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In question 10(b), you are asked to write the equation of a radical function in the form:

\[
y = a\sqrt{b(x - h)} + k
\]

by observing the graph. Let's analyze the graph in part (b) and break down the problem step-by-step.

### Step 1: General form of the equation
The general form is \( y = a\sqrt{b(x - h)} + k \), where:
- \( h \) and \( k \) represent the horizontal and vertical shifts, respectively.
- \( a \) affects the vertical stretch and direction.
- \( b \) affects the horizontal stretch or compression.

### Step 2: Identifying the shifts \( h \) and \( k \)
Looking at the graph:
- The curve starts at the point \( (2, -3) \), which suggests that the graph has been shifted horizontally by 2 units to the right and vertically by -3 units. So, \( h = 2 \) and \( k = -3 \).

### Step 3: Determining the vertical stretch/compression (coefficient \( a \))
Next, we need to identify the vertical stretch/compression factor \( a \). Notice that the graph has an upward slope, but it appears to rise slowly. By checking a point on the graph, say \( (6, -2) \), we can determine \( a \).

Using the equation structure \( y = a\sqrt{b(x - h)} + k \):
- For the point \( (6, -2) \), substitute \( x = 6 \) and \( y = -2 \):
  \[
  -2 = a\sqrt{b(6 - 2)} - 3
  \]
  \[
  -2 + 3 = a\sqrt{b(4)}
  \]
  \[
  1 = a\sqrt{4b}
  \]
  \[
  1 = 2a\sqrt{b}
  \]

At this point, we need further context from the graph to estimate \( b \) and solve for \( a \), but let's assume \( b = 1 \) to simplify. This results in:

\[
1 = 2a \implies a = \frac{1}{2}
\]

### Final Equation:
Thus, the equation that fits this graph would be approximately:

\[
y = \frac{1}{2} \sqrt{x - 2} - 3
\]

Would you like to go over the details further or explore another aspect of this equation? 

Here are 5 related questions you might want to explore:
1. How does changing \( a \) affect the steepness of the graph?
2. What happens to the graph if \( h \) is negative?
3. How do different values of \( b \) affect the horizontal stretch of the graph?
4. What if the graph starts at a different point, how does it change \( h \) and \( k \)?
5. How can you use other points on the graph to verify the correctness of \( a \) and \( b \)?

**Tip**: To check your solution, always test multiple points on the graph to ensure the equation fits the overall behavior.

Explore how to write the equation of a radical function from a graph in form y = a\sqrt{b(x - h)} + k. The analysis includes identifying shifts at (h, k) and determining coefficients for vertical stretch/compression. Perfect for high school students learning function transformations.

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