Dating using measurements of the isotopic $^$C/$^$C ratio can be done even for quite recent dates, but the principle is different to the standard radiocarbon dating.
The method is known as bomb pulse dating and owes its efficacy to the large amount of $^$C that entered the atmosphere as a result of the testing of nuclear weapons.
However, if I interpret the plot above correctly, because the natural $^$C abundance was decreasing between 15 (and perhaps before this too?
When an organism dies, it contains a ratio of carbon 14 to carbon 12.
As the carbon 14 decays with no possibility of replenishment, the ratio decreases at a regular rate. The measurement of carbon 14 decay provides an indication of the age of any carbon-based material.
The dating method is based on the fact that carbon is found in various forms, including the main stable isotope (carbon 12) and an unstable isotope (carbon 14).
Through photosynthesis, plants absorb both forms from carbon dioxide in the atmosphere.
In 1960, Libby was awarded the Nobel Prize in Chemistry for this work.