Transformations of Trig Graphs (B) Worksheet | Printable PDF ... - Free Printable

Let's solve each question step-by-step, using transformations of trigonometric graphs. We'll sketch the graph (conceptually), explain the transformation, and determine the period for each function.

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🔷 General Rules for Transformations:

For a function:
- $ y = a \sin(bx + c) + d $
- $ y = a \cos(bx + c) + d $
- $ y = a \tan(bx + c) + d $

The transformations are:
- Amplitude: $ |a| $ → vertical stretch
- Period:
- For sine/cosine: $ \frac{360^\circ}{b} $
- For tangent: $ \frac{180^\circ}{b} $
- Phase shift (horizontal shift): $ -\frac{c}{b} $
- Vertical shift: $ d $

We’ll apply these rules to each problem.

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## ✔ Question 1: $ y = \sin(2x) $

- This is a horizontal compression by factor 2.
- Original period of $ \sin x $: $ 360^\circ $
- New period: $ \frac{360^\circ}{2} = 180^\circ $
- Graph completes one cycle in $ 0^\circ $ to $ 180^\circ $

✔ Period: $ \boxed{180^\circ} $

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## ✔ Question 2: $ y = \tan(2x) $

- Tangent has original period $ 180^\circ $
- With $ b = 2 $: new period = $ \frac{180^\circ}{2} = 90^\circ $
- Vertical asymptotes every $ 90^\circ $: at $ x = 45^\circ, 135^\circ, 225^\circ, 315^\circ $, etc.

✔ Period: $ \boxed{90^\circ} $

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## ✔ Question 3: $ y = \cos(2x) $

- Same as sine: horizontal compression by factor 2
- Original period: $ 360^\circ $
- New period: $ \frac{360^\circ}{2} = 180^\circ $

✔ Period: $ \boxed{180^\circ} $

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## ✔ Question 4: $ y = 2 \sin(2x) $

Wait — this appears to be written as $ y = 2 \sin 2 $, but that doesn't make sense. It should be $ y = 2 \sin(2x) $. Assuming typo.

So: $ y = 2 \sin(2x) $

- Amplitude: 2 (vertical stretch)
- Period: $ \frac{360^\circ}{2} = 180^\circ $
- So it’s $ \sin(2x) $ stretched vertically by 2

✔ Period: $ \boxed{180^\circ} $

> Note: If it were truly $ y = 2 \sin(2) $, it would just be a constant since $ \sin(2) $ is a number (~0.909), so $ y \approx 1.818 $. But that wouldn’t make sense with the graph grid. So we assume it's $ y = 2 \sin(2x) $

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## ✔ Question 5: $ y = 2 \cos x $

- Amplitude: 2 → vertical stretch
- No horizontal change → period remains same as $ \cos x $
- Period: $ 360^\circ $

✔ Period: $ \boxed{360^\circ} $

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## ✔ Question 6: $ y = \frac{1}{2} \sin x $

- Amplitude: $ \frac{1}{2} $ → vertical compression
- No change in period
- Period: $ 360^\circ $

✔ Period: $ \boxed{360^\circ} $

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## ✔ Question 7: $ y = \tan(x - 30^\circ) $

- Horizontal shift: $ +30^\circ $ → right by $ 30^\circ $
- Period of $ \tan x $ is $ 180^\circ $
- No change in $ b $, so period stays $ 180^\circ $
- Asymptotes shifted right by $ 30^\circ $

✔ Period: $ \boxed{180^\circ} $

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## ✔ Question 8: $ y = \cos(x + 60^\circ) $

- Phase shift: $ -60^\circ $ → left by $ 60^\circ $
- No change in frequency → period unchanged
- Period: $ 360^\circ $

✔ Period: $ \boxed{360^\circ} $

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## ✔ Question 9: $ y = 2 \sin\left(\frac{1}{2}x\right) $

- $ b = \frac{1}{2} $
- Period: $ \frac{360^\circ}{1/2} = 720^\circ $
- So the wave is stretched horizontally — takes twice as long to complete a cycle
- Amplitude: 2 (vertical stretch)

✔ Period: $ \boxed{720^\circ} $

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📌 Summary Table:

| Question | Function | Transformation | Period |
|--------|----------|----------------|--------|
| 1 | $ y = \sin(2x) $ | Horizontal compression ×2 | $ 180^\circ $ |
| 2 | $ y = \tan(2x) $ | Horizontal compression ×2 | $ 90^\circ $ |
| 3 | $ y = \cos(2x) $ | Horizontal compression ×2 | $ 180^\circ $ |
| 4 | $ y = 2\sin(2x) $ | Vertical stretch ×2, horizontal comp ×2 | $ 180^\circ $ |
| 5 | $ y = 2\cos x $ | Vertical stretch ×2 | $ 360^\circ $ |
| 6 | $ y = \frac{1}{2}\sin x $ | Vertical compression ×½ | $ 360^\circ $ |
| 7 | $ y = \tan(x - 30^\circ) $ | Shift right 30° | $ 180^\circ $ |
| 8 | $ y = \cos(x + 60^\circ) $ | Shift left 60° | $ 360^\circ $ |
| 9 | $ y = 2\sin\left(\frac{1}{2}x\right) $ | Vertical stretch ×2, horizontal stretch ×2 | $ 720^\circ $ |

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✔ Final Answers (to fill in boxes):

1) Period: 180°
2) Period: 90°
3) Period: 180°
4) Period: 180°
5) Period: 360°
6) Period: 360°
7) Period: 180°
8) Period: 360°
9) Period: 720°

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🎯 How to Sketch Each Graph (Brief Guide):

1. Start with basic shape ($ \sin x $, $ \cos x $, $ \tan x $).
2. Apply horizontal stretch/compression based on $ b $: use period formula.
3. Apply vertical stretch/compression via amplitude.
4. Apply phase shift (left/right).
5. Draw key points: max, min, zeros, asymptotes (for tan).

Let me know if you'd like detailed sketches or explanations for any specific one!