Trigonometry Graphs for Sine, Cosine and Tangent Functions - Free Printable

It looks like you've uploaded a sample page from a Grade 11 Mathematics worksheet on Trigonometry Graphs, specifically focusing on the Sine Function in the form:

$$
y = a \sin(k\theta + p) + q
$$

However, there is no specific problem or question stated on this page — it's an explanatory guide with a table describing how each parameter affects the sine graph, and some example graphs labeled with their corresponding functions.

Let me help you understand the content and possibly solve a typical problem that might be associated with this material.

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🔍 Understanding the General Form:

$$
y = a \sin(k\theta + p) + q
$$

Each parameter has a specific effect:

| Parameter | Meaning | Effect on Graph |
|---------|--------|------------------|
| $ a $ | Amplitude | Vertical stretch/compression. If $ a < 0 $, reflects over x-axis. |
| $ k $ | Frequency / Period factor | Affects period: $ \text{Period} = \frac{360^\circ}{k} $ (or $ \frac{2\pi}{k} $ in radians). |
| $ p $ | Phase shift (horizontal shift) | Shifts graph left/right: $ -\frac{p}{k} $. |
| $ q $ | Vertical shift | Moves graph up/down by $ q $ units. |

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📈 Example Graphs Explained:

From the image, we can see five graphs:

1. $ f(x) = \sin(x) $ (red)
- Standard sine wave.
- Amplitude = 1, period = $ 360^\circ $, no shifts.

2. $ f(x) = 2\sin(x) $ (blue)
- Amplitude = 2 → stretched vertically.
- Same period and no phase/vertical shift.

3. $ f(x) = \sin(x - 30^\circ) $ (black)
- Horizontal shift right by $ 30^\circ $.
- Phase shift: $ p = -30^\circ $, so shift is $ +30^\circ $.

4. $ f(x) = -3\sin(2x) $ (orange)
- Amplitude = 3, reflected over x-axis (due to negative sign).
- $ k = 2 $ → period = $ \frac{360^\circ}{2} = 180^\circ $ → compressed horizontally.
- No vertical or horizontal shift.

5. $ f(x) = \sin(x) + 3 $ (purple)
- Vertical shift up by 3 units.
- Oscillates between $ y = 2 $ and $ y = 4 $.

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✔ Typical Problem That Might Be Asked:

> Sketch the graph of $ f(x) = 2\sin(3x - 90^\circ) + 1 $, and state its amplitude, period, phase shift, and vertical shift.

Let’s solve this step-by-step using the general form.

#### Step 1: Identify parameters
Compare $ f(x) = 2\sin(3x - 90^\circ) + 1 $ with $ y = a\sin(kx + p) + q $

- $ a = 2 $
- $ k = 3 $
- $ p = -90^\circ $
- $ q = 1 $

#### Step 2: Determine properties

- Amplitude: $ |a| = 2 $
- Period: $ \frac{360^\circ}{k} = \frac{360^\circ}{3} = 120^\circ $
- Phase shift: $ -\frac{p}{k} = -\frac{-90^\circ}{3} = 30^\circ $ → shift right by 30°
- Vertical shift: $ q = 1 $ → shift up by 1 unit

#### Step 3: Sketching Guidelines

1. Start with standard sine wave.
2. Stretch vertically by factor 2 → max = 3, min = -1 (since shifted up by 1).
3. Compress horizontally → period = 120°.
4. Shift right by 30°.
5. Shift up by 1.

So, the graph starts at $ x = 30^\circ $ instead of $ x = 0^\circ $, reaches maximum at $ x = 30^\circ + 30^\circ = 60^\circ $, etc.

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🧠 Summary of How to Solve Similar Problems:

1. Identify $ a, k, p, q $ from the equation.
2. Calculate amplitude $ = |a| $
3. Find period $ = \frac{360^\circ}{k} $
4. Find phase shift $ = -\frac{p}{k} $
5. Vertical shift $ = q $
6. Use these to sketch the graph.

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❓What if you had a question?

If you have a specific problem (e.g., "Write the equation of the graph shown", or "Determine the value of $ k $ given the period"), please provide it, and I’ll solve it for you.

For now, this document is a study guide — not a problem set. But I’ve explained how to use it effectively!

Let me know what you'd like to do next!