You are talking about a cadential 64 chord. A cadential 64 chord is one of the main four types of second-inversion triads. The four types include:
- Passing 64 (Ex. I6 - V64 - I)
- Arpeggiated 64 (Ex. I - 6 - 64 - 53)
- Auxiliary 64 (Ex. I - IV64 - I)
- Cadential 64 (Ex. I64 - V - I)
A cadential 64 is one of the most common dominant embellishment chords. We notate them as I64. It is used to emphasize the dominant-function V chord which must resolve to root position I. Such emphasization is called dominant embellishment.
Here is an SATB-type harmony exercise example: (SAT on treble clef, B on bass clef)
We can notice the authentic cadence at the end. It's extremely important to know that such chords have dominant function like V, and not tonic function. Notice the bass is scale degree 5 and the chord occurs after a subdominant-function IV chord. Also, please please PLEASE don't write an authentic cadence as I64 - I.
It's also very important to know that such dominant expansions are incredibly common to occur in authentic cadences that build up to dramatic climaxes. For example, the finale of Mahler's Symphony No. 1 contains such a passage:
At first, in the dynamic state of mf, we see a tonic-function I, and then a subdominant-function vi chord. Then we see a cadential I64, which is when a crescendo happens. Notice how the chord stretches out for 31/2 bars while the crescendo is in progress. Then we see an inverted V11. Realize how the crescendo extends from I64 to V11(inv.), the two dominant-function chords. This resolves to a tonic-function I in root position, which is when the dynamic finally changes into fff. It is extremely dramatic!
My answer will end up getting too long if I add more information, so I will give a link to https://en.wikipedia.org/wiki/Second_inversion.