Another way to think about it is, an orbit around a single body is entirely defined[0] by your current position and your instantaneous velocity. Six numbers. Where you are, which direction you're going, and how fast. Such orbits are closed, which means wherever you are right now, you'll return to that exact point later.
So if you're at point P in your orbit and suddenly kinetic energy was added, your velocity changes in some arbitrary way. While you aren't sure where you'll go and when you'll get there, as long as the new (position, velocity) pair still defines a closed orbit around the body, you can be damn sure you'll return to the exact point where you are now.
From this follows that the lowest point of your new orbit cannot be higher than where you are now, and the highest point of your orbit cannot be lower.
--
[0] - Ignoring gravitational effects of other bodies, residual drag, magnetic fields, solar pressure, etc. None of these matter on the scale of days to months.
So if you're at point P in your orbit and suddenly kinetic energy was added, your velocity changes in some arbitrary way. While you aren't sure where you'll go and when you'll get there, as long as the new (position, velocity) pair still defines a closed orbit around the body, you can be damn sure you'll return to the exact point where you are now.
From this follows that the lowest point of your new orbit cannot be higher than where you are now, and the highest point of your orbit cannot be lower.
--
[0] - Ignoring gravitational effects of other bodies, residual drag, magnetic fields, solar pressure, etc. None of these matter on the scale of days to months.