To determine which two transformations must be applied to the graph of [tex]\( y = \ln(x) \)[/tex] to result in the graph of [tex]\( y = -\ln(x) + 64 \)[/tex], follow these steps:
1. Reflection over the x-axis:
The original function [tex]\( y = \ln(x) \)[/tex] reflects over the x-axis if we multiply the function by [tex]\(-1\)[/tex]. Mathematically, this is represented as:
[tex]\[
y = -\ln(x)
\][/tex]
This transformation changes the sign of all [tex]\( y \)[/tex]-values, flipping the graph upside down.
2. Vertical Translation:
After the reflection, we have the function [tex]\( y = -\ln(x) \)[/tex]. The next transformation involves shifting the graph vertically. Adding a constant to the function translates the graph vertically. In this case, we add 64 to the function:
[tex]\[
y = -\ln(x) + 64
\][/tex]
This shifts the entire graph of [tex]\( y = -\ln(x) \)[/tex] upwards by 64 units.
Thus, the two transformations that must be applied are:
1. Reflection over the x-axis.
2. Vertical translation upwards by 64 units.
Therefore, the correct choice is:
- Reflection over the [tex]$x$[/tex]-axis, plus a vertical translation.
