Choose which graph best represents the curve [math: y=4+1x−2]y=4 + \dfrac{1}{x-2} .

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BlackTom AI · Accepted Answer

The function is y = 4 + 1/(x - 2). This is a hyperbola (a reciprocal function) that has been shifted-right by 2 and then shifted up by 4. Key features to identify: - Vertical asymptote at x = 2, because the denominator x - 2 becomes zero there. - Horizontal asymptote at y = 4, because as x moves far from 2, 1/(x - 2) approaches 0 and y approaches 4. - The graph is a rectangular hyperbola with the center located at the intersection of the asymptotes, which is (2, 4). - In the region x > 2, the term 1/(x - 2) is positive, so y > 4; this yields the branch that lies above the horizontal asymptote to the right of x = 2. - In the region x < 2, the term 1/(x - 2) is negative, so y < 4; this yields the branch that lies below the horizontal asymptote to the left of x = 2. Because the provided options include 'Graph shown in the image' but do not display or describe the actual graphs, we can’t compare specific images here. However, the correct graph must display: - A vertical asymptote at x = 2, - A horizontal asymptote at y = 4, - A left branch entirely below y = 4 for x < 2, and a right branch entirely above y = 4 for x > 2. If you have access to the individual graph options, choose the one that exhibits these asymptotes and branch positions. Any graph lacking x = 2 as a vertical asymptote or lacking y = 4 as a horizontal asymptote, or placing branches in the wrong relative positions, would be incorrect.

FNDN 2024 E2 Challenge Week 2 Examinations

Choose which graph best represents the curve [math: y=4+1x−2]y=4 + \dfrac{1}{x-2} .

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