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Identify all of the possible equations for the following transformed trig function:




Question 3Select one or more:

a.

f(x)=−sin(4x)𝑓(𝑥)=−𝑠𝑖𝑛(4𝑥)

b.

f(x)=4cos(x+π2)𝑓(𝑥)=4𝑐𝑜𝑠(𝑥+π2)

c.

f(x)=−sin(x−2π)+4𝑓(𝑥)=−𝑠𝑖𝑛(𝑥−2π)+4

d.

f(x)=4sin(x+π)𝑓(𝑥)=4𝑠𝑖𝑛(𝑥+π)

e.

f(x)=−4sin(x)


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.

Learn how to identify equations for transformed trigonometric functions. This analysis covers amplitude, period, reflection, vertical shift, and phase shift, demonstrating their impact on trigonometric function equations. Suitable for high school students studying trigonometry.

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