Improving my violin playing?

The "moving your finger back and forth on long notes" that you're talking about is called vibrato. And you don't just move your finger back and forth, you do it very quickly so its almost like a vibration. It is used to give the music expression and feel. In other words, it makes it sound "prettier". It's a bit of an advanced technique and even many professionals cannot do it perfectly so I suggest you don't attempt it until you have a little more experience. As for the key signature, it is basically a code to tell you what kind of notes to play in the song. C major is the most basic signature and all notes are natural in a song in C major. It is indicated if there are no sharp or flat symbols at the beginning of the score. After that is G major, which has F#'s, but all the other notes are natural. G major is indicated in a song as showing one # at the beginning of the score. Then it is D major with only F# and C#, indicated by two #'s at the beginning of the score, and so on. It is the same with flats, one flat (b flat) is F major, indicated by one flat symbol at the beginning of the score, and so on. The order of the sharps is FCGDAEB and the order of the flats is BEAD. If you want help in general with the violin, there are many great tutorials on youtube you can watch.

If i'm honest it seems that your violin is too big for your hands to cope with, so in the future I would maybe try a smaller size. Playing an E on the A string with your 4th finger shouldn't be anything like dislocating your shoulder, that's stupid from your teacher, it should feel natural. Have you tried resting both arms by your side and holding your violin up with just your head, so it rests naturally by your shoulder and cheek? If it can do that, then your left hand should feel less pressure to support the violin and you can focus more on getting the notes, so i'd work on that. Remember that your wrist should be parallel to your forearm when holding the violin properly, not bent upwards against the neck of the violin because that will create tensions. But all this takes a lot of practice and focus, so i'd say get through this exam as best as you can, and then afterwards, before you go on to grade 3, get your teacher to properly sort out these problems, because these are all fundamental to playing the violin well. As you do more grades, harder problems will come up, where it'll be taken for granted that you can hold the violin properly without tiring, and you can get the fourth finger fine without problem, so work on this with your teacher. If she doesn't know what to do, find a new teacher. Good luck :)

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Key signature - go to musictheory.net and look at the relevant lessons.

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There are individual lessons, and suggestions on what you should be learning, and how to practice. There are videos and printable practice schedules.