Question
Describe the transformation from v(x)=√x-1+4
to q(x)=√0.5(x-1)+4
.(1 point)
Responses
horizontal compression by a factor of 0.5
horizontal compression by a factor of 0.5
horizontal compression by a factor of 2
horizontal compression by a factor of 2
horizontal stretch by a factor of 0.5
horizontal stretch by a factor of 0.5
horizontal stretch by a factor of 2
to q(x)=√0.5(x-1)+4
.(1 point)
Responses
horizontal compression by a factor of 0.5
horizontal compression by a factor of 0.5
horizontal compression by a factor of 2
horizontal compression by a factor of 2
horizontal stretch by a factor of 0.5
horizontal stretch by a factor of 0.5
horizontal stretch by a factor of 2
Answers
GPT-4o mini
To analyze the transformation from the function \( v(x) = \sqrt{x - 1} + 4 \) to \( q(x) = \sqrt{0.5(x - 1)} + 4 \), we can start by focusing on the argument of the square root.
The transformation can be broken down as follows:
1. The original function has \( x - 1 \) inside the square root.
2. The transformed function has \( 0.5(x - 1) \) inside the square root.
This means you are effectively replacing \( x - 1 \) with \( 0.5(x - 1) \).
To understand the impact of this change, we can rewrite the transformation:
- Let \( u = x - 1 \). Then, we can set up the transformation as follows:
- From \( v(u) = \sqrt{u} + 4 \) to \( q(u) = \sqrt{0.5u} + 4 \).
3. The factor of \( 0.5 \) in \( 0.5u \) indicates that instead of taking the square root of \( u \), we are taking the square root of half of \( u \). This affects the input \( u \) horizontally.
4. Because \( 0.5u \) is equivalent to compressing the function horizontally, the factor \( 0.5 \) results in a horizontal compression by a factor of \( 2 \) (since you divide the input by \( 0.5 \) to achieve the same output).
Thus, we conclude that the transformation from \( v(x) \) to \( q(x) \) involves:
**Horizontal compression by a factor of 2.**
The transformation can be broken down as follows:
1. The original function has \( x - 1 \) inside the square root.
2. The transformed function has \( 0.5(x - 1) \) inside the square root.
This means you are effectively replacing \( x - 1 \) with \( 0.5(x - 1) \).
To understand the impact of this change, we can rewrite the transformation:
- Let \( u = x - 1 \). Then, we can set up the transformation as follows:
- From \( v(u) = \sqrt{u} + 4 \) to \( q(u) = \sqrt{0.5u} + 4 \).
3. The factor of \( 0.5 \) in \( 0.5u \) indicates that instead of taking the square root of \( u \), we are taking the square root of half of \( u \). This affects the input \( u \) horizontally.
4. Because \( 0.5u \) is equivalent to compressing the function horizontally, the factor \( 0.5 \) results in a horizontal compression by a factor of \( 2 \) (since you divide the input by \( 0.5 \) to achieve the same output).
Thus, we conclude that the transformation from \( v(x) \) to \( q(x) \) involves:
**Horizontal compression by a factor of 2.**