The text in the image asks to transform the graph of $y = \sin(x)$ to show a new function with:

1. Smaller amplitude
2. Larger period
3. Phase shift of $-\pi$

To achieve this transformation, we can modify the general form of the sine function:

$y = A \sin(B(x - C)) + D$

Where:

• $A$ affects the amplitude.
• $B$ affects the period, which is calculated as $\frac{2\pi}{B}$.
• $C$ is the phase shift (horizontal shift).
• $D$ affects the vertical shift (not relevant here).

Given the conditions:

1. Smaller amplitude: $A < 1$. If the original amplitude is 1 (for $\sin(x)$), we choose $A$ to be smaller, such as $A = \frac{1}{2}$.
2. Larger period: The period is given by $\frac{2\pi}{B}$, so for a larger period, $B$ should be less than 1. For example, if $B = \frac{1}{2}$, the period would be $2\pi \times 2 = 4\pi$, which is larger than the original $2\pi$.
3. Phase shift of $-\pi$: The phase shift means $C = \pi$, since the function shifts to the right by $C$.

Thus, the transformed equation would be:

$y = \frac{1}{2} \sin\left(\frac{1}{2}(x + \pi)\right)$

This satisfies:

• Amplitude: $\frac{1}{2}$ (smaller than 1),
• Period: $4\pi$ (larger than the original $2\pi$),
• Phase shift: $-\pi$ (shifted to the left by $\pi$).

Would you like further details or have any other questions?

Here are 5 related questions:

1. How does changing the amplitude affect the shape of the sine wave?
2. How do you calculate the period of a sine function?
3. What is the difference between a phase shift and a vertical shift in trigonometric functions?
4. Can the amplitude of a sine function ever be negative?
5. How would a vertical shift change the equation?

Tip: Always remember that for a sine function, the period is inversely related to the coefficient $B$ in front of $x$. A smaller $B$ gives a larger period.