whats the eq for a cosine function with max -2, min -10, period of pie over 2, reflection over the x-axis and phase shifr pieover6 right

To express a cosine function with the given properties, we can break down the components and assemble the equation step by step:

### 1. **General form of a cosine function:**
   The standard form of a cosine function is:
   \[
   y = A \cos(B(x - C)) + D
   \]
   where:
   - \( A \) affects the amplitude and reflects the graph over the x-axis if negative.
   - \( B \) affects the period.
   - \( C \) affects the phase shift.
   - \( D \) affects the vertical shift.

### 2. **Maximum and Minimum Values:**
   - The maximum value of the cosine function is \(-2\) and the minimum value is \(-10\).
   - The amplitude is half the difference between the max and min values:
     \[
     \text{Amplitude} = \frac{\text{max} - \text{min}}{2} = \frac{-2 - (-10)}{2} = \frac{8}{2} = 4
     \]
   - The vertical shift \( D \) is the midpoint between the max and min values:
     \[
     D = \frac{\text{max} + \text{min}}{2} = \frac{-2 + (-10)}{2} = \frac{-12}{2} = -6
     \]
   - Therefore, the cosine function has a vertical shift of \(-6\) and an amplitude of \( 4 \).

### 3. **Period:**
   - The period is \( \frac{\pi}{2} \). The formula for the period of a cosine function is:
     \[
     \text{Period} = \frac{2\pi}{|B|}
     \]
   - Given that the period is \( \frac{\pi}{2} \), we can solve for \( B \):
     \[
     \frac{2\pi}{|B|} = \frac{\pi}{2} \quad \Rightarrow \quad |B| = 4
     \]
   - Thus, \( B = 4 \) (since the cosine function is not inverted yet).

### 4. **Reflection Over the x-axis:**
   - Since the cosine function is reflected over the x-axis, the amplitude \( A \) is negative:
     \[
     A = -4
     \]

### 5. **Phase Shift:**
   - The phase shift is \( \frac{\pi}{6} \) to the right, which means the function will shift to the right by \( \frac{\pi}{6} \), and so the \( C \)-value will be \( \frac{\pi}{6} \).
   - The phase shift formula is \( (x - C) \), so in this case, \( C = \frac{\pi}{6} \).

### **Final Equation:**
   Putting all the components together, the equation for the cosine function is:
   \[
   y = -4 \cos\left( 4 \left( x - \frac{\pi}{6} \right) \right) - 6
   \]

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Would you like further explanations or more examples on how to interpret these parameters? Here are some related questions:

1. How does changing the amplitude affect the graph of a cosine function?
2. What would happen if the period were doubled or halved in a cosine function?
3. How would the graph change if the phase shift were to the left instead of the right?
4. How do you determine the range of a cosine function from its equation?
5. What is the impact of a negative vertical shift in a trigonometric function?

**Tip**: When dealing with phase shifts, remember that a phase shift to the right corresponds to \( (x - C) \), and to the left corresponds to \( (x + C) \).

Discover how to construct the equation of a cosine function with maximum -2, minimum -10, period π/2, a reflection over the x-axis, and a phase shift of π/6 to the right. This solution includes key trigonometric concepts such as amplitude, vertical shift, and transformations.

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