Question

Analyze the graphs and describe the transformation that occurs from f(x)=x−3−−−−√
𝑓
(
𝑥
)
=
𝑥
−
3
to h(x)=2(x−3)−−−−−−−√
ℎ
(
𝑥
)
=
2
(
𝑥
−
3
)
. For the first response, enter 1 for stretch and 2 for compression. Enter your response as a fraction. If your second response is not a whole number, enter it as a fraction.(1 point)
The transformation is a horizontal by a factor of .

Answer 1

To analyze the transformation from \( f(x) = \sqrt{x - 3} \) to \( h(x) = 2\sqrt{x - 3} \), let's break down the transformations step by step:

1. **Vertical Stretch/Compression**: The function \( h(x) = 2\sqrt{x - 3} \) can be seen as a vertical stretch of the function \( f(x) \) by a factor of 2. This is because the output of the function is multiplied by 2, which stretches the graph away from the x-axis.

2. **Horizontal Transformation**: The term \( x - 3 \) inside the square root indicates a horizontal shift. Specifically, the graph of \( f(x) \) has been shifted to the right by 3 units. This transformation applies to both \( f(x) \) and \( h(x) \) and is not changed in \( h(x) \).

We can summarze the transformations as follows:

- **Vertical Transformation**:
- The vertical stretch factor is 2.
- So, for the first response (vertical transformation), we enter **1 (for stretch)** and then enter the factor which is **2**.

- **Horizontal Transformation**:
- The function has been shifted right by 3 units.
- Therefore, the factor for the horizontal transformation is **3**.

So the final responses would be:

1. Stretch: **1**
2. Factor: **3**

In summary, the transformations are:
- A vertical stretch by a factor of 2, resulting in the number **1** for stretch and **2** for the factor, and
- A horizontal shift to the right by a factor of **3**.