The provided task involves graphing the trigonometric function

$y = 2\sin\left(\frac{3}{4}x\right) - 1$

Steps to solve:

1. Understand the Function:

   • Amplitude = $2$ (the coefficient of $\sin$).
   • Vertical shift = $-1$ (subtracting 1 moves the graph 1 unit down).
   • Period = $\frac{2\pi}{b}$, where $b = \frac{3}{4}$. Therefore, the period is: $T = \frac{2\pi}{\frac{3}{4}} = \frac{2\pi \cdot 4}{3} = \frac{8\pi}{3}.$

2. Key Points:

   • A full cycle of the sine wave happens between $x = 0$ and $x = \frac{8\pi}{3}$.
   • The midline of the graph is $y = -1$.
   • The maximum value is $-1 + 2 = 1$, and the minimum value is $-1 - 2 = -3$.

3. Divide the Period into Intervals:

   • The key points occur at:
     • $x = 0$ (start at the midline).
     • $x = \frac{T}{4} = \frac{8\pi}{12} = \frac{2\pi}{3}$ (max).
     • $x = \frac{T}{2} = \frac{4\pi}{3}$ (back to the midline).
     • $x = \frac{3T}{4} = \frac{6\pi}{3} = 2\pi$ (min).
     • $x = T = \frac{8\pi}{3}$ (return to the midline).

4. Plot Points:

   • At $x = 0$: $y = -1$ (midline).
   • At $x = \frac{2\pi}{3}$: $y = 1$ (max).
   • At $x = \frac{4\pi}{3}$: $y = -1$ (midline).
   • At $x = 2\pi$: $y = -3$ (min).
   • At $x = \frac{8\pi}{3}$: $y = -1$ (midline).

Graphing:

1. Plot the points $(0, -1)$, $(\frac{2\pi}{3}, 1)$, $(\frac{4\pi}{3}, -1)$, $(2\pi, -3)$, and $(\frac{8\pi}{3}, -1)$.
2. Draw a smooth sine curve through these points.

Would you like me to create the graph or provide more detailed calculations? Let me know! 😊

Related Questions:

1. What is the effect of the amplitude on the graph's shape and range?
2. How does the period of a sine function change when $b$ is increased?
3. How would adding a phase shift, such as $\sin\left(\frac{3}{4}x - \phi\right)$, alter the key points?
4. What is the significance of the midline in a transformed sine function?
5. How would the graph change if the coefficient $-1$ was changed to $+1$?

Tip:

Always label the key points, the midline, and the axis to avoid confusion when plotting trigonometric functions!