To graph the equation \( y - |x| = 6 \), follow these steps:

### 1. Rewrite the equation:
Start by isolating \( y \) on one side:
\[
y = |x| + 6
\]
This is the equation you will graph.

### 2. Understand the structure of the equation:
The equation \( y = |x| + 6 \) represents a vertical translation of the basic absolute value function \( y = |x| \) (a V-shaped graph) upward by 6 units. 

The vertex of this graph occurs where \( x = 0 \), and the graph opens upwards on both sides.

### 3. Plot the vertex:
The vertex of the absolute value function is at \( (0, 6) \).

### 4. Plot points on either side of the vertex:
- For \( x = 1 \), \( y = |1| + 6 = 7 \).
- For \( x = -1 \), \( y = |-1| + 6 = 7 \).
- For \( x = 2 \), \( y = |2| + 6 = 8 \).
- For \( x = -2 \), \( y = |-2| + 6 = 8 \).

### 5. Draw the V-shape:
- From the vertex \( (0, 6) \), draw two straight lines:
  - One going upwards with a slope of 1 for \( x \geq 0 \) (right side of the V).
  - One going upwards with a slope of -1 for \( x \leq 0 \) (left side of the V).

The graph should look like a "V" with the point at \( (0, 6) \).

### 6. Conclusion:
The graph of \( y - |x| = 6 \) is a V-shaped graph with its vertex at \( (0, 6) \), opening upwards symmetrically on both sides.

Do you have any questions or would you like more details?

Here are 5 related questions:
1. How would you shift the graph of \( y = |x| \) downwards instead?
2. How do you reflect the graph \( y = |x| \) over the x-axis?
3. What happens if you add a constant inside the absolute value, e.g., \( y = |x - 3| + 6 \)?
4. How do you determine the slope of the arms of the V-shaped graph?
5. Can you write the equation of a graph that is a reflection of \( y = |x| + 6 \) over the y-axis?

**Tip:** When graphing absolute value functions, always look for the vertex and check for vertical shifts or reflections.

Learn how to graph the absolute value equation y - |x| = 6. This step-by-step guide explains how to rewrite the equation, plot key points, and understand the V-shaped graph structure. Suitable for high school students learning about algebra and graphing functions.

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