Final answer:
The transformations in order compared to y = log₂ x are horizontal reflection, horizontal compression by a factor of 2, and vertical translation down by 4 units.
Step-by-step explanation:
The given equation is y = log₂ (−x³) − 4. We need to identify the transformations in order compared to y = log₂ x.
The transformation from y = log₂ x to y = log₂ (−x³) − 4 can be broken down into the following steps:
- Horizontal Reflection (Reflection about the y-axis)
- Horizontal Compression (by a factor of 2)
- Vertical Translation (down by 4 units)
When comparing the transformations of y = log2(-x3) - 4 to y = log2 x, the first transformation observed is a horizontal reflection, which is a reflection about the y-axis due to the negative sign inside the log function. This changes the domain of the function to x < 0.
Following this, we see a horizontal compression by a factor of 3, caused by the cubing of x before taking the logarithm, which affects the rate at which the log function grows. Lastly, a vertical translation down by 4 units is performed, shifting the entire graph down in the coordinate system on the y-axis.
It is important to note that there is no vertical compression by a factor of 3 because the power is inside the logarithm and affects the x-axis, not the y-axis. Similarly, the horizontal compression is not by a factor of 2, but by a factor of 3, reflecting the power to which x is raised.
Therefore, the correct answer is C) Horizontal Reflection (Reflection about the y-axis), A) Horizontal Compression (by a factor of 2), D) Vertical Translation (down by 4 units).