The official math definition of an odd function, though, is f(–x) = –f(x) for every value of x in the domain. Inspecting the graph, we can determine that the period is π, the midline is y = 1,and the amplitude is 3. Because the graph of the sine function is being graphed on the x–y plane, you rewrite this as f(x) = sin x where x is the measure of the angle in radians. In the given equation, [latex]B =\frac{π}{6}[/latex], so the period will be, [latex]\begin{align}P&=\frac{\frac{2}{\pi}}{|B|} \\ &=\frac{2\pi}{\frac{x}{6}} \\ &=2\pi\times \frac{6}{\pi} \\ &=12 \end{align}[/latex]. With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center. Animation: Graphing the Sine Function Using The Unit Circle . We can use what we know about transformations to determine the period. The graph of a sinusoidal function has the same general shape as a sine or cosine function. So far, our equation is either [latex]y=3\sin(\frac{\pi}{3}x−C)−2[/latex] or [latex]y=3\cos(\frac{\pi}{3}x−C)−2[/latex]. Notice how the sine values are positive between 0 and π, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between π and 2π, which correspond to the values of the sine function in quadrants III and IV on the unit circle. The greater the value of |C|, the more the graph is shifted. where t is in minutes and y is measured in meters. (at time x = 0) to –7in. Determine the direction and magnitude of the vertical shift for [latex]f(x)=3\sin(x)+2[/latex]. Given [latex]y=−2\cos\left(\frac{\pi}{2}x+\pi\right)+3[/latex], determine the amplitude, period, phase shift, and horizontal shift. We have included a tool that will plot the sine graph In the given equation, [latex]D=-3[/latex], so the shift is 3 units downward. More Lessons On Equations For Sine Or Cosine Graphs. Passengers board 2 m above ground level, so the center of the wheel must be located 67.5 + 2 = 69.5 m above ground level. When [latex]x=0[/latex], the graph has an extreme point, [latex](0,0)[/latex]. Figure 21 shows one cycle of the graph of the function. Step 5. The function is stretched. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. While C relates to the horizontal shift, D indicates the vertical shift from the midline in the general formula for a sinusoidal function. See Figure 14. It completes one rotation every 30 minutes. Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation. Find out where the graph of f(x) = sin x crosses the x-axis by finding unit circle angles where sine is 0. Since [latex]|B|=|\frac{π}{4}|=\frac{π}{4}[/latex], we determine the period as follows. When you graph lines in algebra, the x-intercepts occur when y = 0. As we can see in Figure 6, the sine function is symmetric about the origin. Let’s use a cosine function because it starts at the highest or lowest value, while a sine function starts at the middle value. Therefore, the domain of sine is all real numbers, or, On the unit circle, the y values are your sine values — what you get after plugging the value of. How To Identify The Graph Of A Stretched Cosine Curve? If [latex]f(x) = \sin\left(\frac{x}{2} \right)[/latex], then [latex]B=\frac{1}{2}[/latex], so the period is [latex]4π[/latex] and the graph is stretched. Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. The quarter points include the minimum at x = 1 and the maximum at x = 3. Step 3. [latex]C=−\pi[/latex], so we calculate the phase shift as [latex]\frac{C}{B}=\frac{−\pi}{\frac{\pi}{2}}=−\pi\times\frac{2}{\pi}=−2[/latex]. In addition, notice in the example that, [latex]|A|=\text{amplitude}=\frac{1}{2}|\text{maximum}−\text{minimum}|[/latex]. The general forms of sinusoidal functions are. We see that the graph of f(x) = sin x crosses the x-axis three times: You now know that three of the coordinate points are. [latex]g(x)=0.5\cos\left(x\right)+0.5[/latex]. Determine the midline, amplitude, period, and phase shift. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [−1,1]. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. The period is 4. It’s symmetrical about the origin (thus, in math speak, it’s an odd function). Step 3. An equation for the rider’s height would be, [latex]y=−67.5\cos\left(\frac{\pi}{15}t\right)+69.5[/latex]. (at time x = π) below the board. The waves crest and fall over and over again forever, because you can keep plugging in values for. If [latex]f(x) =\sin (2x)[/latex], then [latex]B= 2[/latex], so the period is [latex]π[/latex] and the graph is compressed. The greatest distance above and below the midline is the amplitude. [latex]y = A \sin(Bx−C)+D[/latex] and [latex]y=A\cos(Bx−C)+D[/latex], [latex]y=A\sin(B(x−\frac{C}{B}))+D[/latex] and [latex]y=A\cos(B(x−\frac{C}{B}))+D[/latex], With the highest value at 1 and the lowest value at−5, the midline will be halfway between at −2. Returning to the general formula for a sinusoidal function, we have analyzed how the variable B relates to the period. 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 value [latex]\frac{C}{B}[/latex] for a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. Here is a sine function we will graph. Determining the Equation of a Sine and Cosine Graph - YouTube The graph of sine is called periodic because of this repeating pattern. Any value of D other than zero shifts the graph up or down. The point closest to the ground is labeled P, as shown in Figure 23. Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. Determine the period as [latex]P=\frac{2π}{|B|}[/latex]. Calculate the graph’s maximum and minimum points. The x-intercepts are at the beginning of one period, x = 0, the horizontal midpoints are at x = 2 and at the end of one period at x = 4. Begin by comparing the equation to the general form and use the steps outlined in Example 9. Let’s start with the midline. Determine the midline, amplitude, period, and phase shift of the function [latex]y=3\sin(2x)+1[/latex]. Draw a coordinate plane. Step 3. Determine the equation for the sinusoidal function in Figure 17. So what do they look like on a graph on a coordinate plane? The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). So the phase shift is, [latex]\begin{align}\frac{C}{B}&=−\frac{\frac{x}{6}}{1} \\ &=−\frac{\pi}{6} \end{align}[/latex]. Figure 7. 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 |A|=[latex]\frac{1}{2}[/latex]. into the sine function. When finding the equation for a trig function, try to identify if it is a sine or cosine graph. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection. Try the given examples, or type in your own

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