Solved Name Dale Class LESSON Graphing Absolute Value | Chegg.com - Free Printable

Let’s solve each part of the worksheet step by step.

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## Graphing Absolute Value Functions

The general form is:
f(x) = a|x - h| + k
- Vertex: (h, k)
- If a > 0, opens upward; if a < 0, opens downward.
- Domain: always all real numbers → (-∞, ∞)
- Range: depends on vertex and direction:
- Upward: [k, ∞)
- Downward: (-∞, k]

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1. f(x) = |x| + 2

This is f(x) = 1|x - 0| + 2

✔ Vertex: (0, 2)
✔ Domain: All real numbers → (-∞, ∞)
✔ Range: Since it opens upward, minimum y-value is 2 → [2, ∞)

Graphing tips:
- Plot vertex at (0,2)
- Use slope ±1 from vertex:
- Right: (1,3), (2,4)
- Left: (-1,3), (-2,4)

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2. f(x) = -|x - 4|

This is f(x) = -1|x - 4| + 0

✔ Vertex: (4, 0)
✔ Domain: (-∞, ∞)
✔ Range: Opens downward → maximum y-value is 0 → (-∞, 0]

Graphing tips:
- Plot vertex at (4,0)
- Slope: ±(-1) → from vertex, go down 1 for every 1 right or left:
- Right: (5,-1), (6,-2)
- Left: (3,-1), (2,-2)

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3. f(x) = -3|x| + 5

This is f(x) = -3|x - 0| + 5

✔ Vertex: (0, 5)
✔ Domain: (-∞, ∞)
✔ Range: Opens downward → max y = 5 → (-∞, 5]

Graphing tips:
- Vertex at (0,5)
- Slope: ±(-3) → steeper V-shape
- Right: (1,2), (2,-1)
- Left: (-1,2), (-2,-1)

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4. f(x) = |x + 1| - 1

This is f(x) = 1|x - (-1)| - 1

✔ Vertex: (-1, -1)
✔ Domain: (-∞, ∞)
✔ Range: Opens upward → min y = -1 → [-1, ∞)

Graphing tips:
- Vertex at (-1, -1)
- Slope ±1:
- Right: (0,0), (1,1)
- Left: (-2,0), (-3,1)

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## Charles’ Pricing Function: f(x) = 5|x - 8| + 20

Here, x = number of hours, f(x) = hourly rate in dollars

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5. How much does Charles charge per hour if a customer hires him for 3 hours?

Plug x = 3 into the function:

f(3) = 5|3 - 8| + 20
= 5|-5| + 20
= 5(5) + 20
= 25 + 20
= $45 per hour

✔ Answer: $45 per hour

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6. Find the lowest hourly rate that Charles charges. Show your work.

We are looking for the minimum value of f(x) = 5|x - 8| + 20

Since absolute value is always ≥ 0, the smallest |x - 8| can be is 0, which occurs when x = 8.

So:

f(8) = 5|8 - 8| + 20
= 5(0) + 20
= 20

✔ Lowest hourly rate: $20 per hour

*(This happens when the customer hires him for exactly 8 hours.)*

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## ✔ Final Answers Summary:

Graphing Problems:

1. f(x) = |x| + 2
- Vertex: (0, 2)
- Domain: (-∞, ∞)
- Range: [2, ∞)

2. f(x) = -|x - 4|
- Vertex: (4, 0)
- Domain: (-∞, ∞)
- Range: (-∞, 0]

3. f(x) = -3|x| + 5
- Vertex: (0, 5)
- Domain: (-∞, ∞)
- Range: (-∞, 5]

4. f(x) = |x + 1| - 1
- Vertex: (-1, -1)
- Domain: (-∞, ∞)
- Range: [-1, ∞)

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Charles’ Problem:

5. For 3 hours → $45 per hour

6. Lowest hourly rate → $20 per hour (when x = 8)

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You can now graph each function using the vertices and slopes as described above. Let me know if you want hand-drawn sketches or digital plots!