To rewrite the function \( v(x) = \left( \frac{7^{5x - 3}}{28} \right)^{1/2} + 2 \) and analyze it for its vertical stretch or shrink from its parent function, we can break it down step by step.
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Simplification of the expression: We know that \( 28 = 7^2 \cdot 4 \) (though this isn't immediately necessary). Since the term under the square root involves a division by \( 28 \), we can express it in terms of exponents: \[ v(x) = \left( \frac{7^{5x-3}}{7^2 \cdot 4} \right)^{1/2} \] However, it's simpler to focus directly on the impact of that division.
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Identifying the vertical transformation: The term \( \frac{7^{5x-3}}{28} \) before raising to the \( \frac{1}{2} \) can be considered as a vertical shrink since we are dividing by \( 28 \). Specifically, dividing a function by a value larger than 1 generally leads to vertical shrinkage.
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Vertical shrink factor:
- When we take \( \frac{1}{28} \), this represents a vertical shrink by that factor. However, since the expression is under a square root, we need to take the square root of the shrink factor as well. Thus, when we raise \( \frac{1}{28} \) to the power of \( \frac{1}{2} \), we change the scale of shrinkage.
- The shrink factor from \( 28 \) becomes \( \frac{1}{\sqrt{28}} \).
Now, looking at the choices you provided:
- A. Vertical stretch by a factor of 28 - Incorrect.
- B. Vertical stretch by a factor of 2 - Incorrect.
- C. Vertical shrink by a factor of \( \frac{1}{28} \) - Partially correct, but doesn't consider the square root.
- D. Vertical shrink factor by \( \frac{1}{28} \) raised to the power of \( \frac{1}{2} \) - Correct, as it includes the impact of the square root.
Thus, the correct answer is D. Vertical shrink factor by \( \frac{1}{28} \) raised to the power of \( \frac{1}{2} \).
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