To graph the function $y = 2\sin(-3x - \frac{\pi}{2}) + 2$ step by step, we'll follow a structured process and analyze each transformation. Here's how you can proceed:

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Step 1: Start with the basic sine function

The basic sine function is $y = \sin(x)$. Its graph oscillates between $-1$ and $1$, has a period of $2\pi$, and passes through the points:

1. $(0, 0)$
2. $(\frac{\pi}{2}, 1)$
3. $(\pi, 0)$
4. $(\frac{3\pi}{2}, -1)$
5. $(2\pi, 0)$

We will transform this graph step by step.

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Step 2: Reflect the sine graph horizontally with $-3x$

The negative sign in $\sin(-3x)$ reflects the graph horizontally (around the y-axis). Points such as $(\frac{\pi}{2}, 1)$ become $(-\frac{\pi}{2}, 1)$, and $(\frac{3\pi}{2}, -1)$ becomes $(-\frac{3\pi}{2}, -1)$.

The coefficient $3$ compresses the graph horizontally by a factor of $\frac{1}{3}$, reducing the period. The period of a sine function is given by $\frac{2\pi}{|b|}$, where $b$ is the coefficient of $x$. For $\sin(-3x)$, $b = -3$, so the period becomes:

$\text{New Period} = \frac{2\pi}{3}.$

The key points now adjust as follows:

1. Start at $x = 0$ (still $y = 0$).
2. The first peak is at $x = \frac{\pi}{6}$.
3. Cross zero at $x = \frac{\pi}{3}$.
4. Reach the minimum at $x = \frac{\pi}{2}$.
5. Complete one cycle at $x = \frac{2\pi}{3}$.

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Step 3: Phase shift $-\frac{\pi}{2}$

The $-\frac{\pi}{2}$ term shifts the graph horizontally. Because it is subtracted, the graph shifts right by $\frac{\pi}{2}$.

New key points:

1. Start at $x = \frac{\pi}{2}$ (y-value remains $0$).
2. Peak at $x = \frac{\pi}{2} + \frac{\pi}{6} = \frac{2\pi}{3}$.
3. Zero at $x = \frac{\pi}{2} + \frac{\pi}{3} = \frac{5\pi}{6}$.
4. Minimum at $x = \frac{\pi}{2} + \frac{\pi}{2} = \pi$.
5. Complete the cycle at $x = \frac{\pi}{2} + \frac{2\pi}{3} = \frac{4\pi}{3}$.

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Step 4: Vertical stretch and shift

The coefficient $2$ outside the sine function stretches the graph vertically by a factor of $2$. This means the amplitude increases to $2$. The graph now oscillates between $-2$ and $2$.

The final vertical shift of $+2$ moves the entire graph up by 2 units, so it oscillates between $0$ and $4$.

Final key points:

1. Start at $(\frac{\pi}{2}, 2)$ (middle line).
2. Peak at $(\frac{2\pi}{3}, 4)$.
3. Zero at $(\frac{5\pi}{6}, 2)$.
4. Minimum at $(\pi, 0)$.
5. Complete the cycle at $(\frac{4\pi}{3}, 2)$.

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Formula Recap:

$y = 2\sin(-3x - \frac{\pi}{2}) + 2$

This formula describes:

• A reflection and compression from $\sin(x)$.
• A horizontal shift by $\frac{\pi}{2}$ to the right.
• A vertical stretch by 2.
• A vertical shift by 2.

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Final Tip for Plotting:

1. Use the calculated key points to draw the curve smoothly.
2. Label the axes and note the amplitude ($0$ to $4$) and period ($\frac{2\pi}{3}$).

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Would you like a detailed sketch or further explanation?

Related Questions:

1. How does a negative phase shift differ from a positive one in sine functions?
2. How do you find the amplitude and period for any sine or cosine function?
3. How do vertical shifts impact the graph of sinusoidal functions?
4. What happens if the coefficient of $x$ inside $\sin(kx)$ is fractional?
5. How can this function be applied in real-world periodic phenomena?

Tip: Always start transformations from the basic graph $\sin(x)$, applying one modification at a time.