Brass instruments offer volume and power but are lacking in the realm of range, both above and below the staff. The average brass instrument is lucky to hit an octave and a half, while clarinets, for instance, can play three and a half octaves with ease.
The key to expanding your range—and generally improving as a brass player—is to work on your embouchure and breathing. Practice your scales, gradually working your way up to the notes above the register. For higher notes, tighten up your embouchure and support your sound with plenty of air. For those notes below a low C, avoid loosening your embouchure. Instead, drop your jaw while keeping your embouchure comfortably tight.
If you have trouble playing notes on either end of the staff, don’t sweat it. Your range expands as you play, and playing high or low isn’t as important as playing with good tone and a full sound.