How did the test offensive affect the perceptions of the vietnamese war within the united states

The Tet Offensive of 1968 (as well of US media coverage of the Tet Offensive) had strong negative effect on perceptions of the Vietnam War in the United States.  It became clear to the American public that the US was not making progress in the war, and that the US government had not been honest with the public about US ability to win the war.

Additional explanation:

By 1966, 93% of American households owned a television set. Most Americans now were getting their news from television, and Walter Cronkite was the most trusted anchorman on TV news.  When the North Vietnamese and the Viet Cong launched the Tet Offensive in 1968, Cronkite, known as "the most trusted man in America," offered a television editorial that shaped the nation's mood.  Cronkite said in that broadcast on February 27, 1968:  "It seems now more certain than ever, that the bloody experience of Vietnam is to end in a stalemate. To say that we are closer to victory today is to believe, in the face of the evidence, the optimists who have been wrong in the past.  To say that we are mired in stalemate seems the only realistic, if unsatisfactory conclusion. ...It is increasingly clear to this reporter that the only rational way out then will be to negotiate, not as victors, but as an honorable people who lived up to their pledge to defend democracy, and did the best they could."

In addition to Cronkite's reporting, there were also photo and video images coming from the war front that showed a "credibility gap" between what the US government had been saying about the war and what was actually happening there.

US public perception of the Vietnam War increasingly turned away from support for the war effort.

the idealization of the rural landscape ; admiration for the beauty of nature ; focus on an individual's inner concerns