As an illustration, here's part of an AFM chart for Takeoff Field Length, showing how temperature and altitude interrelate - you enter the chart on the LHS with the ambient temp, read across to the pressure altitude, then downwards into another part of the chart (for weight, in fact). The further to the right you are when you go 'down', the longer the TOFL.
The distinct non-linearity on the chart lines is associated with engine flat-rating limits coming into effect - you can see that above the flat rating, the effect is roughly one major division of distance per 5 deg C, and below the flat rating its more like 20-30 deg C for the same effect.