Question
Analyze the graphs b(x)=23x−9−−−−√+1
and k(x)=23107(x−9)−−−−−−−−√+1
, and determine the transformation that is occurring.(1 point)
Responses
horizontal stretch by a factor of 0.8
horizontal stretch by a factor of 0.8
horizontal stretch by a factor of 710
horizontal stretch by a factor of 7 tenths
horizontal compression by a factor of 0.7
horizontal compression by a factor of 0.7
horizontal compression by a factor of 107
and k(x)=23107(x−9)−−−−−−−−√+1
, and determine the transformation that is occurring.(1 point)
Responses
horizontal stretch by a factor of 0.8
horizontal stretch by a factor of 0.8
horizontal stretch by a factor of 710
horizontal stretch by a factor of 7 tenths
horizontal compression by a factor of 0.7
horizontal compression by a factor of 0.7
horizontal compression by a factor of 107
Answers
GPT-4o mini
To analyze the functions \( b(x) = \sqrt{23x - 9} + 1 \) and \( k(x) = \sqrt{23107(x - 9)} + 1 \), we can look at how the transformations affect the horizontal dilation/compression of the graphs.
The base function in both cases is the square root function \( \sqrt{x} \). The transformations that involve horizontal stretching or compression come from the coefficients of \( x \) inside the square root.
1. For \( b(x) = \sqrt{23x - 9} + 1 \):
- The term inside the square root is \( 23x - 9 \).
- The coefficient of \( x \) is 23, which suggests a horizontal compression. The factor associated with horizontal transformations is the reciprocal of the coefficient, which would be \( \frac{1}{23} \).
2. For \( k(x) = \sqrt{23107(x - 9)} + 1 \):
- The term inside the square root can be rewritten as \( 23107x - 23107 \cdot 9 \).
- The coefficient of \( x \) is 23107, which also suggests a horizontal compression. The factor associated with this transformation would be \( \frac{1}{23107} \).
Therefore, we can summarize the transformations:
- Both functions represent horizontal compressions because their coefficients are greater than 1.
The correct transformation from the options given appears to be a horizontal compression, and specifically, it factors towards \( 0.7 \) for \( b(x) \) and a very significant compression for \( k(x) \).
However, analyzing the options provided, the transformations can imply that we have a horizontal compression by a factor of \( \frac{1}{7} \), but without further clarification of exactly what factor the question is asking in terms of \( 0.7 \).
Thus, the most likely answer from your options depends on how they are formulated.
The response indicating a "horizontal compression by a factor of 0.7" seems to align with the overall idea when you consider that both functions show signs of compression. However, if we want to be precise with the numbers:
- The closest response that aligns with the nature of the transformations would be "horizontal compression by a factor of 0.7."
If the intent is to be precise with the different transformations, please clarify if you are looking for a specific numeric transformation or general behavior.
The base function in both cases is the square root function \( \sqrt{x} \). The transformations that involve horizontal stretching or compression come from the coefficients of \( x \) inside the square root.
1. For \( b(x) = \sqrt{23x - 9} + 1 \):
- The term inside the square root is \( 23x - 9 \).
- The coefficient of \( x \) is 23, which suggests a horizontal compression. The factor associated with horizontal transformations is the reciprocal of the coefficient, which would be \( \frac{1}{23} \).
2. For \( k(x) = \sqrt{23107(x - 9)} + 1 \):
- The term inside the square root can be rewritten as \( 23107x - 23107 \cdot 9 \).
- The coefficient of \( x \) is 23107, which also suggests a horizontal compression. The factor associated with this transformation would be \( \frac{1}{23107} \).
Therefore, we can summarize the transformations:
- Both functions represent horizontal compressions because their coefficients are greater than 1.
The correct transformation from the options given appears to be a horizontal compression, and specifically, it factors towards \( 0.7 \) for \( b(x) \) and a very significant compression for \( k(x) \).
However, analyzing the options provided, the transformations can imply that we have a horizontal compression by a factor of \( \frac{1}{7} \), but without further clarification of exactly what factor the question is asking in terms of \( 0.7 \).
Thus, the most likely answer from your options depends on how they are formulated.
The response indicating a "horizontal compression by a factor of 0.7" seems to align with the overall idea when you consider that both functions show signs of compression. However, if we want to be precise with the numbers:
- The closest response that aligns with the nature of the transformations would be "horizontal compression by a factor of 0.7."
If the intent is to be precise with the different transformations, please clarify if you are looking for a specific numeric transformation or general behavior.