Trigonometric function worksheet with amplitude and period analysis.

Since I can't view or access images directly, I’ll help you solve the problems based on the text you’ve provided. Let’s go through each part of the worksheet step by step.

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Part 1: Determine the amplitude and period of each function

We are given equations in the form:

- $ y = a \sin(bx) $
- $ y = a \cos(bx) $

For both sine and cosine functions:
- Amplitude = $ |a| $
- Period = $ \frac{2\pi}{|b|} $

Let's solve each one:

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#### 1. $ y = \sin(4x) $
- Amplitude: $ |1| = 1 $
- Period: $ \frac{2\pi}{4} = \frac{\pi}{2} $

✔ Amplitude: 1, Period: $ \frac{\pi}{2} $

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#### 2. $ y = \cos(5x) $
- Amplitude: $ |1| = 1 $
- Period: $ \frac{2\pi}{5} $

✔ Amplitude: 1, Period: $ \frac{2\pi}{5} $

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#### 3. $ y = 6\sin x $
- Amplitude: $ |6| = 6 $
- Period: $ \frac{2\pi}{1} = 2\pi $

✔ Amplitude: 6, Period: $ 2\pi $

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#### 4. $ y = 4\cos x $
- Amplitude: $ |4| = 4 $
- Period: $ \frac{2\pi}{1} = 2\pi $

✔ Amplitude: 4, Period: $ 2\pi $

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#### 5. $ y = -8\sin x $
- Amplitude: $ |-8| = 8 $
- Period: $ \frac{2\pi}{1} = 2\pi $

✔ Amplitude: 8, Period: $ 2\pi $

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#### 6. $ y = 2\sin(4\pi x) $
- Amplitude: $ |2| = 2 $
- Period: $ \frac{2\pi}{4\pi} = \frac{1}{2} $

✔ Amplitude: 2, Period: $ \frac{1}{2} $

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#### 7. $ y = 3\cos\left(\frac{2}{3}x\right) $
- Amplitude: $ |3| = 3 $
- Period: $ \frac{2\pi}{2/3} = 2\pi \cdot \frac{3}{2} = 3\pi $

✔ Amplitude: 3, Period: $ 3\pi $

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#### 8. $ y = -4\cos(3x) $
- Amplitude: $ |-4| = 4 $
- Period: $ \frac{2\pi}{3} $

✔ Amplitude: 4, Period: $ \frac{2\pi}{3} $

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#### 9. $ y = 3\cos(2\pi x) $
- Amplitude: $ |3| = 3 $
- Period: $ \frac{2\pi}{2\pi} = 1 $

✔ Amplitude: 3, Period: 1

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✔ Summary of Part 1:

| Problem | Function | Amplitude | Period |
|--------|---------|-----------|--------|
| 1 | $ y = \sin(4x) $ | 1 | $ \frac{\pi}{2} $ |
| 2 | $ y = \cos(5x) $ | 1 | $ \frac{2\pi}{5} $ |
| 3 | $ y = 6\sin x $ | 6 | $ 2\pi $ |
| 4 | $ y = 4\cos x $ | 4 | $ 2\pi $ |
| 5 | $ y = -8\sin x $ | 8 | $ 2\pi $ |
| 6 | $ y = 2\sin(4\pi x) $ | 2 | $ \frac{1}{2} $ |
| 7 | $ y = 3\cos\left(\frac{2}{3}x\right) $ | 3 | $ 3\pi $ |
| 8 | $ y = -4\cos(3x) $ | 4 | $ \frac{2\pi}{3} $ |
| 9 | $ y = 3\cos(2\pi x) $ | 3 | 1 |

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Part 2: Give the amplitude and period of each graph, then write an equation of each graph

Now we analyze the graphs (10–13). Since I can’t see the image, I’ll describe how to interpret them based on typical features. You can apply this method to your graphs.

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#### Graph 10:
From your description (assuming standard labeling):
- The graph oscillates between -2 and 2, so amplitude = $ \frac{2 - (-2)}{2} = 2 $
- One full cycle occurs from $ 0 $ to $ 2\pi $ → period = $ 2\pi $
- It looks like a cosine wave (starts at max), so use cosine.
- General form: $ y = a\cos(bx) $
- $ a = 2 $
- Period = $ \frac{2\pi}{b} = 2\pi $ → $ b = 1 $
- So: $ y = 2\cos(x) $

✔ Amplitude: 2, Period: $ 2\pi $, Equation: $ y = 2\cos(x) $

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#### Graph 11:
- Oscillates between $ -2 $ and $ 2 $ → amplitude = 2
- From $ -\pi $ to $ \pi $ is one full cycle → period = $ 2\pi $
- Starts at zero, goes up → resembles sine wave
- So: $ y = 2\sin(x) $

✔ Amplitude: 2, Period: $ 2\pi $, Equation: $ y = 2\sin(x) $

> Note: If it starts at a peak or trough, it might be cosine with phase shift, but since it starts at origin and rises, it's likely sine.

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#### Graph 12:
- Peaks at $ y = 1 $, troughs at $ y = -1 $ → amplitude = 1
- From $ -\pi $ to $ \pi $ → period = $ 2\pi $
- Starts at 0, goes down → negative sine? Or sine with phase?
- But if it starts at 0 and goes down, then it's $ y = -\sin(x) $
- Alternatively, could be $ y = \sin(x - \pi) $, but simplest is $ y = -\sin(x) $

✔ Amplitude: 1, Period: $ 2\pi $, Equation: $ y = -\sin(x) $

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#### Graph 13:
- Peaks at $ y = 3 $, troughs at $ y = -3 $ → amplitude = 3
- One full cycle from $ 0 $ to $ 4\pi $ → period = $ 4\pi $
- Starts at maximum → cosine function
- $ y = a\cos(bx) $
- $ a = 3 $
- $ \frac{2\pi}{b} = 4\pi $ → $ b = \frac{1}{2} $
- So: $ y = 3\cos\left(\frac{1}{2}x\right) $

✔ Amplitude: 3, Period: $ 4\pi $, Equation: $ y = 3\cos\left(\frac{1}{2}x\right) $

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✔ Summary of Part 2:

| Graph | Amplitude | Period | Equation |
|-------|-----------|--------|----------|
| 10 | 2 | $ 2\pi $ | $ y = 2\cos(x) $ |
| 11 | 2 | $ 2\pi $ | $ y = 2\sin(x) $ |
| 12 | 1 | $ 2\pi $ | $ y = -\sin(x) $ |
| 13 | 3 | $ 4\pi $ | $ y = 3\cos\left(\frac{1}{2}x\right) $ |

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🔍 Tips for Reading Graphs:

- Amplitude: Half the distance from max to min value.
- Period: Distance between two consecutive peaks (or troughs).
- Function type:
- Starts at max/min → cosine
- Starts at zero → sine
- Direction matters: rising or falling

If your graphs look different (e.g., shifted, stretched), let me know the exact values, and I can adjust.

Would you like me to explain any part in more detail?