Graph function. `y=|x+2|+x`

Answer

To graph the function \( y = |x + 2| + x \), we will follow these steps:

### Step 1: Analyze the Absolute Value
The expression \( |x + 2| \) changes based on the value of \( x \). We need to determine the point where the expression inside the absolute value equals zero:
\[ x + 2 = 0 \implies x = -2 \]
This divides our analysis into two cases: when \( x \geq -2 \) and when \( x < -2 \).

### Step 2: Define the Function for Each Case
1. **Case 1: \( x \geq -2 \)**
- Here, \( |x + 2| = x + 2 \).
- Therefore, the function becomes:
\[ y = (x + 2) + x = 2x + 2 \]

2. **Case 2: \( x < -2 \)**
- Here, \( |x + 2| = -(x + 2) = -x - 2 \).
- Therefore, the function becomes:
\[ y = (-x - 2) + x = -2 \]

### Step 3: Write the Piecewise Function
We can now write the function as a piecewise function:
\[ y = \begin{cases} 2x + 2 & \text{if } x \geq -2 \\ -2 & \text{if } x < -2 \end{cases} \]

### Step 4: Plot the Graph
1. **For \( x \geq -2 \)**:
- The line \( y = 2x + 2 \) has a slope of 2 and a y-intercept of 2.
- At \( x = -2 \), \( y = 2(-2) + 2 = -2 \).
- As \( x \) increases, the line will rise steeply.

2. **For \( x < -2 \)**:
- The line \( y = -2 \) is a horizontal line at \( y = -2 \).

### Step 5: Identify Key Points
- The point where the two cases meet is at \( (-2, -2) \).
- For \( x < -2 \), the function remains constant at \( y = -2 \).
- For \( x = -1 \), substituting into the first case gives \( y = 2(-1) + 2 = 0 \).

### Step 6: Draw the Graph
- Draw the horizontal line \( y = -2 \) for \( x < -2 \).
- Draw the line \( y = 2x + 2 \) starting from the point \( (-2, -2) \) and extending to the right.

### Final Graph
The graph consists of a horizontal line at \( y = -2 \) for \( x < -2 \) and a line with a positive slope starting from the point \( (-2, -2) \) for \( x \geq -2 \).

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