Final answer:
The graph of f(x) = log₄(x-1) includes a vertical asymptote at x=1 and passes through the point (2,0). The function's value increases as x increases but at a decreasing rate, highlighting the logarithmic nature of the function.
Step-by-step explanation:
The function f(x) = log₄(x-1) has key features that can be identified when analyzing its graph. Firstly, there is a vertical asymptote at x=1, since the logarithm is undefined for non-positive numbers, and x-1 must be greater than zero. Secondly, the function will pass through the point (2,0), because log₄(1) equals zero. As x increases, the value of f(x) will increase at a decreasing rate, which is a property of logarithmic functions, as per the logarithmic property of log(a) = log b - log c. Looking at the function, it's also evident that there are no x-intercepts, since log₄(x-1) never equals zero except at x=2, as mentioned.