What is the primary effect of increasing boom length and adding directors to a Yagi antenna?
Why A Yagi's forward gain comes mainly from how long the boom is, because a longer boom lets you space more directors out in front of the driven element, each one reinforcing the wave traveling forward. Adding directors and stretching the boom therefore raises gain, roughly a few tenths of a dB per doubling of elements once the boom gets long. Beamwidth is the flip side: as gain rises, the main lobe gets narrower, not wider.
Watch out The choice about beamwidth increasing has the relationship backwards, since gain and beamwidth trade off inversely. Front-to-back ratio depends mostly on reflector tuning and element spacing, and resonant frequency is set by element lengths, not by boom length.
Longer boom, more directors, more gain and a narrower beam. Boom length buys gain; element length sets frequency.