Worksheet for evaluating and graphing greatest integer functions.

Let's solve this Greatest Integer Function Worksheet step by step, explaining each part thoroughly.

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🔷 Part 1: Evaluating Greatest Integer Expressions

The greatest integer function, denoted as $ \lfloor x \rfloor $ or $ [[x]] $, returns the largest integer less than or equal to $ x $. It is also known as the floor function.

Let’s evaluate each:

#### (1) $ [[7.1]] = ? $

- The largest integer ≤ 7.1 is 7.
- ✔ Answer: 7

#### (2) $ [[1.8]] = ? $

- The largest integer ≤ 1.8 is 1.
- ✔ Answer: 1

#### (3) $ [[\pi]] = ? $

- $ \pi \approx 3.14159 $
- The largest integer ≤ π is 3.
- ✔ Answer: 3

#### (4) $ [[-6.8]] = ? $

- We need the largest integer ≤ -6.8
- Numbers less than -6.8: -7, -8, ...
- But we want the greatest one that is still ≤ -6.8 → -7
- Because -7 < -6.8 < -6, and -6 > -6.8, so not allowed.
- ✔ Answer: -7

#### (5) $ [[-2.1]] = ? $

- -3 < -2.1 < -2
- Largest integer ≤ -2.1 is -3
- ✔ Answer: -3

#### (6) $ [[0]] = ? $

- The greatest integer ≤ 0 is 0.
- ✔ Answer: 0

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✔ Answers for Part 1:

| Problem | Answer |
|--------|--------|
| (1) | 7 |
| (2) | 1 |
| (3) | 3 |
| (4) | -7 |
| (5) | -3 |
| (6) | 0 |

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🔷 Part 2: Translating Graphs of Greatest Integer Functions

We are given transformations of the basic greatest integer function $ y = [[x]] $, which looks like a series of horizontal steps at every integer value.

Let’s analyze each pair.

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#### (7)
Graph:
- $ f(x) = [[x]] + 2 $
- $ g(x) = [[x + 2]] $

##### 🔹 Graphing $ f(x) = [[x]] + 2 $

- This is a vertical shift upward by 2 units.
- Every output value increases by 2.
- So if $ [[x]] = n $, then $ f(x) = n + 2 $
- Example: At $ x = 1.3 $, $ [[1.3]] = 1 $, so $ f(1.3) = 3 $
- The graph is the same shape as $ [[x]] $, but shifted up by 2.

##### 🔹 Graphing $ g(x) = [[x + 2]] $

- This is a horizontal shift to the left by 2 units.
- The input is $ x + 2 $, so it "shifts left" by 2.
- Example: $ g(-1.5) = [[-1.5 + 2]] = [[0.5]] = 0 $
- Compared to $ [[x]] $, this graph starts earlier (at $ x = -2 $ instead of $ x = 0 $).

##### 🔸 Explanation of Shifts:
- $ f(x) = [[x]] + 2 $: Vertical translation up by 2 units — all steps move up.
- $ g(x) = [[x + 2]] $: Horizontal translation left by 2 units — entire graph shifts left.
- They differ in direction and type: one is vertical, one is horizontal.

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#### (8)
Graph:
- $ f(x) = 2[[x]] $
- $ g(x) = [[2x]] $

##### 🔹 Graphing $ f(x) = 2[[x]] $

- This is a vertical stretch by factor of 2.
- The output values are doubled.
- Example: If $ [[x]] = 1 $, then $ f(x) = 2 $
- If $ [[x]] = -1 $, then $ f(x) = -2 $
- Each step height is now twice as tall, but the width remains the same.
- The graph has same step width, but doubled step height.

##### 🔹 Graphing $ g(x) = [[2x]] $

- This is a horizontal compression by factor of 1/2.
- Input is multiplied by 2 → faster changes.
- Example: $ g(0.5) = [[1]] = 1 $, $ g(0.4) = [[0.8]] = 0 $
- The function jumps more frequently.
- The step width is halved — each step occurs over an interval of length 0.5 instead of 1.

##### 🔸 Explanation of Dilation:
- $ f(x) = 2[[x]] $: Vertical dilation — outputs scaled by 2 → taller steps.
- $ g(x) = [[2x]] $: Horizontal compression — inputs compressed → steps occur more often.
- They differ: one affects height, the other affects frequency (how often steps occur).

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#### (9)
Graph:
- $ f(x) = -[[x]] $
- $ g(x) = [[-x]] $

##### 🔹 Graphing $ f(x) = -[[x]] $

- This reflects the graph of $ [[x]] $ across the x-axis.
- Example: $ [[2.3]] = 2 $, so $ f(2.3) = -2 $
- All outputs are negated.
- The graph is the mirror image over the x-axis.

##### 🔹 Graphing $ g(x) = [[-x]] $

- This reflects the graph across the y-axis.
- You plug in $ -x $, so the function behaves backward.
- Example: $ g(2.3) = [[-2.3]] = -3 $
- Compare to $ [[x]] $: at $ x = 2.3 $, $ [[x]] = 2 $, but $ [[-x]] = [[-2.3]] = -3 $
- This is a reflection over the y-axis, but note: it’s not the same as $ -[[x]] $

##### 🔸 Comparison:
- $ f(x) = -[[x]] $: Reflects outputs → flips over x-axis.
- $ g(x) = [[-x]] $: Reflects inputs → flips over y-axis.
- These are different transformations:
- $ f(x) $: Flip the graph vertically.
- $ g(x) $: Flip the graph horizontally.

> 📌 Note: $ -[[x]] $ and $ [[-x]] $ are not the same!
> - For example: at $ x = 1.5 $,
> - $ f(1.5) = -[[1.5]] = -1 $
> - $ g(1.5) = [[-1.5]] = -2 $
> → Different values!

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✔ Summary of Graph Transformations

| Function | Transformation Type | Description |
|----------------------|-----------------------------------|-------------|
| $ [[x]] + 2 $ | Vertical shift up by 2 | All outputs increase by 2 |
| $ [[x + 2]] $ | Horizontal shift left by 2 | Graph moves left |
| $ 2[[x]] $ | Vertical stretch by factor of 2 | Step heights double |
| $ [[2x]] $ | Horizontal compression by 1/2 | Steps happen twice as fast |
| $ -[[x]] $ | Reflection over x-axis | Flips upside down |
| $ [[-x]] $ | Reflection over y-axis | Flips left-right |

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✔ Final Answers:

#### Part 1: Evaluations
1. $ [[7.1]] = 7 $
2. $ [[1.8]] = 1 $
3. $ [[\pi]] = 3 $
4. $ [[-6.8]] = -7 $
5. $ [[-2.1]] = -3 $
6. $ [[0]] = 0 $

#### Part 2: Graph Explanations

(7)
- $ f(x) = [[x]] + 2 $: Vertical shift up by 2 units
- $ g(x) = [[x + 2]] $: Horizontal shift left by 2 units
→ One shifts up, one shifts left; different directions.

(8)
- $ f(x) = 2[[x]] $: Vertical stretch — outputs doubled
- $ g(x) = [[2x]] $: Horizontal compression — inputs halved
→ One makes steps taller, one makes them narrower.

(9)
- $ f(x) = -[[x]] $: Reflect over x-axis
- $ g(x) = [[-x]] $: Reflect over y-axis
→ One flips vertically, one flips horizontally.

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If you'd like, I can generate the actual graphs (as ASCII or describe how to draw them on the grid). Let me know!