President Barack Obama walks off the stage after addressing an audience at the University of Queensland in Brisbane, Australia. (Dan Peled/European Pressphoto Agency)

President Obama left Washington a week ago with sagging approval ratings on foreign policy, after a year of crises including the Ebola outbreak, the rise of Islamist militants and violence by Ukrainian separatists.

He’ll return Sunday evening rejuvenated after a swing through the Asia Pacific region that netted a potentially major deal with China on climate change and may provide a renewed sense that the United States still holds sway on the major issues of the day.

Obama’s eight-day, three-nation tour won’t by itself turn around public perception of his global leadership, and the outcomes of the crises that have preoccupied his administration in the past year are very much in doubt. But his trip helped him to rediscover his voice and make a forceful case for his worldview. And his concrete accomplishments provided a rare chance to boast.

In addition to getting China to reduce carbon emissions in a first-of-its-kind agreement, Obama will bring back to Washington a plan with Beijing that could lead to the first significant international tariff reduction in 17 years. He also announced new U.S. commitments — $3 billion to a U.N. fund to help developing countries deal with the effects of global warming and the creation of a new Peace Corps bureau in Burma.

Obama and his aides hope the trip sets the stage for his final two years in office, a time when two-term presidents traditionally turn more attention to foreign policy. The president delivered a 5,400-word speech at the University of Queensland here Saturday — an address billed as a talk about U.S. leadership in the Asia Pacific region, but one that touched on a wide variety of issues: security, economics, trade, climate change, gay rights, women’s rights, health care and human rights.

“Our world is getting smaller,” Obama said, emphasizing that the United States is determined to help shape events across the globe. Standing in front of American and Australian flags, Obama then quoted local writer David Malouf: “In that shrinking of distance that is characteristic of our contemporary world, even the Pacific, largest of oceans, has become a lake.”

Asked about the broad nature of Obama’s remarks, a senior administration official said the goal was to “underscore or highlight the agenda for the next two years. The bulk of the speech covered areas where we are going to continue to invest.”

Obama’s critics have said the president has overseen a foreign policy that has diminished the role of the United States in global affairs and, as a consequence, ­allowed other actors — Russian President Vladi­mir Putin, Chinese President Xi Jinping, the Islamic State — to feel emboldened to expand their power and influence.

But the president took on Xi directly during the first stop on his trip, a three-day visit to Beijing, and more than held his own. In addition to the climate and tariff deals, the United States and China hammered out an agreement to relax short-term visa restrictions for students, tourists and businesses. Obama’s first bilateral meeting in Beijing was a symbolic rebuke to China — he met with Indonesian President Joko Widodo, elected this summer on an anticorruption platform in a nation transitioning to democracy.

Obama also took the opportunity at the 21-nation Asia-Pacific Economic Cooperation summit in Beijing to talk with Putin. They chatted informally three times on the sidelines of the sessions, speaking about Ebola, the Islamic State and the fighting in eastern Ukraine, for a total of 15 to 20 minutes, White House aides said. The aides did not elaborate on the talks, the first face-to-face interaction between Obama and Putin in five months.

In his speech at the University of Queensland, Obama acknowledged that some have looked skeptically at his administration’s “pivot to Asia,” a strategic shift of military, economic and diplomatic resources away from the Middle East and Europe.

People question “whether America has the staying power to sustain it,” Obama said. “And it’s true that in recent years, pressing events around the world demand our attention. As the world’s only superpower, the United States has unique responsibilities that we gladly embrace.”

President Obama, right, speaks with Xi Jinping, China's president, before a family photograph at the Group of 20 (G-20) summit in Brisbane, Australia, on Nov. 15. (Ian Waldie/Bloomberg)

He ticked off the international crises of the past year. “So, yes, we have a range of responsibilities,” he said. “It’s a burden we gladly shoulder.”

In Burma, Obama gave voice to American values such as free speech, democratic elections and tolerance for religious and ethnic diversity — a theme that human rights advocates say has gotten short shrift on Obama’s international agenda. He was greeted warmly by Burmese Nobel Peace laureate Aung San Suu Kyi, who hugged him as he stood with her on her porch to address the media. And Obama met with Vietnamese Prime Minister Nguyen Tan Dung in hopes of deepening ties with a fast-growing country that, like Burma and other Southeast Asian nations, has felt threatened by a rising China.

Obama aides said the larger goal of the week was to make the point that the United States’ Asia pivot is not only aimed at increasing America’s influence in Asia but also is intended to enlist U.S. allies — and even some traditional adversaries — to increase Asia’s role in global problem-solving. Obama praised China, for example, for contributing $130 million to help fight Ebola.

Feeling good about the trip, White House aides distributed ­e-mails full of positive news coverage, and the president appeared energized as he took the stage at Queensland to the soundtrack from his 2008 campaign, including Brooks and Dunn’s “Only in America.”

In a bid to charm his audience, Obama noted that he had sent word last year about a study on climate change conducted by the university to his 31 million Twitter followers. (Obama now has almost 50 million followers).

“Just bragging a little bit,” the president said, though he acknowledged that his influence only goes so far.

“I don’t think that’s quite as much as Lady Gaga,” he noted, “but it’s pretty good.”