Task Four: Write down the equation of each of the trig graphs.

Let's solve Task Four: Write down the equation of each of the trig graphs.

We are given four trigonometric graphs, and we need to determine their equations. The general form of a sine or cosine function is:

$$
y = A \sin(Bx + C) + D \quad \text{or} \quad y = A \cos(Bx + C) + D
$$

But since all graphs appear to be sine or cosine waves with no horizontal shift (phase shift), and they start at specific points, we’ll analyze them one by one.

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Key Concepts:

- Amplitude (A): Half the distance between the maximum and minimum values.
$$
A = \frac{\text{max} - \text{min}}{2}
$$

- Midline (D): The average of max and min.
$$
D = \frac{\text{max} + \text{min}}{2}
$$

- Period: The length of one full cycle. For standard sine/cosine, period = $360^\circ$, but here all graphs complete one cycle in $360^\circ$, so $B = 1$ (since $ \text{period} = \frac{360^\circ}{B} $).

- Phase Shift (C): If the graph starts at origin like sine, it’s likely a sine function; if it starts at max/min, it might be cosine.

Let’s go through each graph.

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Graph 1

- Starts at origin $(0, 0)$, goes up → Sine wave
- Max value ≈ 7, Min value ≈ -7
- So:
$$
A = \frac{7 - (-7)}{2} = \frac{14}{2} = 7
$$
$$
D = \frac{7 + (-7)}{2} = 0
$$
- No vertical shift.
- Period = $360^\circ$, so $B = 1$
- Starts at zero and increases → Sine function

✔ Equation:
$$
\boxed{y = 7 \sin x}
$$

---

Graph 2

- Starts at $x=0$, $y=4$, then decreases → Cosine wave
- Max = 4, Min = -4
- So:
$$
A = \frac{4 - (-4)}{2} = \frac{8}{2} = 4
$$
$$
D = \frac{4 + (-4)}{2} = 0
$$
- No vertical shift.
- Period = $360^\circ$, so $B = 1$
- Starts at maximum → Cosine function

✔ Equation:
$$
\boxed{y = 4 \cos x}
$$

---

Graph 3

- Starts at $x=0$, $y=10$, then decreases → Cosine wave
- Max = 10, Min = -10
- So:
$$
A = \frac{10 - (-10)}{2} = \frac{20}{2} = 10
$$
$$
D = \frac{10 + (-10)}{2} = 0
$$
- No vertical shift.
- Period = $360^\circ$, so $B = 1$
- Starts at maximum → Cosine function

✔ Equation:
$$
\boxed{y = 10 \cos x}
$$

---

Graph 4

- Starts at origin $(0, 0)$, goes up → Sine wave
- Max ≈ 0.2, Min ≈ -0.2
- So:
$$
A = \frac{0.2 - (-0.2)}{2} = \frac{0.4}{2} = 0.2
$$
$$
D = \frac{0.2 + (-0.2)}{2} = 0
$$
- No vertical shift.
- Period = $360^\circ$, so $B = 1$
- Starts at zero and increases → Sine function

✔ Equation:
$$
\boxed{y = 0.2 \sin x}
$$

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✔ Final Answers:

1. $ y = 7 \sin x $
2. $ y = 4 \cos x $
3. $ y = 10 \cos x $
4. $ y = 0.2 \sin x $

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📌 Summary:

| Graph | Function Type | Amplitude | Midline | Equation |
|-------|----------------|-----------|---------|----------|
| 1 | Sine | 7 | 0 | $ y = 7 \sin x $ |
| 2 | Cosine | 4 | 0 | $ y = 4 \cos x $ |
| 3 | Cosine | 10 | 0 | $ y = 10 \cos x $ |
| 4 | Sine | 0.2 | 0 | $ y = 0.2 \sin x $ |

Let me know if you'd like these rewritten in degrees explicitly (e.g., $ \sin(x^\circ) $).