Plunging temperatures this week will present Muriel E. Bowser (D) with one of her first big challenges as D.C. mayor: Will she be able to shelter all of the homeless families that come to the city for help?

Family homelessness is on track to increase 16 percent in the District this year, according to a recent analysis by Kate Coventry of the DC Fiscal Policy Institute, meaning more than 840 families will seek shelter. Even as the city has rented out two motels, questions still remain as to whether it is prepared to fulfill its legal obligation to provide shelter for its most vulnerable families when the temperature dips below freezing.

Bowser has inherited a stable position to avoid the travails of last winter, when so many homeless families sought city services that officials had not choice but to place them in motels outside the District. According to a Jan 5. count from the Community Partnership for the Prevention of Homelessness, 459 families were in a shelter, about 20 percent fewer than at this point last year. And about 200 empty units are available for homeless families in the District, said Dora Taylor, spokeswoman for the Department of Human Services.

Nonetheless, local advocates fear that those extra units might not be adequate to deal with an influx expected by month’s end. The city’s “exit rate” — it’s ability to move families out of shelters and into housing voucher program, so other families can move into shelters — has begun to slip, raising concern again that the city might run out of space.

In its final months, the Vincent C. Gray administration — after being hammered by many, including Bowser, for not doing enough to address the homeless crisis — increased its efforts. City officials focused more on prevention services and speeding up the rate at which they placed homeless families into more stable housing. The exit rate jumped from about 40 to 64 per month, inching toward the optimal rate of 100 per month.

But Coventry and other advocates were alarmed during a meeting last month when officials disclosed that the exit rate had begun to slip. Since Nov. 1, 79 families have left the city shelter system.

“Things are better than they were last year,” Coventry said. “But we still might not have enough.”

Meanwhile, legal advocates are concerned that the city’s attempts to keep numbers low have resulted in families unfairly being denied shelter. Case managers have been trained to exhaust all options before placing families into the shelter system, often encouraging them to seek the help of friends and relatives.

Amber Harding, a staff lawyer at the Washington Legal Clinic for the Homeless, said she had spent hours trying to help two families who had been denied shelter multiple times.

“They are asking families to go over so many hurdles that it seems impossible to get shelter without the help of a lawyer,” she said. “I’m hopeful that this new administration will admit that there is a problem.”

At a news conference last Thursday, Brenda Donald, the city’s new deputy mayor for health and human services, said she is confident that her staff will be able to reduce homelessness. Officials said the Bowser and Gray administrations are having ongoing conversations about best practices.

Said LaToya Foster, Bowser’s spokeswoman: “This issue is a top priority for the mayor. Whatever has been done in the past, she wants to make sure that she does it better.”