Question
Analyze the effect on the x- and y-values of the original function f(x)=x3
when it is stretched vertically by a factor of a that is greater than 0 but less than 1. (1 point)
Responses
The x-values remain the same, and the y-values increase when multiplied by a factor of a.
The x -values remain the same, and the y -values increase when multiplied by a factor of a .
The x-values remain the same while the magnitudes of the y-values decrease when multiplied by a factor of a.
The x -values remain the same while the magnitudes of the y -values decrease when multiplied by a factor of a .
The x-values decrease when multiplied by a factor of a, and the y- values remain the same.
The x -values decrease when multiplied by a factor of a , and the y - values remain the same.
The x-values increase by a factor of a, and the y-values decrease when multiplied by a factor of a.
when it is stretched vertically by a factor of a that is greater than 0 but less than 1. (1 point)
Responses
The x-values remain the same, and the y-values increase when multiplied by a factor of a.
The x -values remain the same, and the y -values increase when multiplied by a factor of a .
The x-values remain the same while the magnitudes of the y-values decrease when multiplied by a factor of a.
The x -values remain the same while the magnitudes of the y -values decrease when multiplied by a factor of a .
The x-values decrease when multiplied by a factor of a, and the y- values remain the same.
The x -values decrease when multiplied by a factor of a , and the y - values remain the same.
The x-values increase by a factor of a, and the y-values decrease when multiplied by a factor of a.
Answers
Answered by
GPT-4o mini
When the function \( f(x) = x^3 \) is stretched vertically by a factor of \( a \) (where \( 0 < a < 1 \)), the new function can be expressed as \( g(x) = a \cdot f(x) = a \cdot x^3 \).
Now, let's analyze the effect on the x- and y-values:
1. **Effect on x-values**: The x-values remain unchanged because the transformation does not affect the input to the function. We simply take the same x-values from the original function \( f(x) \).
2. **Effect on y-values**: The y-values of the original function \( f(x) \) are given by \( f(x) = x^3 \). When we multiply by \( a \), the new y-values become \( g(x) = a \cdot x^3 \). Since \( 0 < a < 1 \), this means that the magnitudes of the y-values decrease. For positive y-values, this results in a contraction towards the x-axis, and for negative y-values, the y-values will be less negative (closer to zero).
Putting it all together:
- The x-values remain the same.
- The magnitudes of the y-values decrease when multiplied by a factor of \( a \).
Thus, the correct response is:
**The x-values remain the same while the magnitudes of the y-values decrease when multiplied by a factor of a.**
Now, let's analyze the effect on the x- and y-values:
1. **Effect on x-values**: The x-values remain unchanged because the transformation does not affect the input to the function. We simply take the same x-values from the original function \( f(x) \).
2. **Effect on y-values**: The y-values of the original function \( f(x) \) are given by \( f(x) = x^3 \). When we multiply by \( a \), the new y-values become \( g(x) = a \cdot x^3 \). Since \( 0 < a < 1 \), this means that the magnitudes of the y-values decrease. For positive y-values, this results in a contraction towards the x-axis, and for negative y-values, the y-values will be less negative (closer to zero).
Putting it all together:
- The x-values remain the same.
- The magnitudes of the y-values decrease when multiplied by a factor of \( a \).
Thus, the correct response is:
**The x-values remain the same while the magnitudes of the y-values decrease when multiplied by a factor of a.**
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