The equation you provided is $y = 2\sin(2x) + 1$. This represents a sinusoidal function, specifically a sine wave with the following modifications compared to the basic sine function $y = \sin(x)$:

1. Amplitude: The coefficient of 2 in front of the sine function means the amplitude (the vertical stretch) is 2. So, the wave will oscillate between $y = 1 - 2 = -1$ and $y = 1 + 2 = 3$.
2. Period: The factor of 2 inside the sine function affects the period. The period of a sine wave is given by $\frac{2\pi}{\text{frequency}}$. Here, the frequency is 2, so the period is $\frac{2\pi}{2} = \pi$.
3. Vertical Shift: The "+1" at the end of the function shifts the entire graph of the sine wave upwards by 1 unit.

Key Properties:

• Amplitude: 2
• Period: $\pi$
• Vertical Shift: +1 (shifted 1 unit up)

If you want to explore this graph or solve for specific values of $y$ for given values of $x$, I can help with that.

Would you like to see the graph of this function or explore something else about it?

Related Questions:

1. How does changing the coefficient of $\sin(x)$ affect the amplitude?
2. What happens to the graph of the function if the vertical shift changes to +3?
3. How do you find the x-intercepts of this sine function?
4. How would the graph change if the coefficient inside the sine function was greater than 2?
5. How can we solve for $x$ when $y = 0$ in this equation?

Tip:

When working with sinusoidal functions, always remember to analyze the amplitude, period, and phase shift to understand how the graph will behave!