To understand the translation represented by the transformation [tex]\( f(x) \rightarrow f(x) + 6 \)[/tex], let's analyze its impact on the function [tex]\( f(x) \)[/tex].
1. Understanding [tex]\( f(x) + 6 \)[/tex]:
- When we add a constant to a function [tex]\( f(x) \)[/tex], it results in a vertical translation of the graph.
- Specifically, adding 6 to [tex]\( f(x) \)[/tex] means that every value of [tex]\( f(x) \)[/tex] is increased by 6 units.
2. Graphical Interpretation:
- Consider the original point [tex]\((x, f(x))\)[/tex] on the graph of [tex]\( f(x) \)[/tex].
- After the transformation [tex]\( f(x) \rightarrow f(x) + 6 \)[/tex], this point becomes [tex]\((x, f(x) + 6)\)[/tex].
- Thus, each point on the graph of [tex]\( f(x) \)[/tex] is moved 6 units upwards.
3. Translation Type:
- The modification [tex]\( f(x) + 6 \)[/tex] does not involve reflecting across any axis:
- A reflection across the [tex]\( x \)[/tex]-axis would be represented by [tex]\(-f(x)\)[/tex].
- A reflection across the [tex]\( y \)[/tex]-axis would be represented by [tex]\( f(-x) \)[/tex].
- It also does not involve rotation around the origin, which would modify the function's symmetry properties.
Therefore, the translation represented by [tex]\( f(x) \rightarrow f(x) + 6 \)[/tex] is a vertical translation, specifically sliding the graph upwards by 6 units.
The corresponding choice for this translation is:
sliding it up
