The parent function f(x) = \sqrt{x} passes through the points (0, 0) and (4,2). Suppose g(x) = a\sqrt{b(x-h)} + k is a transformed function that comes from the parent function f(x). The
following is also given: the point (0,0) on f(x) becomes the point (0, -1) on g(x) and the point (4, 2)
on f(x) becomes the point (2, -3) on g(x). Find A B H K

To solve for the values of \( a \), \( b \), \( h \), and \( k \) for the transformed function \( g(x) = a\sqrt{b(x - h)} + k \), we need to use the information about how specific points on the parent function \( f(x) = \sqrt{x} \) are transformed to new points on \( g(x) \).

### Step-by-step Solution:

1. **Identify the transformations from points:**

   - The point \((0, 0)\) on \(f(x)\) transforms to \((0, -1)\) on \(g(x)\).
   - The point \((4, 2)\) on \(f(x)\) transforms to \((2, -3)\) on \(g(x)\).

2. **Set up equations using the transformations:**

   Using the transformation formula \(g(x) = a\sqrt{b(x - h)} + k\), we substitute the given points to form equations:

   - **For point \((0, 0) \rightarrow (0, -1)\):**

     Substitute \( x = 0 \) and \( g(x) = -1 \):

     \[
     g(0) = a\sqrt{b(0 - h)} + k = -1
     \]

     Since \( \sqrt{b(0 - h)} = \sqrt{-bh} \), for this to be valid, \(h = 0\) (otherwise it would involve the square root of a negative number if \(b > 0\)), simplifying the equation:

     \[
     g(0) = a\sqrt{b \cdot 0} + k = -1 \implies k = -1
     \]

   - **For point \((4, 2) \rightarrow (2, -3)\):**

     Substitute \( x = 2 \) and \( g(x) = -3 \):

     \[
     g(2) = a\sqrt{b(2 - 0)} - 1 = -3
     \]

     \[
     a\sqrt{2b} - 1 = -3
     \]

     \[
     a\sqrt{2b} = -2 \implies a\sqrt{2b} = -2
     \]

3. **Determine values of \( a \) and \( b \):**

   - To solve for \(b\), we use the fact that \(f(4) = 2\) transforms to \(g(2) = -3\).

     \[
     g(2) = -3
     \]

     From the equation \(a\sqrt{2b} = -2\), we square both sides to isolate \(b\):

     \[
     (-2)^2 = a^2 \cdot 2b \implies 4 = 2a^2b
     \]

     \[
     2 = a^2b \implies b = \frac{2}{a^2}
     \]

   - Use the equation \(a\sqrt{2b} = -2\):

     Plug \(b = \frac{2}{a^2}\) into the equation:

     \[
     a\sqrt{2 \cdot \frac{2}{a^2}} = -2
     \]

     \[
     a\sqrt{\frac{4}{a^2}} = -2
     \]

     \[
     a \cdot \frac{2}{|a|} = -2
     \]

     \[
     2 \frac{a}{|a|} = -2
     \]

   - This implies that \(a\) is negative and \(|a| = 1\), so \(a = -1\).

4. **Find \(b\) and \(h\):**

   With \(a = -1\), solve for \(b\):

   \[
   b = \frac{2}{a^2} = \frac{2}{1} = 2
   \]

   We already found that \(h = 0\) and \(k = -1\).

### Final Values:

- \( a = -1 \)
- \( b = 2 \)
- \( h = 0 \)
- \( k = -1 \)

Thus, the transformed function \( g(x) = -\sqrt{2x} - 1 \).

### Summary:

1. The transformation shifts the graph down by 1 unit (\(k = -1\)).
2. The function is also compressed vertically and reflected over the x-axis (\(a = -1\)).
3. The function is horizontally compressed by a factor of \( \sqrt{2} \) (\(b = 2\)).
4. The horizontal shift (\(h = 0\)) indicates no horizontal shift. 

Would you like more details on these transformations, or have any other questions?

### Related Questions:

1. How does changing the value of \(a\) affect the graph of \(g(x)\)?
2. What does the value of \(b\) represent in the transformation of \(f(x)\)?
3. How does the value of \(h\) shift the graph horizontally?
4. How do vertical and horizontal compressions differ?
5. How can you determine if a graph is reflected over the x-axis?

### Tip:

When dealing with transformations of functions, always check how key points on the original function move to help identify the transformations.

Learn how to find the values of a, b, h, and k for the transformed function g(x) = a*sqrt(b*(x - h)) + k using specific point transformations from the parent function f(x) = sqrt(x). This problem involves vertical and horizontal transformations of functions and is suitable for high school students.

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