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Write a function for the sinusoid shown below. EXAMPLE 1. Solve a multi-step problem. SOLUTION. Find the maximum value M and minimum value m . From the graph, M = 5 and m = –1. STEP 1. 5 + ( – 1). M + m. 2 π. 2. k. =. =. =. =. 2. 2. b. π. 4. 2. 2. =. EXAMPLE 1.
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Write a function for the sinusoid shown below. EXAMPLE 1 Solve a multi-step problem SOLUTION Find the maximum value Mand minimum value m. From the graph, M = 5 and m = –1. STEP 1
5 + (–1) M + m 2π 2. k = = = = 2 2 b π 4 2 2 = EXAMPLE 1 Solve a multi-step problem Identify the vertical shift, k. The value of kis the mean of the maximum and minimum values. The vertical shift is STEP 2 So, k = 2. Decide whether the graph should be modeled by a sine or cosine function. Because the graph crosses the midline y = 2 on the y-axis, the graph is a sine curve with no horizontal shift. STEP 3 So, h = 0. Find the amplitude and period. The period is STEP 4 So, b = 4.
= a = = 5 – (–1) M–m 2 2 ANSWER 6 The function is y = 3 sin 4x + 2. 2 EXAMPLE 1 Solve a multi-step problem = 3. The amplitude is The graph is not a reflection, so a > 0. a = 3. Therefore,
Jump Rope ROPE At a Double Dutch competition, two people swing jump ropes as shown in the diagram below. The highest point of the middle of each rope is 75 inches above the ground, and the lowest point is 3 inches. The rope makes 2 revolutions per second. Write a model for the height h(in feet) of a rope as a function of the time t(in seconds) if the rope is at its lowest point when t = 0. EXAMPLE 2 Model circular motion
EXAMPLE 2 Model circular motion SOLUTION STEP 1 Find the maximum and minimum values of the function. A rope’s maximum height is 75 inches, so M = 75. A rope’s minimum height is 3 inches, so m = 3.
= = = k M + m 78 2 2 75 + 3 2 EXAMPLE 2 Model circular motion STEP 2 Identify the vertical shift. The vertical shift for the model is: = 39 STEP 3 Decide whether the height should be modeled by a sine or cosine function. When t = 0, the height is at its minimum. So, use a cosine function whose graph is a reflection in the x-axis with no horizontal shift (h = 0).
75 –3 a = = 2 M–m 2π 2 b ANSWER A model for the height of a rope is h = –36 cos 4π t + 39. = 0.5, and b = 4π. EXAMPLE 2 Model circular motion Find the amplitude and period. STEP 4 The amplitude is = 36. Because the graph is a reflection, a < 0. So, a = –36. Because a rope is rotating at a rate of 2 revolutions per second, one revolution is completed in 0.5 second. So, the period is
1. for Examples 1 and 2 GUIDED PRACTICE Write a function for the sinusoid. SOLUTION Find the maximum value Mand minimum value m. From the graph, M = 2 and m = –2. STEP 1
1. 2 + (–2) M + m 0. k = = = = 2 2 0 2 for Examples 1 and 2 GUIDED PRACTICE Identify the vertical shift, k. The value of kis the mean of the maximum and minimum values. The vertical shift is STEP 2 So, k = 0.
1. 2π b 2π 3 = for Examples 1 and 2 GUIDED PRACTICE Decide whether the graph should be modeled by a sine or cosine function. Because the graph peaks at y = 2 on the y-axis, the graph is a cos curve with no horizontal shift. So, h = 0. STEP 3 Find the amplitude and period. The period is STEP 4 So, b = 3.
= a = = 1. 2 – (–2) M–m 2 2 ANSWER 4 The function is y = 2 cos 3x. 2 for Examples 1 and 2 GUIDED PRACTICE = 2. The amplitude is The graph is not a reflection, so a > 0. a = 2. Therefore,
2. ANSWER y = 2 sinπx – 1 for Examples 1 and 2 GUIDED PRACTICE Write a function for the sinusoid.
ANSWER The amplitude changes to 32.5 and the vertical shift becomes 37.5, but the period is not affected for Examples 1 and 2 GUIDED PRACTICE 3. WHAT IF?Describe how the model in Example 2 would change if the lowest point of a rope is 5 inches above the ground and the highest point is 70 inches above the ground.
