The effects of varying key physical parameters of a source emitting synchrotron self-Compton (SSC) radiation are examined in relation to the broadband spectrum of blazars, with emphasis on the high-energy γ- ray spectrum. We also explore the connection between observational parameters, such as angular size, and the predicted spectra. Model spectra show that the relationship of the predicted variations of high- energy flux densities to those at radio to infrared wavelengths is sensitive to whether magnetic field or relativistic electron density dominates the variations. Furthermore, models that involve the scattering of external seed photons can be discerned from SSC models based on the relative variability amplitudes of the high- and low-energy emission. Second-order self-Compton scattering tends to dominate the γ-ray emission in sources with steeper spectral indices, provided that the other observed properties (flux, spectral turnover, etc.) are roughly equal. Also, if the emission at the highest energies is due to second- order scattering, and if the quantity νFnu_ at X-ray frequencies is less than that in the radio-IR domain, then the value of νFnu_ in the γ-ray range must be less than that in the X-ray range. In general, the second-order spectra show strong curvature, contrary to observations, while the νFnu_ spectrum of first-order scattering has a broad, flat peak that could be mistaken for a power law of slope near zero. These considerations lead us to conclude that, in general, second-order SSC emission does not dominate the hard γ-ray emission from γ-ray-bright blazars. However, first-order SSC emission remains a prime candidate for explaining the hard γ-rays observed in these sources.