Certainly! Let's break down the transformation step by step:
1. Understanding the given functions:
- We start with the function [tex]\( y = f(x) \)[/tex].
- We transform this function to [tex]\( y = f(x) - 7 \)[/tex].
2. Identify the type of transformation:
- The change from [tex]\( y = f(x) \)[/tex] to [tex]\( y = f(x) - 7 \)[/tex] involves subtracting 7 from the original function’s output.
- Subtracting a constant from the function [tex]\( f(x) \)[/tex] results in a vertical shift. Specifically, each point on the graph of [tex]\( f(x) \)[/tex] is moved downward by 7 units.
3. Expressing the transformation in vector form:
- In vector form, a translation is represented by how much the graph moves in the horizontal and vertical directions.
- For the function [tex]\( y = f(x) - 7 \)[/tex], there is no horizontal movement (no change to the [tex]\( x \)[/tex]-value), only a vertical movement downward by 7 units.
4. Form the translation vector:
- The horizontal shift is 0 units (no change).
- The vertical shift is -7 units (downward).
- Therefore, the translation can be represented by the vector:
[tex]\[
\begin{bmatrix}
0 \\
-7
\end{bmatrix}
\][/tex]
Here's the translation in vector form:
[tex]\[
\begin{bmatrix}
0 \\
-7
\end{bmatrix}
\][/tex]
This vector fully describes the translation from [tex]\( y = f(x) \)[/tex] to [tex]\( y = f(x) - 7 \)[/tex], indicating a downward shift by 7 units and no shift horizontally.
