Absolute Value Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem step by step, analyzing the absolute value functions and determining their characteristics, including domain, range, transformations, axis of symmetry, y-intercept, and zeros.

We'll also sketch the graph for problems 3 and 4 using transformations.

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Problem 1:

Function:
$$ f(x) = -|x| - 3 $$

#### Step-by-step Analysis:

- Parent function: $ f(x) = |x| $
- Transformations:
- Reflection over x-axis: The negative sign in front of $|x|$ flips the V-shape upside down.
- Vertical shift down by 3 units: The $-3$ at the end shifts the entire graph down by 3.

So:
- Vertex moves from $(0, 0)$ to $(0, -3)$
- Opens downward

#### Characteristics:

- Domain: All real numbers → $ (-\infty, \infty) $
- Range: Since it opens downward and vertex is at $ y = -3 $, maximum value is $-3$, so range is $ (-\infty, -3] $
- Transformations:
- Reflection over x-axis
- Vertical translation down 3 units
- Axis of symmetry: $ x = 0 $ (the y-axis)
- y-intercept: Plug $ x = 0 $:
$ f(0) = -|0| - 3 = -3 $ → $ (0, -3) $
- Zero(s): Solve $ f(x) = 0 $:
$$
-|x| - 3 = 0 \\
-|x| = 3 \\
|x| = -3
$$
No solution since absolute value can't be negative.
→ No zeros

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ (-\infty, -3] $
- Transformations: Reflection over x-axis; vertical shift down 3 units
- Axis of symmetry: $ x = 0 $
- y-intercept: $ (0, -3) $
- Zeros: None

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Problem 2:

Function:
$$ f(x) = 2|x - 1| $$

#### Step-by-step Analysis:

- Parent function: $ |x| $
- Transformations:
- Horizontal shift right 1 unit: $ x - 1 $ → vertex at $ x = 1 $
- Vertical stretch by factor of 2: Makes the V-shape steeper
- Vertex: $ (1, 0) $

#### Characteristics:

- Domain: All real numbers → $ (-\infty, \infty) $
- Range: Opens upward, minimum at $ y = 0 $ → $ [0, \infty) $
- Transformations:
- Horizontal shift right 1 unit
- Vertical stretch by a factor of 2
- Axis of symmetry: $ x = 1 $
- y-intercept: Plug $ x = 0 $:
$ f(0) = 2|0 - 1| = 2(1) = 2 $ → $ (0, 2) $
- Zero(s): Solve $ f(x) = 0 $:
$ 2|x - 1| = 0 $ → $ |x - 1| = 0 $ → $ x = 1 $
→ Zero at $ x = 1 $

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ [0, \infty) $
- Transformations: Horizontal shift right 1 unit; vertical stretch by factor of 2
- Axis of symmetry: $ x = 1 $
- y-intercept: $ (0, 2) $
- Zero(s): $ x = 1 $

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Problem 3:

Function:
$$ f(x) = |x + 2| $$

#### Step-by-step Analysis:

- Parent function: $ |x| $
- Transformation:
- Horizontal shift left 2 units: $ x + 2 = x - (-2) $ → vertex at $ x = -2 $
- Vertex: $ (-2, 0) $
- Opens upward (no reflection)

#### Graphing Steps:
1. Start with $ |x| $
2. Shift left 2 units → vertex at $ (-2, 0) $
3. Plot points:
- $ x = -2 $ → $ f(-2) = 0 $
- $ x = -1 $ → $ f(-1) = |-1 + 2| = |1| = 1 $
- $ x = 0 $ → $ f(0) = |0 + 2| = 2 $
- $ x = -3 $ → $ f(-3) = |-3 + 2| = |-1| = 1 $
- $ x = -4 $ → $ f(-4) = |-4 + 2| = 2 $

Plot these points and draw the V-shape.

#### Characteristics:

- Domain: $ (-\infty, \infty) $
- Range: $ [0, \infty) $ (opens up, minimum at 0)
- Transformations: Horizontal shift left 2 units
- Axis of symmetry: $ x = -2 $
- y-intercept: $ f(0) = |0 + 2| = 2 $ → $ (0, 2) $
- Zero(s): Solve $ |x + 2| = 0 $ → $ x + 2 = 0 $ → $ x = -2 $

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ [0, \infty) $
- Transformations: Horizontal shift left 2 units
- Axis of symmetry: $ x = -2 $
- y-intercept: $ (0, 2) $
- Zero(s): $ x = -2 $

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Problem 4:

Function:
$$ f(x) = -|x| + 4 $$

#### Step-by-step Analysis:

- Parent function: $ |x| $
- Transformations:
- Reflection over x-axis: Negative sign
- Vertical shift up 4 units: $+4$
- Vertex: $ (0, 4) $
- Opens downward

#### Graphing Steps:
1. Start with $ |x| $
2. Reflect over x-axis → opens down
3. Shift up 4 units → vertex at $ (0, 4) $
4. Points:
- $ x = 0 $ → $ f(0) = -|0| + 4 = 4 $
- $ x = 1 $ → $ f(1) = -|1| + 4 = 3 $
- $ x = 2 $ → $ f(2) = -2 + 4 = 2 $
- $ x = -1 $ → same as $ x = 1 $: $ f(-1) = 3 $
- $ x = -2 $ → $ f(-2) = 2 $
5. Draw V-shape opening downward with vertex at $ (0, 4) $

#### Characteristics:

- Domain: $ (-\infty, \infty) $
- Range: Opens downward, max at $ y = 4 $ → $ (-\infty, 4] $
- Transformations:
- Reflection over x-axis
- Vertical shift up 4 units
- Axis of symmetry: $ x = 0 $
- y-intercept: $ f(0) = -|0| + 4 = 4 $ → $ (0, 4) $
- Zero(s): Solve $ -|x| + 4 = 0 $:
$$
-|x| = -4 \\
|x| = 4 \\
x = \pm 4
$$
→ Zeros at $ x = -4 $ and $ x = 4 $

✔ Answer:
- Domain: $ (-\infty, \infty) $
- Range: $ (-\infty, 4] $
- Transformations: Reflection over x-axis; vertical shift up 4 units
- Axis of symmetry: $ x = 0 $
- y-intercept: $ (0, 4) $
- Zero(s): $ x = -4 $, $ x = 4 $

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✔ Final Answers Summary:

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#### Problem 1: $ f(x) = -|x| - 3 $
- Domain: $ (-\infty, \infty) $
- Range: $ (-\infty, -3] $
- Transformations: Reflection over x-axis; vertical shift down 3 units
- Axis of symmetry: $ x = 0 $
- y-intercept: $ (0, -3) $
- Zeros: None

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#### Problem 2: $ f(x) = 2|x - 1| $
- Domain: $ (-\infty, \infty) $
- Range: $ [0, \infty) $
- Transformations: Horizontal shift right 1 unit; vertical stretch by 2
- Axis of symmetry: $ x = 1 $
- y-intercept: $ (0, 2) $
- Zeros: $ x = 1 $

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#### Problem 3: $ f(x) = |x + 2| $
- Domain: $ (-\infty, \infty) $
- Range: $ [0, \infty) $
- Transformations: Horizontal shift left 2 units
- Axis of symmetry: $ x = -2 $
- y-intercept: $ (0, 2) $
- Zeros: $ x = -2 $

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#### Problem 4: $ f(x) = -|x| + 4 $
- Domain: $ (-\infty, \infty) $
- Range: $ (-\infty, 4] $
- Transformations: Reflection over x-axis; vertical shift up 4 units
- Axis of symmetry: $ x = 0 $
- y-intercept: $ (0, 4) $
- Zeros: $ x = -4 $, $ x = 4 $

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📌 Graphs (Description):

You should sketch each graph based on transformations:

- Problem 3: V-shape with vertex at $ (-2, 0) $, opening up
- Problem 4: Inverted V-shape with vertex at $ (0, 4) $, opening down

Use grid paper to plot key points as shown above.

Let me know if you'd like a visual sketch or further explanation!