Question

Analyze the graphs and describe the transformation that occurs from f(x)=x−3−−−−√
𝑓
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=
𝑥
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3
to h(x)=2(x−3)−−−−−−−√
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=
2
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−
3
)
.

Answer 1

To analyze the transformation from the function \( f(x) = \sqrt{x - 3} \) to \( h(x) = 2\sqrt{x - 3} \), let's break down each function and the modifications applied:

1. **Original Function \( f(x) \)**:
- The function \( f(x) = \sqrt{x - 3} \) represents the square root function shifted to the right by 3 units. The vertex of this graph is at the point (3,0), and the graph only exists for \( x \geq 3 \) because the expression under the square root must be non-negative.

2. **Transformed Function \( h(x) \)**:
- The function \( h(x) = 2\sqrt{x - 3} \) takes the original function and scales it vertically by a factor of 2.

### Transformation Steps
1. **Vertical Scaling**:
- The main transformation from \( f(x) \) to \( h(x) \) is the multiplication of the output by 2. This means that for every \( y \)-value of \( f(x) \), the corresponding \( y \)-value of \( h(x) \) is double. Hence, the shape of the graph remains the same, but it is stretched vertically.

### Key Features
- The domain of both functions remains the same: \( x \geq 3 \).
- The range of the original function \( f(x) \) is \( [0, \infty) \), and for the transformed function \( h(x) \), the range becomes \( [0, \infty) \) as well, but the output values are scaled.
- The vertex of \( h(x) \) is still at the point (3,0), but for any other point, the \( y \)-values will be twice their corresponding values in \( f(x) \). For example, if \( f(4) = 1 \), then \( h(4) = 2 \).

### Summary
Through the transformation from \( f(x) = \sqrt{x - 3} \) to \( h(x) = 2\sqrt{x - 3} \):
- **Vertical Stretch**: The graph is stretched vertically by a factor of 2, which doubles the height of the graph at every point while maintaining the same horizontal position and basic shape of the square root function.