Final answer:
A continuous rational function with a horizontal asymptote at y=8 is transformed by squaring the x-values and then subtracting 1. The transformed function will still have a horizontal asymptote at y=8, but the graph will be narrower and shifted downward by 1 unit compared to the original function.
Step-by-step explanation:
A continuous rational function with a horizontal asymptote at y=8 implies that as x approaches positive or negative infinity, the function approaches y=8. The equation s(x) = r(x^2)-1 represents a transformation of the original function r(x), where x is squared and then subtracted by 1. To understand the key features of s(x), we can consider the effects of these transformations:
- If the original function had a horizontal asymptote at y=8, the transformed function s(x) will also have a horizontal asymptote at y=8, since the transformations do not change the horizontal asymptote value.
- The transformation x^2 will affect the shape of the graph by compressing the x-values closer to the y-axis. This means that the graph of s(x) will be narrower compared to the graph of r(x).
- The transformation -1 will shift the graph of s(x) downward by 1 unit. This means that all the points on the graph of r(x) will be shifted downward by 1 unit to create the graph of s(x).