Question 1190402: Sketch, on separate diagrams, the following graphs for 0° ≤ x ≤ 360°. For each function, write down
the amplitude, the period, the coordinates of the maximum and minimum points and the corresponding
range of y.
a)y = 4sin3x + 2
b)y =3|cos2x|
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Unfortunately, I can't directly sketch graphs here. However, I can provide you with all the information you need to sketch them yourself, including the key features and transformations.
**a) y = 4sin3x + 2**
* **Amplitude:** 4 (the coefficient of the sine function)
* **Period:** (360°) / 3 = 120° (the period of the basic sine function divided by the coefficient of x)
* **Vertical Shift:** 2 units upward (the constant term)
* **Maximum Points:** The maximum value of the sine function is 1. So, the maximum value of y is 4(1) + 2 = 6. This occurs when sin3x = 1, which happens at x = 30°, 150°, 270° within the given range. The maximum points are (30°, 6), (150°, 6), and (270°, 6).
* **Minimum Points:** The minimum value of the sine function is -1. So, the minimum value of y is 4(-1) + 2 = -2. This occurs when sin3x = -1, which happens at x = 90°, 210°, 330° within the given range. The minimum points are (90°, -2), (210°, -2), and (330°, -2).
* **Range:** -2 ≤ y ≤ 6
**To sketch the graph:**
1. Start with the basic sine graph.
2. Compress it horizontally by a factor of 3 (so that it completes one cycle within 120°).
3. Stretch it vertically by a factor of 4.
4. Shift the entire graph 2 units upward.
**b) y = 3|cos2x|**
* **Amplitude:** 3 (the coefficient in front of the absolute value)
* **Period:** (360°) / 2 = 180° (the period of the basic cosine function divided by the coefficient of x)
* **Absolute Value:** The absolute value ensures that the function is always non-negative.
* **Maximum Points:** The maximum value of |cos2x| is 1. So, the maximum value of y is 3(1) = 3. This occurs when cos2x = 1 or -1, which happens at x = 0°, 90°, 180°, 270°, 360° within the given range. The maximum points are (0°, 3), (90°, 3), (180°, 3), (270°, 3), and (360°, 3).
* **Minimum Points:** The minimum value of |cos2x| is 0. So, the minimum value of y is 3(0) = 0. This occurs when cos2x = 0, which happens at x = 45°, 135°, 225°, 315° within the given range. The minimum points are (45°, 0), (135°, 0), (225°, 0), and (315°, 0).
* **Range:** 0 ≤ y ≤ 3
**To sketch the graph:**
1. Start with the basic cosine graph.
2. Compress it horizontally by a factor of 2 (so that it completes one cycle within 180°).
3. Reflect any parts of the graph that are below the x-axis to above the x-axis (because of the absolute value).
4. Stretch the graph vertically by a factor of 3.
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