Problem

Analyze the displayed graph to determine the long-term ordering of \(Q\) and \(E\) for large positive inputs, justify it from their end behavior and structure, and state what crossings or earlier ordering remain undetermined.

Graph of a downward-opening quadratic Q with a finite maximum and an increasing exponential E that continues upward as x increases.

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long-term = E(x)>Q(x) for every real x; evidence = E→∞ and Q→-∞ as x→∞; structural contrast = exponential growth; undetermined = no earlier-input qualification is needed

long-term = E(x)>Q(x) for all sufficiently large positive x; evidence = Q(x)→-∞ while E(x)→∞ as x→∞; structural contrast = Q has a finite maximum then falls, E keeps growing; undetermined = exact crossing inputs and earlier dominance without equations

long-term = E(x)>Q(x) after the vertex; evidence = the quadratic's vertex is its maximum; structural contrast = E grows; undetermined = exact crossing is the quadratic vertex

long-term = Q(x)>E(x) for large positive x; evidence = quadratics have vertices; structural contrast = Q has a finite maximum while E grows; undetermined = no crossing information

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Analyze the displayed graph to determine the long-term ordering of \(Q\) and \(E\) for large positive inputs, justify it from their end behavior and structure, and state what crossings or earlier ordering remain undetermined.

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