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

Question

BlackTom AI · Accepted Answer

To identify the correct graph, we start from the equation y = 4 − 1/(x − 2).

- First, determine the asymptotes: the term 1/(x − 2) has a vertical asymptote at x = 2, and the constant 4 shifts the whole curve up so that the horizontal asymptote is y = 4.

- Next, examine the behavior on either side of the vertical asymptote:
  • As x approaches 2 from the right (x → 2+), (x − 2) is a small positive number, so 1/(x − 2) is a large positive number, and y = 4 − (large positive) tends to −∞. Therefore, the right-hand branch heads downward toward −∞ near x = 2 and approaches the horizontal line y = 4 from below as x → ∞.
  • As x approaches 2 from the left (x → 2−), (x − 2) is a small negative number, so 1/(x − 2) is a large negative number, and y = 4 − (negative large) tends to +∞. Thus, the left-hand branch shoots upward to +∞ near x = 2 and approaches y = 4 from above as x → −∞.

- From these trends, the correct graph must show:
  • a vertical asymptote at x = 2,
  • a horizontal asymptote at y = 4,
  • a left branch that lies above y = 4 for x < 2 and heads toward +∞ near x = 2, then curves toward y = 4 from above as x → −∞,
  • a right branch that lies below y = 4 for x > 2 and heads toward −∞ near x = 2, then approaches y = 4 from below as x → ∞.

- Now evaluate the options:
  • Option A (first graph) displays a vertical asymptote at x = 2 and a horizontal asymptote at y = 4, with the left-hand branch located above y = 4 and the right-hand branch below y = 4, consistent with the derived behavior. This matches y = 4 − 1/(x − 2).
  • Option B (second graph) shows asymptotes at different locations (e.g., a vertical line at a different x-value) and the branches positioned inconsistently with the required y = 4 horizontal asymptote, so it does not represent the given equation.
  • Option C (third graph) also places the asymptotes and branches in incorrect quadrants or wrong side of the horizontal line y = 4, contradicting the sign behavior of the function.
  • Option D (fourth graph) likewise misplaces the vertical asymptote or flips the branch positions relative to y = 4, failing to depict the correct end-behavior.

- In summary, the correct graph is the one that shows a vertical asymptote at x = 2, a horizontal asymptote at y = 4, with the left branch above y = 4 and the right branch below y = 4, approaching y = 4 from the appropriate sides as x moves away from 2.

FNDN 2025 O1 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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