Worksheet on graphing quadratic functions using transformations, featuring four exercises with equations and blank coordinate grids for plotting.

You're working on graph transformations of quadratic functions. All these problems involve transforming the parent function `f(x) = x²`, which is a parabola opening upward with vertex at `(0, 0)`.

Let’s solve each one step by step using transformations, and I’ll explain how to graph them.

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🔹 Problem 1: `g(x) = x² - 6`

✔ Transformation: Vertical shift down 6 units

- Start with the parent function `f(x) = x²` → vertex at `(0, 0)`
- Subtracting 6 outside the function means every y-value is decreased by 6.
- So, the new vertex is at `(0, -6)`
- The shape (width, direction) remains the same — still opens upward.

📌 How to graph:
- Plot the vertex at `(0, -6)`
- Use key points from `x²`: e.g., `(1, 1)`, `(2, 4)` → now become `(1, -5)`, `(2, -2)`
- Draw a smooth U-shaped curve through those points.

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🔹 Problem 2: `g(x) = x² - 5`

✔ Transformation: Vertical shift down 5 units

- Parent: `f(x) = x²` → vertex `(0, 0)`
- Subtract 5 → vertex moves to `(0, -5)`
- Shape unchanged.

📌 How to graph:
- Vertex at `(0, -5)`
- Key points: `(1, 1-5) = (1, -4)`, `(2, 4-5) = (2, -1)`
- Sketch the parabola.

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🔹 Problem 3: `g(x) = x² + 6`

✔ Transformation: Vertical shift up 6 units

- Parent: `f(x) = x²` → vertex `(0, 0)`
- Add 6 → vertex moves to `(0, 6)`
- Still opens upward.

📌 How to graph:
- Vertex at `(0, 6)`
- Key points: `(1, 1+6) = (1, 7)`, `(2, 4+6) = (2, 10)`
- Draw the parabola.

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🔹 Problem 4: `g(x) = (x + 5)²`

✔ Transformation: Horizontal shift left 5 units

- Parent: `f(x) = x²` → vertex `(0, 0)`
- Inside the parentheses: `x + 5` means we replace `x` with `x + 5`
- Rule: `f(x - h)` shifts right `h`, so `f(x + 5) = f(x - (-5))` → shift left 5
- New vertex: `(-5, 0)`

📌 How to graph:
- Vertex at `(-5, 0)`
- Key points: from `x²`, `(1, 1)` → becomes `(-5 + 1, 1) = (-4, 1)`
- `(2, 4)` → `(-3, 4)`
- `(-1, 1)` → `(-6, 1)`
- Sketch the parabola centered at `(-5, 0)`

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## ✔ Summary of Transformations:

| Function | Transformation | Vertex |
|------------------|----------------------------|--------------|
| `x² - 6` | Down 6 | `(0, -6)` |
| `x² - 5` | Down 5 | `(0, -5)` |
| `x² + 6` | Up 6 | `(0, 6)` |
| `(x + 5)²` | Left 5 | `(-5, 0)` |

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## 🖍️ How to Draw on the Grids (for your worksheet):

For each problem:

1. Locate the vertex based on the transformation.
2. Plot 2–3 symmetric points around the vertex (e.g., ±1, ±2 from vertex x-value).
3. Connect with a smooth, U-shaped curve (parabola).

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✔ You’re done! These are all vertical or horizontal shifts of the basic parabola. No stretching, compressing, or flipping involved here — just translations.

Let me know if you want to see sketches or coordinate tables for each graph!