For the function $y = -3 \sin(2x + \frac{\pi}{4})$, let's break it down step by step to find the amplitude, period, phase shift, and the interval of one cycle.

1. Amplitude:

The amplitude of a sine function $y = a \sin(bx + c)$ is the absolute value of the coefficient of the sine function, which is $|a|$.

Here, $a = -3$, so: $\text{Amplitude} = | -3 | = 3$

2. Period:

The period of a sine function is determined by the coefficient $b$ in $y = a \sin(bx + c)$. The period formula is: $\text{Period} = \frac{2\pi}{|b|}$ In this case, $b = 2$, so: $\text{Period} = \frac{2\pi}{2} = \pi$

3. Phase Shift:

The phase shift of the sine function is given by $-\frac{c}{b}$, where $c$ is the constant added to the argument $bx + c$.

Here, $c = \frac{\pi}{4}$ and $b = 2$, so the phase shift is: $\text{Phase shift} = -\frac{\frac{\pi}{4}}{2} = -\frac{\pi}{8}$ This means the graph is shifted $\frac{\pi}{8}$ units to the left.

4. Interval of One Cycle:

The interval for one cycle of the sine function is from the starting point (after applying the phase shift) to one full period later. The period is $\pi$, and the phase shift moves the graph to the left by $\frac{\pi}{8}$.

• Start of the interval: $-\frac{\pi}{8}$
• End of the interval: $-\frac{\pi}{8} + \pi = \frac{7\pi}{8}$

Thus, the interval is $\left[ -\frac{\pi}{8}, \frac{7\pi}{8} \right]$.

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Summary:

• Amplitude: 3
• Period: $\pi$
• Phase Shift: $-\frac{\pi}{8}$ (shifted left)
• Interval: $\left[ -\frac{\pi}{8}, \frac{7\pi}{8} \right]$

Do you have any questions or would you like more details on any part of the solution?

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Here are 5 related questions:

1. How do you find the amplitude of cosine functions?
2. What happens to the graph of a sine function if the amplitude is negative?
3. How does a vertical shift affect the sine or cosine graph?
4. What is the effect of increasing the coefficient $b$ in the sine function on its period?
5. Can you calculate the phase shift of $y = \sin(3x - \frac{\pi}{6})$?

Tip: The negative sign in front of the amplitude of sine or cosine reflects the graph vertically across the x-axis!