Describe the transformation that takes the graph of
y x 4 log
to the graph of
2
y log4 16x .
The transformation that takes the graph of y=log4(x) to the graph of y=log4(16x) involves two main changes: a horizontal compression and a translation.
First, let's focus on the expression inside the logarithm function. In y=log4(16x), the 16x is 16 times the x in y=log4(x). This means that the graph of y=log4(16x) is horizontally compressed by a factor of 16 compared to the graph of y=log4(x).
To understand this, let's look at some points on the graph. For example, when x=1, y=log4(1)=0 in y=log4(x). However, in y=log4(16x), when x=1, y=log4(16(1))=log4(16)=2. Therefore, the x-values in y=log4(16x) get multiplied by 16, resulting in a horizontal compression.
Next, let's look at the translation. In y=log4(16x), there is no additional constant term inside the logarithm function. This means that the graph of y=log4(16x) is shifted towards the left compared to the graph of y=log4(x).
To see this, let's consider the point (1, 0) on the graph of y=log4(x). In y=log4(16x), the point becomes (1/16, 0). This means that the graph is shifted to the left by a distance of 1/16 units compared to the graph of y=log4(x).
In summary, the transformation that takes the graph of y=log4(x) to the graph of y=log4(16x) involves a horizontal compression by a factor of 16 and a translation to the left by a distance of 1/16 units.