PreCalculus worksheet focusing on sinusoidal functions, including problems to find amplitude and period, analyze given graphs, and sketch functions.

Here are the step-by-step solutions for each problem on the worksheet.

Part 1: Determine Amplitude and Period

For functions in the form $y = a \sin(bx)$ or $y = a \cos(bx)$:
* Amplitude is the absolute value of $a$ ($|a|$). It represents how far the graph goes up and down from the center.
* Period is calculated as $\frac{2\pi}{|b|}$. It represents how long it takes for the wave to repeat itself.

1. $y = \sin 4x$
* Here, $a = 1$ and $b = 4$.
* Amplitude: $|1| = 1$
* Period: $\frac{2\pi}{4} = \frac{\pi}{2}$

2. $y = \cos 5x$
* Here, $a = 1$ and $b = 5$.
* Amplitude: $|1| = 1$
* Period: $\frac{2\pi}{5}$

3. $y = 2 \sin x$
* Here, $a = 2$ and $b = 1$.
* Amplitude: $|2| = 2$
* Period: $\frac{2\pi}{1} = 2\pi$

4. $y = -4 \sin 3x$
* Here, $a = -4$ and $b = 3$.
* Amplitude: $|-4| = 4$ (Amplitude is always positive)
* Period: $\frac{2\pi}{3}$

5. $y = 2 \sin (-4x)$
* Here, $a = 2$ and $b = -4$.
* Amplitude: $|2| = 2$
* Period: $\frac{2\pi}{|-4|} = \frac{2\pi}{4} = \frac{\pi}{2}$

6. $y = 3 \sin \frac{2}{3} x$
* Here, $a = 3$ and $b = \frac{2}{3}$.
* Amplitude: $|3| = 3$
* Period: $\frac{2\pi}{2/3} = 2\pi \cdot \frac{3}{2} = 3\pi$

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Part 2: Graph Analysis

To find the equation from a graph:
1. Find the Amplitude ($A$): Look at the maximum height ($Max$) and minimum depth ($Min$). $A = \frac{Max - Min}{2}$.
2. Find the Period ($P$): Measure the distance between two peaks or two corresponding points. Calculate $B = \frac{2\pi}{P}$.
3. Choose Sine or Cosine:
* If the graph starts at the middle (0) and goes up, use positive Sine ($y = A \sin Bx$).
* If the graph starts at a peak (max), use positive Cosine ($y = A \cos Bx$).
* If it starts at a valley (min), use negative Cosine ($y = -A \cos Bx$).

7. Graph 7
* Amplitude: The graph goes from $-3$ to $3$. Amplitude = $3$.
* Period: One full cycle happens between $x=0$ and $x=\pi$. Period = $\pi$.
* Calculate $B$: $\frac{2\pi}{\pi} = 2$.
* Equation Type: At $x=0$, $y=0$ and it goes up. This is a positive sine wave.
* Answer: Amplitude: 3, Period: $\pi$, Equation: $y = 3 \sin 2x$

8. Graph 8
* Amplitude: The graph goes from $-4$ to $4$. Amplitude = $4$.
* Period: One full cycle happens between $x=0$ and $x=2\pi$. Period = $2\pi$.
* Calculate $B$: $\frac{2\pi}{2\pi} = 1$.
* Equation Type: At $x=0$, the graph is at its peak ($4$). This is a positive cosine wave.
* Answer: Amplitude: 4, Period: $2\pi$, Equation: $y = 4 \cos x$

9. Graph 9
* Amplitude: The graph goes from $-2$ to $2$. Amplitude = $2$.
* Period: One full cycle ends at $x=4\pi$. Period = $4\pi$.
* Calculate $B$: $\frac{2\pi}{4\pi} = \frac{1}{2}$.
* Equation Type: At $x=0$, $y=0$ and it goes up. This is a positive sine wave.
* Answer: Amplitude: 2, Period: $4\pi$, Equation: $y = 2 \sin \frac{1}{2}x$

10. Graph 10
* Amplitude: The graph goes from $-5$ to $5$. Amplitude = $5$.
* Period: The graph hits a peak at $-\pi$ and the next peak at $\pi$. The distance is $2\pi$. Period = $2\pi$.
* Calculate $B$: $\frac{2\pi}{2\pi} = 1$.
* Equation Type: At $x=0$, the graph is at its lowest point ($-5$). This is a reflected (negative) cosine wave.
* Answer: Amplitude: 5, Period: $2\pi$, Equation: $y = -5 \cos x$

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Part 3: Sketching Graphs

To sketch these, mark key points on the x-axis: $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$ (and negative counterparts).

11. $y = 4 \sin x$
* Amplitude: 4 (Go up to 4, down to -4).
* Period: $2\pi$ (Standard speed).
* Key Points to Plot:
* $(0, 0)$
* $(\frac{\pi}{2}, 4)$ -> Peak
* $(\pi, 0)$ -> Middle
* $(\frac{3\pi}{2}, -4)$ -> Valley
* $(2\pi, 0)$ -> End of cycle
* Repeat this pattern backwards for negative x values.

12. $y = 2 \cos x$
* Amplitude: 2 (Go up to 2, down to -2).
* Period: $2\pi$ (Standard speed).
* Key Points to Plot:
* $(0, 2)$ -> Start at Peak
* $(\frac{\pi}{2}, 0)$ -> Middle
* $(\pi, -2)$ -> Valley
* $(\frac{3\pi}{2}, 0)$ -> Middle
* $(2\pi, 2)$ -> End at Peak
* Repeat this pattern backwards for negative x values.

Final Answer:
1. Amp: 1, Per: $\pi/2$
2. Amp: 1, Per: $2\pi/5$
3. Amp: 2, Per: $2\pi$
4. Amp: 4, Per: $2\pi/3$
5. Amp: 2, Per: $\pi/2$
6. Amp: 3, Per: $3\pi$
7. Amp: 3, Per: $\pi$, Eq: $y = 3 \sin 2x$
8. Amp: 4, Per: $2\pi$, Eq: $y = 4 \cos x$
9. Amp: 2, Per: $4\pi$, Eq: $y = 2 \sin \frac{1}{2}x$
10. Amp: 5, Per: $2\pi$, Eq: $y = -5 \cos x$
11 & 12. See plotting instructions above.