https://file.aiquickdraw.com/mathbot/file/1729349732015ir1s7c2h.jpg

The problem involves a transformation of the function \( f(x) = \sqrt{x} \). You are asked to determine the equation of the function \( k(x) \), which is a translation of \( f(x) \).

From the image, we can observe that the graph of \( f(x) \) is shifted down and to the right, meaning it has undergone a **horizontal** and **vertical translation**.

Let's break this down:

- The function \( f(x) = \sqrt{x} \) starts at the origin (0, 0).
- The graph of \( k(x) \), however, appears to be shifted both horizontally and vertically.
  
  From the graph:
  - The graph of \( f(x) \) is shifted **4 units to the right** (positive horizontal shift).
  - It is also shifted **3 units down** (negative vertical shift).

Using the general form for transformations:

\[
k(x) = f(x - h) + k
\]

where \( h \) is the horizontal shift and \( k \) is the vertical shift, we have:

\[
k(x) = \sqrt{x - 4} - 3
\]

This reflects the translation of the square root function by 4 units to the right and 3 units down.

So, the expression for \( k(x) \) is:

\[
k(x) = \sqrt{x - 4} - 3
\]

Would you like further clarification on this transformation?

### Related Questions:
1. How would the function change if it was translated up instead of down?
2. What happens to the domain of \( f(x) \) after this translation?
3. How would you write the function if it were shifted 5 units to the left and 2 units up?
4. Can you identify the vertex of the translated function \( k(x) \)?
5. How does the horizontal translation affect the square root function's range?

**Tip:** Horizontal shifts inside the function affect the input, while vertical shifts outside the function affect the output directly!

The function f(x) = sqrt(x) is translated horizontally and vertically. What is the equation for the translated function k(x)?

Learn how to find the equation for a square root function f(x) = sqrt(x) after a horizontal and vertical translation. This problem covers key concepts such as function transformations, translations, and square root functions. Suitable for Grades 9-10.

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