So I have a question regarding advanced functions, logarithm stuff.
here is the question:
Beginning with the function f(x) = log[base(a)]x ,state what transformations were used on this to obtain the functions given below:
a) p(x) = -(5/8) log [base(a)]x
b) r(x) = log[base(a)](5-x)
c) t(x) = 2log[base(a)]2z
My answers are the following, need them checked please.
a) vertical compression (5/8)
reflect across the vertical axis (not sure how to explain this transformation, do I say it before or after the shift?)
b) vertically stretched by 2
horiz. stretch by 2 as well
Thx
crap, i screwed up writing my answers. Ignore them please.
Let's go through each function and determine the transformations applied to the original function f(x) = log[base(a)]x.
a) p(x) = -(5/8) log[base(a)]x:
In function p(x), there is a vertical compression by a factor of 5/8 applied to the original function. This means that each value of f(x) is multiplied by 5/8, making it closer to the x-axis. Additionally, there is a reflection across the vertical axis. This reflection occurs before any shifts in the function.
b) r(x) = log[base(a)](5-x):
In function r(x), there is a horizontal translation or shift to the right by 5 units applied to the original function. This means that each x-value in the original function is decreased by 5. There are no vertical transformations like stretching or compression in this case.
c) t(x) = 2log[base(a)](2z):
In function t(x), there is a vertical stretch by a factor of 2 applied to the original function. This means that each value of f(x) is multiplied by 2, making it farther from the x-axis. However, this transformation is not applied to the variable x itself, but rather to the value inside the logarithm, which is 2z. So, it can be considered as a vertical stretch of the log function with respect to the z-axis. There are no horizontal transformations in this case.
Overall, your answers for a) and c) are correct. However, for b), there is only a horizontal shift but no vertical stretching involved.