Trigonometry Lecture Notes_part1-1

We can stretch or shrink the graph by adding a

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We can stretch or shrink the graph by adding a coefficient other than one to the sine graph: sin y A x = This is the graph of y = sinx, compare it to the graph below of y = 2 sinx:
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In general the graph of y = Asinx ranges between A - and A . We call A the amplitude of the graph. If it is A > 1 the graph is stretched, if A < 1 it is shrunk. Graphing Variations of y=sinx Identify the amplitude and the period. Find the values of x for the five key points – the three x -intercepts, the maximum point, and the minimum point. Start with the value of x where the cycle begins and add quarter-periods – that is, period/4 – to find successive values of x . Find the values of y for the five key points by evaluating the function at each value of x from step 2. Connect the five key points with a smooth curve and graph one complete cycle of the given function. Extend the graph in step 4 to the left or right as desired. Example 24 Determine the amplitude of y = 1/2 sin x . Then graph y = sin x and y = 1/2 sin x for 0 < x < 2 π . Step 1 Identify the amplitude and the period. The equation y = 1/2 sin x is of the form y = A sin x with A = 1/2. Thus, the amplitude | A | = 1/2. This means that the maximum value of y is 1/2 and the minimum value of y is -1/2. The period for both y = 1/2 sin x and y = sin x is 2 π . Step 2 Find the values of x for the five key points. We need to find the three x-intercepts, the maximum point, and the minimum point on the interval [0, 2 π ]. To do so, we begin by dividing the period, 2 π , by 4.
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Period/4 = 2 π /4 = π /2 We start with the value of x where the cycle begins: x = 0. Now we add quarter periods, π / 2 , to generate x-values for each of the key points. The five x-values are x = 0, x = π /2, x = π , x = 3 π /2, x = 2 π Step 3 Find the values of y for the five key points. We evaluate the function at each value of x from step 2. (0,0), ( π /2, 1/2), ( π ,0), (3 π /2, -1/2), (2 π , 0) Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the given function. The five key points for y = 1/2sin x are shown below. By connecting the points with a smooth curve, the figure shows one complete cycle of y = 1/2sin x . Also shown is graph of y = sin x . The graph of y = 1/2sin x shrinks the graph of y = sin x . Amplitudes and Periods The graph of y = A sin Bx has amplitude = | A | and period = 2 B π .
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Example 25 Determine the amplitude and period of y = 3sin2x, then graph the function for 0 2 x π . Steps 1. Determine the amplitude and the period using the form sin y A Bx = . 2. Divide the period by four and find the five key points using 1 4 i i period x x - = + . 3. Get your resulting y-values after plugging in the above x-values. 4. Extend the graph as you wish The Graph of y = Asin(Bx - C) The graph of y = A sin ( Bx C ) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C B . The number C B is called the phase shift . Example 26
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Determine the amplitude, period, and phase shift of y = 2sin(3x- π ) Solution: y = A sin ( Bx C ) Amplitude = |A| = 2 period = 2 π /B = 2 π /3 phase shift = C/B = π /3 y = 2sin(3x- π ) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -3 -2 -1 1 2 3 The Graph of y = AcosBx We graph y = cos x by listing some points on the graph. Because the period of the cosine function is 2 π
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