SOLUTION: Kuta Software Infinite Algebra 1: Graphing Absolute ... - Free Printable

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 $: shifts the graph right by $ h $ units if $ h > 0 $, left if $ h < 0 $, and up by $ k $ units if $ k > 0 $, down if $ k < 0 $.
- The vertex becomes $ (h, k) $.

We will apply this logic to each problem.

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

- This is a transformation of $ y = |x| $:
- Shift right by 2 units → $ x - 2 $
- Shift down by 4 units → $ -4 $
- Vertex: $ (2, -4) $
- Plot the vertex at $ (2, -4) $
- Use the slope of ±1 from the vertex:
- To the right: go up 1, right 1 → point $ (3, -3) $
- To the left: go up 1, left 1 → point $ (1, -3) $
- Draw two rays from the vertex through these points.

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

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2) $ y = |x + 1| $

- This is $ y = |x - (-1)| $ → shift left by 1 unit
- Vertex: $ (-1, 0) $
- From vertex:
- Right: $ (-1+1, 1) = (0, 1) $
- Left: $ (-1-1, 1) = (-2, 1) $
- Draw V-shape with vertex at $ (-1, 0) $

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

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3) $ y = |x| + 1 $

- Shift up by 1 unit
- Vertex: $ (0, 1) $
- From vertex:
- Right: $ (1, 2) $
- Left: $ (-1, 2) $
- Draw V-shape with vertex at $ (0, 1) $

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

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4) $ y = |x| + 2 $

- Shift up by 2 units
- Vertex: $ (0, 2) $
- From vertex:
- Right: $ (1, 3) $
- Left: $ (-1, 3) $
- Draw V-shape with vertex at $ (0, 2) $

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

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✔ Summary of Graphs:

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

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📌 How to Graph Each:

For each equation:
1. Identify the vertex.
2. Plot the vertex.
3. Use the fact that the graph has two lines with slopes of +1 and -1 from the vertex.
4. Plot one point on each side using slope.
5. Draw the two rays forming a "V".

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🔍 Final Answer (Instructions for Graphing):

#### 1) $ y = |x - 2| - 4 $
- Vertex: $ (2, -4) $
- Points: $ (3, -3), (1, -3) $
- Graph V-shape from $ (2, -4) $ with arms going up-right and up-left.

#### 2) $ y = |x + 1| $
- Vertex: $ (-1, 0) $
- Points: $ (0, 1), (-2, 1) $
- Graph V-shape from $ (-1, 0) $

#### 3) $ y = |x| + 1 $
- Vertex: $ (0, 1) $
- Points: $ (1, 2), (-1, 2) $
- Graph V-shape from $ (0, 1) $

#### 4) $ y = |x| + 2 $
- Vertex: $ (0, 2) $
- Points: $ (1, 3), (-1, 3) $
- Graph V-shape from $ (0, 2) $

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If you were to draw these on the grid provided:
- Make sure the vertex is correctly placed.
- The lines should have slope ±1.
- The graph opens upward in all cases since there’s no negative sign in front of the absolute value.

Let me know if you'd like me to generate or describe the graphs visually!