeans. Any cosine function can be written as a sine function. The value A in front of sin or cos affects the amplitude height. The amplitude half the distance between the maximum and minimum values of the function will be A , since distance is always positive.
A mathematical model is a function that describes some phenomenon. For objects that exhibit periodic behavior, a sinusoidal function can be used as a model since these functions are periodic. However, the concept of frequency is used in some applications of periodic phenomena instead of the period. The frequency of a sinusoidal function is the number of periods or cycles per unit time. A typical unit for frequency is the hertz.
One hertz Hz is one cycle per second. This unit is named after Heinrich Hertz — Since frequency is the number of cycles per unit time, and the period is the amount of time to complete one cycle, we see that frequency and period are related as follows:.
The volume of the average heart is milliliters ml , and it pushes out about one-half its volume 70 ml with each beat. In addition, the frequency of the for a well-trained athlete heartbeat for a well-trained athlete is 50 beats cycles per minute.
We will model the volume, V. Our function is. For example:. Suppose that we want to know at what times after the heart is full that there will be milliliters of blood in the heart. Although we will learn other methods for solving this type of equation later in the book, we can use a graphing utility to determine approximate solutions for this equation.
To solve the equation, we need to use a graphing utility that allows us to determine or approximate the points of intersection of two graphs. This can be done using most Texas Instruments calculators and Geogebra.
We really only need to find the coordinates of one of those points since we can use properties of sinusoids to find the others. The summer solstice in was on June 21 and the winter solstice was on December The maximum hours of daylight occurs on the summer solstice and the minimum hours of daylight occurs on the winter solstice. According to the U. Naval Observatory website, aa. This means that in Grand Rapids,. Since we have the coordinates for a high and low point, we first do the following computations:.
We must now decide whether to use a sine function or a cosine function to get the phase shift.
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