Absolute Value Equations Graphs worksheet for practicing graphing linear equations.

It looks like you've uploaded a worksheet titled "Absolute Value Equations Graphs" that asks students to graph pairs of absolute value equations on the same coordinate plane. Since I can't view images directly, I'll explain how to solve this type of problem step by step based on the typical format of such worksheets.

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🔍 Understanding the Task

Each problem on the worksheet presents two equations involving absolute values, and you are asked to:

1. Graph both equations on the same coordinate grid.
2. Identify points of intersection (if any).

The general form of an absolute value equation is:
\[
y = |x - h| + k
\]
This represents a V-shaped graph with the vertex at (h, k).

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✔ Step-by-Step Solution Strategy

Let’s go through each pair using common examples (since the actual equations aren’t visible in text):

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#### Example Problem:
Graph:
1. \( y = |x| \)
2. \( y = 2 \)

##### Step 1: Understand Each Equation
- \( y = |x| \): This is the basic V-shape with vertex at (0, 0).
- For \( x \geq 0 \), \( y = x \)
- For \( x < 0 \), \( y = -x \)
- \( y = 2 \): A horizontal line passing through all points where y = 2.

##### Step 2: Graph Both
- Draw the V-shape of \( y = |x| \)
- Draw a straight horizontal line at \( y = 2 \)

##### Step 3: Find Points of Intersection
Set \( |x| = 2 \)
- Then \( x = 2 \) or \( x = -2 \)
- So, points of intersection are: \( (-2, 2) \) and \( (2, 2) \)

✔ Answer: The graphs intersect at \( (-2, 2) \) and \( (2, 2) \)

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#### Another Example:
Graph:
4. \( y = |x| + 2 \)
\( y = -2 \)

##### Step 1: Analyze
- \( y = |x| + 2 \): V-shape shifted up 2 units → vertex at (0, 2)
- \( y = -2 \): Horizontal line below the x-axis

##### Step 2: Graph
- Draw the upward V starting at (0, 2)
- Draw horizontal line at \( y = -2 \)

##### Step 3: Solve for Intersections
Set \( |x| + 2 = -2 \)
→ \( |x| = -4 \)

✘ No solution — absolute value cannot be negative.

✔ Answer: No intersection; graphs do not cross.

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📌 General Tips for Solving These Problems

1. Find the vertex of each absolute value function:
- For \( y = |x - h| + k \), vertex is at \( (h, k) \)

2. Sketch each graph:
- Use the vertex and slope (usually ±1 unless scaled)
- For example, \( y = 2|x| \) has steeper sides (slope ±2)

3. Solve algebraically to find intersections:
- Set the two equations equal and solve.
- Remember to consider cases for absolute values.

4. Check if solutions make sense:
- Absolute value expressions ≥ 0
- So, if solving \( |x| = -a \), no solution if \( a > 0 \)

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🔁 Let's Apply to Common Worksheet Problems

Assuming your worksheet has problems like these:

| Problem | Equations | Solution |
|--------|----------|---------|
| 1 | \( y = |x| \), \( y = 0 \) | Intersect at (0, 0) |
| 2 | \( y = |x| \), \( y = 3 \) | Intersect at (3, 3), (-3, 3) |
| 3 | \( y = |x - 4| \), \( y = 2 \) | Solve: \( |x - 4| = 2 \) → x = 6 or 2 → points: (6, 2), (2, 2) |
| 4 | \( y = |x| + 2 \), \( y = -2 \) | No solution (no intersection) |
| 5 | \( y = |x + 1| \), \( y = 3 \) | Solve: \( |x + 1| = 3 \) → x = 2 or -4 → points: (2, 3), (-4, 3) |
| 6 | \( y = |x| - 3 \), \( y = 0 \) | Solve: \( |x| - 3 = 0 \) → \( |x| = 3 \) → x = ±3 → (3, 0), (-3, 0) |

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✔ Final Instructions for Completing the Worksheet

For each problem:

1. Label the axes (usually from -10 to 10).
2. Plot the vertex of the absolute value function.
3. Draw the V-shape using the correct slopes.
4. Draw the second line (horizontal or slanted).
5. Mark points of intersection clearly.
6. Write the coordinates of intersection points (if any).

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If you can provide the specific equations from the image (e.g., "Problem 1: y = |x|, y = 0"), I can walk you through each one exactly.

Would you like me to help solve specific problems from your worksheet? Just list them!