If the broader Tet offensive revealed chaos where the government was trying to project control, Adams’s photo made people question whether the United States was fighting for a just cause. Together, they undermined the argument for the war on two fronts, leading many Americans to conclude not only that it could not be won, but also that, perhaps, it shouldn’t be.

The photo “fed into a developing narrative in the wake of the Tet offensive that the Vietnam War was looking more and more like an unwinnable war,” said Robert J. McMahon, a historian at the Ohio State University. “And I think more people began to question whether we were, in fact, the good guys in the war or not.”

A police chief had fired a bullet, point-blank, into the head of a handcuffed man, in likely violation of the Geneva Conventions. And the official was not a Communist, but a member of South Vietnam’s government, the ally of the United States.

“It raised a different kind of question to Americans than whether or not the war was winnable,” said Christian G. Appy, a professor of history at the University of Massachusetts, Amherst. “It really introduced a set of moral questions that would increasingly shape debate about the Vietnam War: Is our presence in Vietnam legitimate or just, and are we conducting the war in a way that is moral?”

In the months after the Tet offensive, public opinion shifted more rapidly than at any other point in the war, Dr. McMahon said. Adams’s photo won a Pulitzer Prize, and Time magazine called it one of the 100 most influential ever taken.

“You can talk about ‘the execution photograph from the Vietnam War,’ and not just the generation who lived through it but multiple generations can call that image to mind,” said Susan D. Moeller, the author of “Shooting War: Photography and the American Experience of Combat,” and a professor of media and international affairs at the University of Maryland. “It was immediately understood to be an icon.”

Yet the decisions on how to display this photo and other graphic Vietnam war imagery were matters of debate in the newsroom of The Times. “I remember certain pictures,” the influential photo editor John G. Morris, who died at 100 last year, said, “which I was just determined to get on page one.” This was one, as was the 1972 picture of a naked 9-year-old, Phan Thi Kim Phuc, fleeing a napalm attack. That ran at the bottom of the page.