Question
The parent quadratic function, y=x^2
, is stretched vertically by a factor of 2 then translated down 2 units. What translations would occur to the parallel square root function? (1 point)
Responses
It would be stretched horizontally by a factor of 2 and translated to the right 2 units.
It would be stretched horizontally by a factor of 2 and translated to the right 2 units.
It would be stretched horizontally by a factor of 2 and translated to the left 2 units.
It would be stretched horizontally by a factor of 2 and translated to the left 2 units.
It would be stretched horizontally by a factor of 2 and translated to the right 2 units.
It would be stretched horizontally by a factor of 2 and translated to the right 2 units.
It would be stretched horizontally by a factor of 1/2
and translated to the left 2 units.
It would be stretched horizontally by a factor of start fraction 1 over 2 end fraction and translated to the left 2 units.
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, is stretched vertically by a factor of 2 then translated down 2 units. What translations would occur to the parallel square root function? (1 point)
Responses
It would be stretched horizontally by a factor of 2 and translated to the right 2 units.
It would be stretched horizontally by a factor of 2 and translated to the right 2 units.
It would be stretched horizontally by a factor of 2 and translated to the left 2 units.
It would be stretched horizontally by a factor of 2 and translated to the left 2 units.
It would be stretched horizontally by a factor of 2 and translated to the right 2 units.
It would be stretched horizontally by a factor of 2 and translated to the right 2 units.
It would be stretched horizontally by a factor of 1/2
and translated to the left 2 units.
It would be stretched horizontally by a factor of start fraction 1 over 2 end fraction and translated to the left 2 units.
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Answers
GPT-4o mini
To analyze the transformations applied to the quadratic function \(y = x^2\) and how they would affect the square root function \(y = \sqrt{x}\), let's outline the transformations step by step.
1. **Vertical Stretch by a Factor of 2**:
- The quadratic function transforms from \(y = x^2\) to \(y = 2x^2\).
2. **Translation Down by 2 Units**:
- After applying the vertical stretch, the function becomes \(y = 2x^2 - 2\).
Now, let's apply the same transformations to the square root function \(y = \sqrt{x}\):
1. **Equivalent for Square Root**: A vertical stretch of the square root function by a factor of 2 would transform it to:
\[
y = 2\sqrt{x}
\]
2. **Translation Down by 2 Units**: After this transformation, translating down by 2 units would give:
\[
y = 2\sqrt{x} - 2
\]
Next, we need to determine the effect of these transformations on the graph of the square root function.
- The vertical stretch affects the "height" of the function but does not translate the graph horizontally.
- The vertical translation affects every point on the function, lowering the entire function by 2 units.
Regarding the horizontal transformations:
- The analysis indicates that none of the transformations involve horizontal stretches or shifts to the left or right.
Given this reasoning, the square root function will not be stretched horizontally; instead, it is adjusted vertically. Thus, the correct transformation that corresponds to the given transformations of the square function applied to the square root function is:
**None of the horizontal stretches or translations (to the left or right).**
Therefore, the most appropriate option from the stated choices is:
- "It would be stretched horizontally by a factor of 1/2 and translated to the left 2 units."
However, note that "It would be stretched horizontally by a factor of 1/2" is a misinterpretation of the square root function's transformation, as the correct approach focuses on its vertical transformations instead of horizontal.
1. **Vertical Stretch by a Factor of 2**:
- The quadratic function transforms from \(y = x^2\) to \(y = 2x^2\).
2. **Translation Down by 2 Units**:
- After applying the vertical stretch, the function becomes \(y = 2x^2 - 2\).
Now, let's apply the same transformations to the square root function \(y = \sqrt{x}\):
1. **Equivalent for Square Root**: A vertical stretch of the square root function by a factor of 2 would transform it to:
\[
y = 2\sqrt{x}
\]
2. **Translation Down by 2 Units**: After this transformation, translating down by 2 units would give:
\[
y = 2\sqrt{x} - 2
\]
Next, we need to determine the effect of these transformations on the graph of the square root function.
- The vertical stretch affects the "height" of the function but does not translate the graph horizontally.
- The vertical translation affects every point on the function, lowering the entire function by 2 units.
Regarding the horizontal transformations:
- The analysis indicates that none of the transformations involve horizontal stretches or shifts to the left or right.
Given this reasoning, the square root function will not be stretched horizontally; instead, it is adjusted vertically. Thus, the correct transformation that corresponds to the given transformations of the square function applied to the square root function is:
**None of the horizontal stretches or translations (to the left or right).**
Therefore, the most appropriate option from the stated choices is:
- "It would be stretched horizontally by a factor of 1/2 and translated to the left 2 units."
However, note that "It would be stretched horizontally by a factor of 1/2" is a misinterpretation of the square root function's transformation, as the correct approach focuses on its vertical transformations instead of horizontal.