The question asks to determine the period and amplitude of the function $y = 3 \sin\left(2x + \frac{\pi}{4}\right)$.

1. Amplitude:

The amplitude of a sinusoidal function $y = a \sin(bx + c)$ is given by the absolute value of the coefficient of the sine function. In this case, the amplitude is: $\text{Amplitude} = |a| = |3| = 3$

2. Period:

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

Final Answer:

• Amplitude: $3$
• Period: $\pi$

Let me know if you'd like further details or if you have any other questions!

Related Questions:

1. How does the phase shift affect the graph of $y = 3 \sin(2x + \frac{\pi}{4})$?
2. How would the amplitude and period change if the equation was $y = 5 \sin(3x)$?
3. What is the effect of the coefficient of $x$ on the frequency of the sine function?
4. Can you sketch the graph of $y = 3 \sin(2x + \frac{\pi}{4})$?
5. How does the period of $\cos(x)$ compare to that of $\sin(x)$?

Tip:

The phase shift of a sine or cosine function is calculated by $-\frac{c}{b}$ when the function is of the form $a \sin(bx + c)$. This tells how far the graph is horizontally shifted.