What effect does translating the radical function $$ k(x) = \sqrt{x} $$ vertically downward by 3 units have on its graph? Provide the new equation.

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Translating the radical function $$ k(x) = \sqrt{x} $$ vertically downward by 3 units means shifting every point on the graph of $$ k(x) $$ downwards by 3 units along the y-axis.

**Effect on the Graph:**
- **Vertical Shift Downward:** Each point $$(x, y)$$ on the original graph $$ y = \sqrt{x} $$ moves to $$(x, y - 3)$$. This results in the entire graph moving downward by 3 units.

**New Equation:**
$$ y = \sqrt{x}\ -\ 3 $$

This equation represents the transformed function after being shifted vertically downward by 3 units.

**Graphical Illustration:**

- **Original Function:** $$ y = \sqrt{x} $$ starts at the origin $$(0,0)$$ and increases to the right.
- **Translated Function:** $$ y = \sqrt{x} - 3 $$ starts at $$(0, -3)$$ and maintains the same shape, shifted downward by 3 units.

**Example Points:**

| $$ x $$ | Original $$ y = \sqrt{x} $$ | Translated $$ y = \sqrt{x} - 3 $$ |
|---------|-----------------------------|-----------------------------------|
| 0       | 0                           | -3                                |
| 1       | 1                           | -2                                |
| 4       | 2                           | -1                                |
| 9       | 3                           | 0                                 |

As illustrated, every y-coordinate of the original graph is decreased by 3 in the translated graph.

**Summary:**
The graph of $$ k(x) = \sqrt{x} $$ is shifted downward by 3 units, resulting in the new equation:
$$ y = \sqrt{x}\ -\ 3 $$
Translating the function $$ k(x) = \sqrt{x} $$ downward by 3 units shifts every point on its graph 3 units down. The new equation is: $$ y = \sqrt{x} - 3 $$

What effect does translating the radical function \( k(x) = \sqrt{x} \) vertically downward by 3 units have on its graph? Provide the new equation.

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