How does turbofan engine performance depend on speed and density altutude?
The dependence on density is straightforward: Thrust depends on the mass of air pushed out at the back, and this is changing linearly with density if the speeds don't change.
$$T \propto \frac{1}{\rho}$$
The dependency on speed has two factors, the intake precompression and the speed difference between flight speed and exhaust speed. First intake precompression:
$$T \propto \left(\frac{\kappa-1}{2}\cdot Ma^2\cdot\left(1-\left(\frac{Ma_{intake}}{Ma_{\infty}}\right)^2\right) + 1\right)^{\frac{\kappa}{\kappa-1}}$$
or, if we insert $\kappa = 1.405$ for diatomic gasses:
$$T \propto \left(0.2025\cdot Ma^2 \cdot\left(1-\left(\frac{Ma_{intake}}{Ma_{\infty}}\right)^2\right) + 1\right)^{3.469}$$
The term which captures the thrust proportionality to the speed difference looks like this:
$$T \propto \frac{v_{nozzle} - v_{\infty}}{v_{nozzle}}$$
Please see this answer for an explanation of the variables. The plot below shows how little the intake precompression matters at subsonic speed, but how much it does at higher Mach numbers. Note that the plot sets the pressure at the engine face to 1 when flight speed equals the assumed intake speed of Mach 0.5. This pressure recovery lets thrust climb above the static value once Concorde flies supersonic.



Explanation of the plot: This shows relative ideal intake pressure over speed, assuming that the speed at the compressor face is Mach 0.5. Note that in static conditions the air needs to be accelerated, so the intake pressure is only 84% of ambient pressure, and at Mach 0.85, the maximum speed of airliners, the intake pressure is 1.37 times higher than ambient pressure. But at supersonic speed things take really off: Pressure recovery for the Concorde was already 6 at Mach 2.0, and for the SR-71 it was 40 at Mach 3.2.

If the engine has no variable nozzle, it is fair to assume that the nozzle speed will not change over the flight speed if the throttle setting is constant. If the nozzle speed is low (= high bypass ratio), the drop over speed is steeper, which explains the difference between turbojets and turbofans.