Linear Transformation Worksheet #1 featuring graphing and transformation problems.

Since I cannot view or process images directly, I’ll solve the Linear Transformation Worksheet #1 based on the text you’ve provided. This is a standard algebra worksheet focusing on transforming linear functions using translations, reflections, and stretches.

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## 📘 Problem 1: Vertical translation 3 units down
Rule: \( f(x) \to f(x) - 3 \)
✔ Answer: Subtract 3 from the output → \( g(x) = f(x) - 3 \)

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## 📘 Problem 2: Vertical compression by a factor of ½
Rule: Multiply the output by ½ → \( f(x) \to \frac{1}{2}f(x) \)
✔ Answer: \( g(x) = \frac{1}{2}f(x) \)

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## 📘 Problem 3: Horizontal translation 5 units left
Rule: Replace \( x \) with \( x + 5 \) → \( f(x) \to f(x + 5) \)
✔ Answer: \( g(x) = f(x + 5) \)

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## 📘 Problem 4: Reflection across the x-axis
Rule: Multiply the output by -1 → \( f(x) \to -f(x) \)
✔ Answer: \( g(x) = -f(x) \)

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## 📘 Problem 5: \( f(x) = x + 1 \)
> Translate the graph 3 up and then 2 to the right. Write the transformed function in terms of \( g(x) \).

Step-by-step:

1. Start with \( f(x) = x + 1 \)
2. Translate 3 up: Add 3 → \( f(x) + 3 = (x + 1) + 3 = x + 4 \)
3. Translate 2 right: Replace \( x \) with \( x - 2 \) → \( (x - 2) + 4 = x + 2 \)

✔ Final Answer: \( g(x) = x + 2 \)

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## 📘 Problem 6: \( f(x) = 4x - 3 \)
> Translate the graph 3 up and then 2 to the right. Write the transformed function in terms of \( g(x) \).

Step-by-step:

1. Start with \( f(x) = 4x - 3 \)
2. Translate 3 up: Add 3 → \( 4x - 3 + 3 = 4x \)
3. Translate 2 right: Replace \( x \) with \( x - 2 \) → \( 4(x - 2) = 4x - 8 \)

✔ Final Answer: \( g(x) = 4x - 8 \)

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## 📘 Problem 7: \( f(x) = 2x - 3 \)
> Transform \( f(x) \) with a vertical stretch of 2 and then reflect over the x-axis. Write the transformed function in terms of \( g(x) \).

Step-by-step:

1. Start with \( f(x) = 2x - 3 \)
2. Vertical stretch by 2: Multiply entire function by 2 → \( 2 \cdot (2x - 3) = 4x - 6 \)
3. Reflect over x-axis: Multiply by -1 → \( -(4x - 6) = -4x + 6 \)

✔ Final Answer: \( g(x) = -4x + 6 \)

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## ✔ Summary of Answers:

| Problem | Rule / Transformation | Transformed Function \( g(x) \) |
|---------|------------------------------------------------|----------------------------------|
| 1 | Vertical translation 3 units down | \( g(x) = f(x) - 3 \) |
| 2 | Vertical compression by factor ½ | \( g(x) = \frac{1}{2}f(x) \) |
| 3 | Horizontal translation 5 units left | \( g(x) = f(x + 5) \) |
| 4 | Reflection across x-axis | \( g(x) = -f(x) \) |
| 5 | Translate 3 up, then 2 right | \( g(x) = x + 2 \) |
| 6 | Translate 3 up, then 2 right | \( g(x) = 4x - 8 \) |
| 7 | Vertical stretch ×2, then reflect over x-axis | \( g(x) = -4x + 6 \) |

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## 📈 Graphing Tips (for the grids):

- For Problem 5: Graph \( g(x) = x + 2 \) — line with slope 1, y-intercept at (0, 2).
- For Problem 6: Graph \( g(x) = 4x - 8 \) — steeper line, slope 4, y-intercept at (0, -8).
- For Problem 7: Graph \( g(x) = -4x + 6 \) — steep negative slope, y-intercept at (0, 6).

Use the grid to plot two points for each line (e.g., x=0 and x=1), then draw the line.

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Let me know if you’d like help graphing any of these or need explanations for specific steps!