graph the y=2-2 Cos X, using the transformations of functions

To graph the function \( y = 2 - 2\cos(x) \) using transformations, let's break it down step by step. The parent function here is \( y = \cos(x) \), and we'll apply a series of transformations to obtain the desired graph.

### 1. **Start with the parent function: \( y = \cos(x) \)**
- The graph of \( y = \cos(x) \) has a period of \( 2\pi \), an amplitude of 1, and it oscillates between 1 and -1. It crosses the x-axis at \( x = 0 \), \( x = \pi \), and \( x = 2\pi \).

### 2. **Apply vertical stretch: \( y = -2\cos(x) \)**
- The coefficient \( -2 \) affects both the amplitude and the direction of the graph:
  - **Amplitude change:** The amplitude becomes 2, meaning the function now oscillates between -2 and 2 instead of -1 and 1.
  - **Reflection:** The negative sign causes a reflection over the x-axis. So, the crests and troughs of the cosine wave are inverted.

### 3. **Apply vertical shift: \( y = 2 - 2\cos(x) \)**
- The constant \( +2 \) shifts the entire graph upward by 2 units. This means the graph now oscillates between 0 and 4, instead of between -2 and 2.
  - The midline of the graph is now \( y = 2 \) instead of \( y = 0 \).
  - The highest value is 4 and the lowest value is 0.

### Final characteristics of the graph:
- **Amplitude:** 2
- **Period:** \( 2\pi \)
- **Vertical shift:** Up 2 units
- **Reflection over the x-axis** (due to the negative sign)
- **Range:** [0, 4]

### Graph summary:
- The wave still repeats every \( 2\pi \), with the maximum value at \( y = 4 \) and the minimum value at \( y = 0 \).
- The x-axis intercepts are shifted upward, and the crests and troughs are reflected due to the negative sign.

Would you like a detailed graph of this function, or more explanation?

### 5 Related Questions:
1. How would the graph change if the cosine function was multiplied by a positive number?
2. What would happen if we added a horizontal shift to this function?
3. How does the period of the graph change with different frequencies?
4. How does changing the amplitude affect the range of the function?
5. How do transformations affect the graph of other trigonometric functions?

**Tip:** When applying transformations, it's easiest to start with amplitude changes and reflections before applying shifts.

Graph the y=2-2 Cos X, using the transformations of functions.

Understand how to graph the function y = 2 - 2cos(x) using transformations. This step-by-step solution covers amplitude changes, reflections, vertical shifts, and periodicity. Suitable for students in Grades 10-12, focusing on trigonometric graph transformations.

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