Worksheet for practicing greatest integer function evaluations and graphing transformations.

Let's solve this Greatest Integer Function Worksheet step by step, including evaluations and explanations for graphing transformations.

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🔷 Part 1: Evaluating Greatest Integer Expressions

The greatest integer function, denoted $ \lfloor x \rfloor $ (also called the floor function), returns the largest integer less than or equal to $ x $.

Let’s evaluate each:

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(1) $ \lfloor 7.1 \rfloor = 7 $
→ The largest integer ≤ 7.1 is 7

(2) $ \lfloor 1.8 \rfloor = 1 $
→ The largest integer ≤ 1.8 is 1

(3) $ \lfloor \pi \rfloor = 3 $
→ $ \pi \approx 3.1415 $, so the greatest integer ≤ π is 3

(4) $ \lfloor -6.8 \rfloor = -7 $
→ The largest integer ≤ -6.8 is -7 (since -7 < -6.8 < -6)

(5) $ \lfloor -2.1 \rfloor = -3 $
→ -3 < -2.1 < -2 → largest integer ≤ -2.1 is -3

(6) $ \lfloor 0 \rfloor = 0 $
→ 0 is an integer, so floor(0) = 0

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✔ Answers:
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’ll analyze and sketch transformations of the basic greatest integer function $ y = \lfloor x \rfloor $, which looks like a series of horizontal line segments with jumps at integers.

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#### (7)
Graph:
- $ f(x) = \lfloor x \rfloor + 2 $
- $ g(x) = \lfloor x + 2 \rfloor $

##### ✔ Graphing:
- $ f(x) = \lfloor x \rfloor + 2 $: This shifts the entire graph up by 2 units.
- Every output value increases by 2.
- Example: At $ x=1.3 $, $ \lfloor 1.3 \rfloor = 1 $, so $ f(1.3) = 1+2 = 3 $
- So, instead of jumping from 0 to 1 at x=1, it jumps from 2 to 3.

- $ g(x) = \lfloor x + 2 \rfloor $: This shifts the graph left by 2 units.
- You're adding 2 inside the function → horizontal shift left.
- Example: $ g(0) = \lfloor 0+2 \rfloor = \lfloor 2 \rfloor = 2 $
- Compare to original: $ \lfloor 0 \rfloor = 0 $, but now it starts at x=-2.

##### 📌 Explanation:
- $ f(x) = \lfloor x \rfloor + 2 $: Vertical shift up by 2
- $ g(x) = \lfloor x + 2 \rfloor $: Horizontal shift left by 2
- They differ in direction and type of transformation: one moves vertically, the other horizontally.

> 💡 Key Rule:
> - $ f(x) + k $: vertical shift (up if $ k > 0 $)
> - $ f(x + h) $: horizontal shift (left if $ h > 0 $)

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#### (8)
Graph:
- $ f(x) = 2\lfloor x \rfloor $
- $ g(x) = \lfloor 2x \rfloor $

##### ✔ Graphing:
- $ f(x) = 2\lfloor x \rfloor $: Multiply the output by 2 → vertical stretch by factor of 2
- Each "step" is twice as tall.
- For example:
- $ x \in [0,1) $: $ \lfloor x \rfloor = 0 $ → $ f(x) = 0 $
- $ x \in [1,2) $: $ \lfloor x \rfloor = 1 $ → $ f(x) = 2 $
- So the steps go from 0 to 2, then 2 to 4, etc.

- $ g(x) = \lfloor 2x \rfloor $: Input is doubled → horizontal compression by factor of 1/2
- The jump points occur more frequently.
- For example:
- $ x \in [0, 0.5) $: $ 2x \in [0,1) $ → $ g(x) = 0 $
- $ x \in [0.5, 1) $: $ 2x \in [1,2) $ → $ g(x) = 1 $
- So jumps happen every 0.5 units instead of every 1 unit.

##### 📌 Explanation:
- $ f(x) = 2\lfloor x \rfloor $: Vertical dilation (stretch) — outputs are scaled by 2
- $ g(x) = \lfloor 2x \rfloor $: Horizontal compression — input is compressed by 1/2 → graph becomes "narrower"
- They differ: one affects height of steps, the other affects frequency of steps.

> 💡 Rule Reminder:
> - $ a \cdot f(x) $: vertical stretch/compression
> - $ f(bx) $: horizontal stretch/compression (by factor $ 1/b $)

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#### (9)
Graph:
- $ f(x) = -\lfloor x \rfloor $
- $ g(x) = \lfloor -x \rfloor $

##### ✔ Graphing:
- $ f(x) = -\lfloor x \rfloor $: Reflect the graph over the x-axis
- Take the floor value and negate it.
- Example:
- $ x = 1.3 $: $ \lfloor x \rfloor = 1 $ → $ f(x) = -1 $
- $ x = 2.7 $: $ \lfloor x \rfloor = 2 $ → $ f(x) = -2 $
- So the steps go down: from -1 to -2, etc.

- $ g(x) = \lfloor -x \rfloor $: Reflect over y-axis (because of $ -x $)
- But note: $ \lfloor -x \rfloor $ is not the same as $ -\lfloor x \rfloor $
- Example:
- $ x = 1.3 $: $ -x = -1.3 $ → $ \lfloor -1.3 \rfloor = -2 $
- $ x = 2.7 $: $ -x = -2.7 $ → $ \lfloor -2.7 \rfloor = -3 $
- So values are not just negated; they follow different behavior.

##### 📌 Comparison:
- $ f(x) = -\lfloor x \rfloor $: Reflect over x-axis → negative of output
- $ g(x) = \lfloor -x \rfloor $: Reflect over y-axis → input negated first
- These are not the same!

> 💡 Example: At $ x = 1.3 $
> - $ f(1.3) = -\lfloor 1.3 \rfloor = -1 $
> - $ g(1.3) = \lfloor -1.3 \rfloor = -2 $

So the graphs are different — one is a reflection over x-axis, the other over y-axis.

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✔ Final Summary:

#### Evaluations:
| Expression | Value |
|----------|-------|
| $ \lfloor 7.1 \rfloor $ | 7 |
| $ \lfloor 1.8 \rfloor $ | 1 |
| $ \lfloor \pi \rfloor $ | 3 |
| $ \lfloor -6.8 \rfloor $ | -7 |
| $ \lfloor -2.1 \rfloor $ | -3 |
| $ \lfloor 0 \rfloor $ | 0 |

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#### Graph Explanations:

(7)
- $ f(x) = \lfloor x \rfloor + 2 $: Up 2 units
- $ g(x) = \lfloor x + 2 \rfloor $: Left 2 units
- Difference: One is vertical, one is horizontal shift.

(8)
- $ f(x) = 2\lfloor x \rfloor $: Vertical stretch by 2 → taller steps
- $ g(x) = \lfloor 2x \rfloor $: Horizontal compression by ½ → more frequent jumps
- Difference: One changes height, one changes spacing of steps.

(9)
- $ f(x) = -\lfloor x \rfloor $: Reflection over x-axis
- $ g(x) = \lfloor -x \rfloor $: Reflection over y-axis
- Difference: Not the same! $ -\lfloor x \rfloor \neq \lfloor -x \rfloor $

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🖼️ How to Sketch the Graphs (Quick Guide):

For $ y = \lfloor x \rfloor $:
- Draw horizontal lines on each interval $[n, n+1)$ at height $ n $, where $ n $ is integer.
- Use closed dot at left endpoint, open dot at right.

Then apply transformations accordingly.

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Would you like me to generate the actual sketches (as text diagrams) for each function?