Final answer:
The transformation from the parent function to y=1/2log7(x-6)-9 includes a vertical compression by 1/2, horizontal shift right by 6, and vertical shift down by 9.
Step-by-step explanation:
The transformation of the parent function to y=1/2log7(x-6)-9 involves several steps. Firstly, the factor of 1/2 indicates a vertical compression by a factor of 2. Secondly, the subtraction of 6 from x inside the logarithmic function suggests a horizontal shift to the right by 6 units. Lastly, subtracting 9 from the entire function results in a downward vertical shift by 9 units. These transformations demonstrate how the parent function is altered to create the given function y.
It is crucial to remember that the logarithmic function and its inverse, the exponential function, undo each other. This means that taking the natural logarithm of both sides of an equation can simplify the solution process, especially when dealing with exponential growth models or decay, as shown in the logistic curve example given in the question.