You can graph any trig function in four or five steps. Determine the period of the function [latex]g(x)=\cos\left(\frac{x}{3}\right)[/latex]. In the given equation, notice that B = 1 and [latex]C=−\frac{π}{6}[/latex]. The sine and cosine functions have several distinct characteristics: As we can see, sine and cosine functions have a regular period and range. So. If we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms, [latex]y=A\sin(Bx)[/latex] and [latex]y=A\cos(Bx)[/latex], The amplitude is A, and the vertical height from the midline is |A|. Notice in Figure 8 how the period is indirectly related to [latex]|B|[/latex]. Then graph the function. The graph of the function has a maximum y-value of 2 and a minimum y-value of -2. Step 1. The midline of the oscillation will be at 69.5 m. The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes. or [latex]\frac{\pi}{6}[/latex] units to the left. Begin by comparing the equation to the general form and use the steps outlined in Example 9. Let’s start with the sine function. Step 1. Both y=sin⁡(x){\displaystyle y=\sin(x)} and y=cos⁡(x){\displaystyle y=\cos(x)} repeat the same shape from negative infinity to positive infinity on the x-axis (you'll generally only graph a portion of it). Again, these functions are equivalent, so both yield the same graph. 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. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection. Now let’s take a similar look at the cosine function. Step 3. for the rest of your life. The period is 4. Putting these transformations together, we find that. 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]. [latex]y=A\sin\left(Bx-C\right)+D[/latex], [latex]y=A\cos\left(Bx-C\right)+D[/latex]. In the given equation, [latex]D=-3[/latex], so the shift is 3 units downward. Sketch a graph of the function, and then find a cosine function that gives the position y in terms of x. The value [latex]\frac{C}{B}[/latex] in the general formula for a sinusoidal function indicates the phase shift. Express the function in the general form [latex]y=A\sin(Bx−C)+D[/latex] or [latex]y=A\cos(Bx−C)+D[/latex]. Enter in the values for f(x) = Asin(B(x-h))+k into the sine graph calculator to check your answer. We can see from the equation that A=−2,so the amplitude is 2. Sine functions are perfect ways of expressing this type of movement, because their graphs are repetitive and they oscillate (like a wave). Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation. The sine function has 180-degree-point symmetry about the origin. 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. Again, we can create a table of values and use them to sketch a graph. Instead of focusing on the general form equations. As with the sine function, we can plots points to create a graph of the cosine function as in Figure 4. problem and check your answer with the step-by-step explanations. In interval notation, you write this as [–1, 1]. Sketch a graph of [latex]f(x)=−2\sin(\frac{πx}{2})[/latex]. See Figure 14. Look at the graph on the left to see that curve as well as the curve of the equation y = sin (x + 2). Determine the direction and magnitude of the phase shift for [latex]f(x)=\sin(x+\frac{π}{6})−2[/latex]. Figure 21 shows one cycle of the graph of the function. Determine the phase shift as [latex]\frac{C}{B}[/latex]. The waves crest and fall over and over again forever, because you can keep plugging in values for. D = 3, so the midline is y = 3, and the vertical shift is up 3. The function is stretched. Next, B = 2, so the period is [latex]P=\frac{2π}{|B|}=\frac{2π}{2}=π[/latex]. Because A is negative, the graph descends as we move to the right of the origin. So in the x-direction, the wave (or sinusoid, in math language) goes on forever, and in the y-direction, the sinusoid oscillates only between –1 and 1, including these values. 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. Since A is negative, the graph of the cosine function has been reflected about the x-axis. Sketch a graph of this function. Find the values for domain and range. Determine the midline, amplitude, period, and phase shift of the function [latex]y=3\sin(2x)+1[/latex]. 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. This graph will have the shape of a sine function, starting at the midline and increasing to the right. Any value of D other than zero shifts the graph up or down. This value, which is the midline, is D in the equation, so D=0.5. Determining the amplitude and period of sine and cosine functions. 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. As the spring oscillates up and down, the position y of the weight relative to the board ranges from –1 in. Given a transformed graph of sine or cosine, determine a possible equation. A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. The local maxima will be a distance |A| above the vertical midline of the graph, which is the line x = D; because D = 0 in this case, the midline is the x-axis. 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]. In the first equation, y = sin x, c is equal to zero. If |A| > 1, the function is stretched. Now we can use the same information to create graphs from equations. Figure 5 shows several periods of the sine and cosine functions. Assume the position of y is given as a sinusoidal function of x. Find the amplitude which is half the distance between the maximum and minimum. The table below lists some of the values for the sine function on a unit circle. Express a rider’s height above ground as a function of time in minutes. 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. For example, the amplitude of [latex]f(x)=4\sin\left(x\right)[/latex] is twice the amplitude of. Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. [latex]\frac{C}{B}=\frac{\frac{\pi}{4}}{\frac{\pi}{4}}=1[/latex]. Now we can see from the graph that [latex]\cos(−x)=\cos x[/latex]. Scroll down the page for examples The negative value of A results in a reflection across the x-axis of the sine function, as shown in Figure 10. The graph of a sinusoidal function has the same general shape as a sine or cosine function. If C > 0, the graph shifts to the right. The general equation of a sine graph is y = A sin (B (x - D)) + C The general equation of a cosine graph is y = A cos (B (x - D)) + C The official math definition of an odd function, though, is f(–x) = –f(x) for every value of x in the domain. This makes the amplitude equal to |2| or simply 2. Now let’s turn to the variable A so we can analyze how it is related to the amplitude, or greatest distance from rest. For example. The greatest distance above and below the midline is the amplitude. The function [latex]y=\cos(x)+D[/latex] has its midline at [latex]y=D[/latex]. The quarter points include the minimum at x = 1 and the maximum at x = 3. For the shape and shift, we have more than one option. [latex]|B|=\frac{\pi}{2}[/latex], so the period is [latex]P=\frac{2π}{|B|}=\frac{2\pi}{\frac{\pi}{2}}\times2\pi=4[/latex]. A represents the vertical stretch factor, and its absolute value |A| is the amplitude. Plotting values of the sine function. You know that the highest value that sin x can be is 1. Periodic functions repeat after a given value. Write a formula for the function graphed in Figure 18. two possibilities are: [latex]y=4\sin(\frac{π}{5}x−\frac{π}{5})+4[/latex] or [latex]y=−4sin(\frac{π}{5}x+4\frac{π}{5})+4[/latex]. More Algebra 2 Lessons. Let’s begin by comparing the equation to the general form [latex]y=A\sin(Bx−C)+D[/latex].

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