Question

Which of the following uses the properties of exponents to correctly rewrite v(x)=( 7^5x-3/28)^1/2 +2to determine the vertical stretch or shrink of the function v(x) from its parent function?
A. Vertical stretch by a factor of 28
B. Vertical stretch by a factor of 2
C. Vertical shrink by a factor of 1/28
D. Vertical shrink factor by 1/28 raised to the power of 1/2
​

Answer 1

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.

1. **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.

2. **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.

3. **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 \( \frac{1}{2} \)**.