About induced current
The question asks for the current, not the voltage.

The voltage (emf) generated would follow a straight line (not a curve) as the rate at which the area grows is proportional to the length of the wire - and the length of the wire grows linearly.

The current that flows as a result of this depends on the resistance of the loop. The circumference of the loop grows linearly with the horizontal position of the wire (similar triangles - when the wire has moved twice as far to the right, all the sides are twice as big).

The current is voltage divided by resistance. Both grow linearly; their ratio is constant.