graph of `y=x+|x|`

Options

A.
B.
C.
D.

Answer

To graph the equation \( y = x + |x| \), we will analyze the equation step by step.

### Step 1: Understand the Absolute Value
The absolute value function \( |x| \) is defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

### Step 2: Break the Equation into Cases
We will consider two cases based on the value of \( x \):

**Case 1:** When \( x \geq 0 \)
- Here, \( |x| = x \)
- Substitute into the equation:
\[ y = x + |x| = x + x = 2x \]

**Case 2:** When \( x < 0 \)
- Here, \( |x| = -x \)
- Substitute into the equation:
\[ y = x + |x| = x - x = 0 \]

### Step 3: Create a Table of Values
Now, we can create a table of values for both cases to plot the graph.

| \( x \) | \( y \) (for \( x \geq 0 \)) | \( y \) (for \( x < 0 \)) |
|---------|-------------------------------|-----------------------------|
| -2 | 0 | 0 |
| -1 | 0 | 0 |
| 0 | 0 | 0 |
| 1 | 2 | 0 |
| 2 | 4 | 0 |

### Step 4: Plot the Points
- For \( x < 0 \) (from -2 to 0), the value of \( y \) is always 0.
- For \( x = 0 \), \( y = 0 \).
- For \( x \geq 0 \) (from 0 to 2), the value of \( y \) increases linearly as \( y = 2x \).

### Step 5: Draw the Graph
- Draw a horizontal line along the x-axis for \( x < 0 \) where \( y = 0 \).
- From \( x = 0 \) onwards, draw a line with a slope of 2, starting from the origin (0,0).

### Conclusion
The graph of \( y = x + |x| \) consists of a horizontal line along the x-axis for \( x < 0 \) and a line with slope 2 for \( x \geq 0 \).