We had our very first ultrasound today! How accurate are they at telling your due date?

Elicia • Hi! (-: My name is Elicia. Wife. Momma to a precious 2 yr old. Expecting 7/13
Today we had our very first ultrasound for baby #2. The ultrasound technician said we were 6 weeks and about 1 day... Which isn't accurate to the first day of my last period, it actually says I got pregnant before my period even started.. Not many days off.. Three days before the first day of my last period is when they are calculating I got pregnant (based on the 6 wks.)
I am curious if this could be severely off.. Has anyone ever been told they were 6wks and found out later it was more like 9 or 10wks or  would that be a huge difference on the ultrasound? She said I had to be 6wks because you could see the heart fluttering..
Thanks! (=