Worksheet displaying four absolute value functions with their corresponding graphs.

Let's solve each of the absolute value function graphing problems step by step.

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General Form of Absolute Value Function:

The parent function is $ y = |x| $, which is a V-shaped graph with its vertex at the origin (0, 0).
Transformations affect this graph in the following ways:

- $ y = |x - h| + k $: The vertex moves to $ (h, k) $
- $ h $: horizontal shift (right if positive, left if negative)
- $ k $: vertical shift (up if positive, down if negative)

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Now let's go through each problem.

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1) $ y = |x - 2| - 4 $

- This is a transformation of $ y = |x| $
- $ h = 2 $ → shift right 2 units
- $ k = -4 $ → shift down 4 units
- So the vertex is at $ (2, -4) $

#### Steps:
- Start from the vertex $ (2, -4) $
- The graph opens upward (since no negative sign in front of absolute value)
- Use two points:
- When $ x = 1 $: $ y = |1 - 2| - 4 = 1 - 4 = -3 $ → point $ (1, -3) $
- When $ x = 3 $: $ y = |3 - 2| - 4 = 1 - 4 = -3 $ → point $ (3, -3) $
- Draw lines from $ (2, -4) $ to $ (1, -3) $ and $ (3, -3) $, forming a V

✔ Graph should have a V-shape with vertex at $ (2, -4) $, opening upward.

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2) $ y = |x + 1| $

- $ y = |x + 1| = |x - (-1)| $
- So $ h = -1 $ → shift left 1 unit
- No vertical shift ($ k = 0 $)
- Vertex at $ (-1, 0) $

#### Steps:
- Vertex: $ (-1, 0) $
- Points:
- $ x = 0 $: $ y = |0 + 1| = 1 $ → $ (0, 1) $
- $ x = -2 $: $ y = |-2 + 1| = 1 $ → $ (-2, 1) $
- Draw lines from $ (-1, 0) $ to $ (0, 1) $ and $ (-2, 1) $

✔ Graph is a V-shape with vertex at $ (-1, 0) $, opening upward.

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3) $ y = |x| + 1 $

- $ h = 0 $, $ k = 1 $
- Shift up 1 unit
- Vertex at $ (0, 1) $

#### Steps:
- Vertex: $ (0, 1) $
- Points:
- $ x = 1 $: $ y = |1| + 1 = 2 $ → $ (1, 2) $
- $ x = -1 $: $ y = |-1| + 1 = 2 $ → $ (-1, 2) $
- Draw lines from $ (0, 1) $ to $ (1, 2) $ and $ (-1, 2) $

✔ V-shape, vertex at $ (0, 1) $, opening upward.

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4) $ y = |x| + 2 $

- $ h = 0 $, $ k = 2 $
- Shift up 2 units
- Vertex at $ (0, 2) $

#### Steps:
- Vertex: $ (0, 2) $
- Points:
- $ x = 1 $: $ y = |1| + 2 = 3 $ → $ (1, 3) $
- $ x = -1 $: $ y = |-1| + 2 = 3 $ → $ (-1, 3) $
- Connect to form V-shape

✔ V-shape, vertex at $ (0, 2) $, opening upward.

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✔ Summary of Graphs:

| Equation | Vertex | Shift | Shape |
|--------|--------|-------|-------|
| 1) $ y = |x - 2| - 4 $ | $ (2, -4) $ | Right 2, Down 4 | V up |
| 2) $ y = |x + 1| $ | $ (-1, 0) $ | Left 1 | V up |
| 3) $ y = |x| + 1 $ | $ (0, 1) $ | Up 1 | V up |
| 4) $ y = |x| + 2 $ | $ (0, 2) $ | Up 2 | V up |

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🔍 Final Answer:

You should draw each graph as follows:

1. $ y = |x - 2| - 4 $: V-shape with vertex at $ (2, -4) $
2. $ y = |x + 1| $: V-shape with vertex at $ (-1, 0) $
3. $ y = |x| + 1 $: V-shape with vertex at $ (0, 1) $
4. $ y = |x| + 2 $: V-shape with vertex at $ (0, 2) $

Each graph is symmetric about the vertex and opens upward.

If you need me to sketch these graphs or provide coordinates for plotting, let me know!