QuestionAugust 24, 2025

63. Multiple Choice Which of the following functions could model a physical process in which and y change in tandem so that their product is always constant? (A) a dilation of the identity function (B) a translation of the identity function (C) a dilation of the reciprocal function (D) a translation of the reciprocal function

63. Multiple Choice Which of the following functions could model a physical process in which and y change in tandem so that their product is always constant? (A) a dilation of the identity function (B) a translation of the identity function (C) a dilation of the reciprocal function (D) a translation of the reciprocal function
63. Multiple Choice Which of the following functions could model a physical process in which and y change in tandem so that their product is always constant? (A) a dilation of the identity function (B) a translation of the identity function (C) a dilation of the reciprocal function (D) a translation of the reciprocal function

Solution
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Answer

(C) a dilation of the reciprocal function Explanation 1. Identify the relationship The problem describes a situation where x \cdot y = k, a constant. This is characteristic of an inverse variation, modeled by the reciprocal function y = \frac{k}{x}. 2. Analyze options (A) and (B) involve the identity function y = x, which does not satisfy x \cdot y = k. (C) A dilation of the reciprocal function y = \frac{k}{x} fits the condition since it maintains the product x \cdot y = k. (D) A translation of the reciprocal function would change the form to y = \frac{k}{x} + c, which does not maintain a constant product.

Explanation

1. Identify the relationship<br /> The problem describes a situation where $x \cdot y = k$, a constant. This is characteristic of an inverse variation, modeled by the reciprocal function $y = \frac{k}{x}$.<br /><br />2. Analyze options<br /> (A) and (B) involve the identity function $y = x$, which does not satisfy $x \cdot y = k$. <br /> (C) A dilation of the reciprocal function $y = \frac{k}{x}$ fits the condition since it maintains the product $x \cdot y = k$.<br /> (D) A translation of the reciprocal function would change the form to $y = \frac{k}{x} + c$, which does not maintain a constant product.
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