If she had stuck with basketball, Sophia Smith would have been an excellent Division I player. That assessment comes from former University of Wyoming basketball player Kenny Smith, who happens to be her father.

“She would have been a D-I player,” he said. “Not to brag, she would have been on the high end in D-I.”

When she dropped basketball in favor of soccer, Sophia Smith was a ninth-grader at Fossil Ridge High School in Fort Collins, Colo. Her dad, an assistant coach on the girls basketball team, was heartbroken by Sophia’s choice.

“It caught me by surprise,” he said, “especially after all the time and effort she had put into basketball. But seeing how happy and successful she was in soccer, it was a smooth transition.”

She explains her love of soccer like this: “I like the freedom that you have when you’re playing, to do what you want. There’s no set plays or structure, except on free kicks and corners.”

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One look at her on a soccer field and it’s clear why she’s been on U.S. national teams since she was 14. She can maneuver with the ball in tight quarters, she’s extremely fast and knows how to hit the back of the net. She became a key part of Stanford’s defending national champions the moment she arrived on the Farm.

Sunday’s match Who: No. 2 USC (9-0-1, 2-0 Pac-12) at No. 1 Stanford (9-0-1, 2-0) When: 1 p.m. TV: Pac-12 Networks Smith game log Date Opp. G A 8/17 at UC Davis DNP 8/24 USF 1 1 8/30 at BYU 0 0 9/2 at Minn. 0 1 9/7 Notre Dame 1 0 9/9 N. Carolina 1 0 9/13 Cal Poly 0 0 9/16 Santa Clara 1 0 9/21 Arizona 0 0 Thu. UCLA 1 0 5 2 Source: Stanford athletics

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Consider her first few collegiate games (she missed one to play in the Under-20 World Cup in France):

It took her 17 minutes to score in her debut against USF; she later had an assist. Then came an assist on the game-winning goal against Minnesota, a goal against Notre Dame, a did-you-see-that 20-yard strike against then-No. 2 North Carolina and Stanford’s goal in a 1-1 draw with Santa Clara.

She did all this before attending her first Stanford class.

She scored another goal in Thursday night’s 3-2 win over No. 16 UCLA. With five goals and 12 points, she is second on a loaded team in scoring.

She’ll be one of the players to watch when the No. 1 Cardinal (9-0-1, 2-0 Pac-12) host No. 2 USC (9-0-1, 2-0) at 1 p.m. Sunday. The Cardinal are riding a program-record 32-game unbeaten streak.

Stanford’s other top scoring threats are forward Catarina Macario (six goals, 16 points), midfielder Jordan DiBiasi (three, 11) and forward Madison Haley (three, eight).

Lorne Donaldson, who coached Smith for three years with the Real Colorado club team in suburban Denver, said he’s surprised by her performance in college. That is, he’s surprised she hasn’t scored more goals.

“You have to give her a lot of credit,” he said. “She spent a lot of time with me, hours working on different types of finishes. She hits the ball harder than most guys.”

When Smith fired her goal against North Carolina, first dribbling past a defender on the left side, creating space and bending the ball just inside the far post past a diving keeper, many onlookers’ jaws dropped.

Donaldson had witnessed that same type of shot many times. He wasn’t surprised by that goal, but Smith said she was.

“I practice putting a little curve on it, but I don’t know how it bent like that. I hit it with the instep. I didn’t purposely curve it.”

In one club game, Donaldson said, “She hit the ball so hard, it hit the crossbar, came back, hit the keeper in the head — and knocked her out. She’s a tremendous striker of the ball.”

Donaldson was coaching the U20 team for his native Jamaica when his team ran into Smith and the U.S. in the CONCACAF tournament. Smith scored the first goal on a powerful free kick, then assisted on the decisive goal in a 2-1 win that kept Jamaica from going to the World Cup.

Afterward, she told him, “I told you I was going to score.” Donaldson said, “There was no sympathy. She wasn’t going to hold back.”

The first time Stanford head coach Paul Ratcliffe saw her, Smith was in ninth grade. “It’s rare to see a player who scored every time I saw her,” he said. “If she was trying to impress me, it worked very well.”

She combines “great speed with great finishing ability,” he said. He called her “a really sweet person off the field, very soft-spoken and caring. On the field, she’s very aggressive but respectful.”

Smith’s two older sisters played basketball, including Savannah, a fifth-year senior at Northern Colorado who was Big Sky Conference MVP last season while leading her team to its first NCAA Tournament berth .

Sophia, however, fell in love with soccer in sixth grade. Six years later, she scored goals for the U.S. U18s, U20s and U23s — an age-group hat trick, all in one year.

She and teammate Naomi Girma, a freshman defender from San Jose, have been pals since they met at the U14 camp and have been through the U17 and U20 ranks together.

“I think we’re up to 10 countries so far,” Girma said of their travels. “We went to the Eiffel Tower; that was probably the highlight. We went zip-lining in Trinidad, and we both loved that. She’s pretty adventurous and loves doing new things like that.”

A fan of Brazil’s Marta, perhaps the best female player of all time, Smith hopes to play for the U.S. against her in next summer’s World Cup in France. Whether she makes it or not, basketball will have to do without her.

Tom FitzGerald is a San Francisco Chronicle staff writer. Email: tfitzgerald@sfchronicle.com Twitter: @tomgfitzgerald