Lesson 2 1 Graphing Absolute Value Functions Answer Key - Fill ... - Free Printable

Let's solve the problem step by step based on the image you provided. The task involves graphing absolute value functions, understanding transformations, and identifying key features like vertex, domain, range.

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🔷 Understanding Absolute Value Functions

The basic absolute value function is:

$$
f(x) = |x|
$$

Its graph is a V-shape with the vertex at (0, 0).

Transformations of this function follow the general form:

$$
g(x) = a|x - h| + k
$$

Where:
- $ h $: horizontal shift (right if $ h > 0 $, left if $ h < 0 $)
- $ k $: vertical shift (up if $ k > 0 $, down if $ k < 0 $)
- $ a $: vertical stretch/compression and reflection
- If $ |a| > 1 $: stretch
- If $ 0 < |a| < 1 $: compression
- If $ a < 0 $: reflection over x-axis

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We are given three functions to graph and analyze:

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✔ Problem 1:

$$
g(x) = |x - 2|
$$

#### Step-by-step Analysis:

This is a transformation of $ f(x) = |x| $.

- $ g(x) = |x - 2| $
- This means shift right by 2 units
- So the vertex moves from $ (0, 0) $ to $ (2, 0) $

#### Graph:
- V-shaped graph with vertex at $ (2, 0) $
- Two rays: one going up-right, one up-left from the vertex

#### Features:
- Vertex: $ (2, 0) $
- Domain: All real numbers → $ (-\infty, \infty) $
- Range: Since it opens upward, minimum value is 0 → $ [0, \infty) $

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✔ Problem 2:

$$
g(x) = |x - 3| + 1
$$

#### Step-by-step Analysis:

Compare to $ f(x) = |x| $

- $ |x - 3| $: shift right by 3 units
- $ +1 $: shift up by 1 unit

So, vertex moves to (3, 1)

#### Graph:
- V-shape centered at $ (3, 1) $
- Opens upward

#### Features:
- Vertex: $ (3, 1) $
- Domain: $ (-\infty, \infty) $
- Range: Minimum value is 1 → $ [1, \infty) $

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✔ Problem 3:

$$
g(x) = 3|x - 1|
$$

#### Step-by-step Analysis:

- $ |x - 1| $: shift right by 1 unit
- Multiply by 3: vertical stretch by factor of 3

So, the V-shape becomes narrower (steeper), and vertex shifts to $ (1, 0) $

#### Graph:
- Vertex at $ (1, 0) $
- Slopes of arms: $ \pm 3 $ instead of $ \pm 1 $

#### Features:
- Vertex: $ (1, 0) $
- Domain: $ (-\infty, \infty) $
- Range: Minimum value is 0 → $ [0, \infty) $

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📝 Final Answers:

| Function | Vertex | Domain | Range |
|--------|--------|--------|-------|
| 1. $ g(x) = |x - 2| $ | $ (2, 0) $ | $ (-\infty, \infty) $ | $ [0, \infty) $ |
| 2. $ g(x) = |x - 3| + 1 $ | $ (3, 1) $ | $ (-\infty, \infty) $ | $ [1, \infty) $ |
| 3. $ g(x) = 3|x - 1| $ | $ (1, 0) $ | $ (-\infty, \infty) $ | $ [0, \infty) $ |

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✔ How to Graph Each:

1. Start with the basic V-shape of $ y = |x| $
2. Apply transformations in order:
- Horizontal shift (inside absolute value)
- Vertical stretch/compression (multiply outside)
- Vertical shift (add/subtract outside)

For example, for $ g(x) = 3|x - 1| $:
- Move vertex to $ (1, 0) $
- From there, go up 3 units for every 1 unit right/left

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💡 Tips:

- Always find the vertex first.
- Use the slope to draw the arms.
- Remember: absolute value graphs are never negative unless reflected (but here none are).

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Let me know if you'd like a visual sketch or help plotting points!