The Event Horizon Telescope (EHT) has produced the first image of the 1.3 mm-wavelength emission around the black hole "shadow" at the heart of M87. Because the EHT's dynamic range is currently limited, this image does not show emission from the famous relativistic jet which is prominent in VLBI images at longer wavelengths. I will discuss how large-scale numerical simulations connect VLBI images of the shadow at 1.3 mm to images of the jet at longer wavelengths and constrain the physics of the jet launching region. M87's wide jet opening angle and large jet power suggest that its accretion flow is magnetically arrested and that the jet is powered by the black hole spin. The balance of radiative cooling and dissipative heating in the plasma flow is critical in setting the temperature of the emitting electrons and affects image morphology at all wavelengths. By including radiative feedback and electron-ion thermodynamics self-consistently in simulations of magnetically arrested disks, we are able to produce images of M87 that are consistent with VLBI observations from EHT 1 mm images of the black hole shadow to 1 cm images of the jet. Finally, I will briefly discuss new algorithms for reconstructing images from VLBI data developed for the EHT and describe how these methods will contribute to future EHT results that map the near-horizon emission in polarization and monitor its variability. In the longer term, adding additional short baselines to the EHT network will provide the dynamic range necessary to image the faint emission from the jet base at 1.3 mm and directly observe the connection between a supermassive black hole and its relativistic jet.