In fact, he was the one who made his father accidentally kill Syndrome.
Jack-Jack is smart and intuitive for an infant as he recognized Syndrome as an enemy and identified the exploding plane as a danger. The few things that can calm him down are his family, specifically his mother. He also becomes enraged if he is given a cookie and is not allowed more. For instance, Jack-Jack loves Mozart's music to the point where he is stimulated by it. Like many infants, Jack-Jack's emotions are unpredictable and since his powers are stimulated by them, being with him when the baby uses his powers becomes very difficult when attempting to pacify or restrain him. He laughs and giggles a lot but cries whenever he is disturbed.