The graph of function `y=1+cosx` will be

Options

A.
B.
C.
D. None of these

Answer

To analyze the graph of the function \( y = 1 + \cos x \), we can follow these steps:

### Step 1: Understand the basic graph of \( \cos x \)
The cosine function, \( \cos x \), oscillates between -1 and 1. Its graph is a wave that crosses the y-axis at \( (0, 1) \) and has a period of \( 2\pi \).

### Step 2: Shift the graph of \( \cos x \)
Since we are looking at the function \( y = 1 + \cos x \), we need to shift the entire graph of \( \cos x \) upwards by 1 unit. This means every point on the graph of \( \cos x \) will be increased by 1.

### Step 3: Determine the new range of the function
The original range of \( \cos x \) is from -1 to 1. By adding 1 to each value, the new range becomes:
- Minimum value: \( -1 + 1 = 0 \)
- Maximum value: \( 1 + 1 = 2 \)

Thus, the range of \( y = 1 + \cos x \) is from 0 to 2.

### Step 4: Sketch the graph
1. Start by plotting the key points of \( \cos x \):
- At \( x = 0 \), \( y = 1 + \cos(0) = 2 \)
- At \( x = \pi \), \( y = 1 + \cos(\pi) = 0 \)
- At \( x = 2\pi \), \( y = 1 + \cos(2\pi) = 2 \)
2. The graph will oscillate between the points (0, 2), (π, 0), and (2π, 2).
3. The graph will be a wave-like shape that touches the y-axis at 0 and 2, never going below 0.

### Step 5: Confirm the characteristics of the graph
- The graph of \( y = 1 + \cos x \) will not have any negative values since the minimum value is 0.
- The graph will repeat every \( 2\pi \) due to the periodic nature of the cosine function.

### Final Graph Description
The graph of \( y = 1 + \cos x \) is a wave that oscillates between 0 and 2, starting at the maximum value of 2 when \( x = 0 \) and reaching the minimum value of 0 at \( x = \pi \).

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