Let's analyze the given graph to identify the possible equations:

1. Amplitude:

• The graph's maximum and minimum values are 4 and -4, respectively.
• This means the amplitude is 4.

2. Period:

• The graph completes a full cycle from $-\pi$ to $\pi$, meaning the period is $2\pi$.
• This indicates that the coefficient of $x$ in the sine or cosine function should be 1.

3. Reflection:

• The graph is reflected over the x-axis because it starts at a minimum point rather than a maximum.
• This implies that the function is negative, i.e., $f(x) = -\sin(x)$ or $f(x) = -\cos(x)$.

4. Vertical Shift:

• There is no vertical shift since the graph oscillates between 4 and -4 around the x-axis (i.e., the midline is at $y = 0$).

5. Phase Shift:

• The graph appears to be shifted horizontally, and we need to consider this in the equation.
• The phase shift depends on where the graph starts, and in this case, it seems to be aligned with $\sin(x)$, $\cos(x)$, or their transformations.

Analysis of Given Options:

• Option (a): $f(x) = -\sin(4x)$

  • This has an incorrect period, since the period here would be $\frac{2\pi}{4} = \frac{\pi}{2}$, which is not consistent with the graph.

• Option (b): $f(x) = 4\cos\left(x + \frac{\pi}{2}\right)$

  • This function has the correct amplitude (4) and the correct period ($2\pi$).
  • The phase shift is $-\frac{\pi}{2}$ (left), which aligns well with the graph if considered as a cosine graph starting at $x = -\frac{\pi}{2}$.
  • This option is valid.

• Option (c): $f(x) = -\sin(x - 2\pi) + 4$

  • This has an incorrect vertical shift (up by 4 units) and a phase shift that does not match the graph.
  • This option is not valid.

• Option (d): $f(x) = 4\sin(x + \pi)$

  • This has the correct amplitude and period.
  • The phase shift $\pi$ to the left makes this equivalent to $-4\sin(x)$.
  • This option is valid.

• Option (e): $f(x) = -4\sin(x)$

  • This has the correct amplitude, period, and reflection over the x-axis.
  • No phase shift, and the graph is consistent with this.
  • This option is valid.

Conclusion:

The valid equations based on the analysis are:

• b. $f(x) = 4\cos\left(x + \frac{\pi}{2}\right)$
• d. $f(x) = 4\sin(x + \pi)$
• e. $f(x) = -4\sin(x)$

If you have any more questions or need further clarification, feel free to ask!

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Related Questions:

1. How would the graph change if the amplitude were reduced to 2?
2. What would the function be if the graph was shifted upward by 3 units?
3. How would the graph differ if it was reflected over the y-axis instead?
4. What would be the new equation if the period was doubled?
5. How would a phase shift of $\frac{\pi}{4}$ to the right affect the equation?

Tip: When analyzing trigonometric functions, always check the amplitude, period, reflection, vertical shift, and phase shift to match the graph accurately with its equation.