Ceirin’s teacher has promised a quiz for the next day, so Ceirin calls Adel to review what they had done in class. “Suppose I have $y = \text{sin}\left(2x\right)$,” says Ceirin, “what will its graph look like?”

“It will be horizontally compressed by a factor of $2$,” replies Adel, “so the period must be $π$.”

“Okay, now let’s say I want to shift it $1$ unit to the right. Do I just subtract $1$ from $x$, like always?”

“I think so,” says Adel, “but let’s check on the graphing calculator.” They proceeded to check on their calculators. After a few moments they both speak at the same time.

“Rats,” says Ceirin, “it isn’t right.”

“Cool,” says Adel, “it works.”

When they arrive at school the next morning, they compare the equations they had put into their graphing calculators while they talked on the phone. One has
$y = \text{sin}\left(2x – 1\right)$, while the other had $y = \text{sin}2\left(x – 1\right)$.

Which equation is correct? Had they both subtracted $1$ from $x$? Explain. Describe the rule for shifting a graph $1$ unit to the right in a way that avoids this confusion.