- Both functions are linear with the same slope \(m=7\), implying the transformation is a translation.
- Set up the vertical translation equation \(g(x) = f(x) + k\) as \(7x - 4 = 7x + 6 + k\).
- Solve for \(k\) to get \(k = -10\), which signifies a vertical shift of 10 units down.
- The correct transformation is a translation of 10 units down, which corresponds to option \(\boxed{b}\).
Explanation
- Analysis of the Functions
The functions given are linear functions of the form \(f(x) = mx + b\). Specifically, \(f(x) = 7x + 6\) and \(g(x) = 7x - 4\). Both functions have the same slope, \(m = 7\), which indicates that the graph of \(g(x)\) is a translation of the graph of \(f(x)\). A translation can be either horizontal (shifting along the x-axis) or vertical (shifting along the y-axis), or a combination of both. Our goal is to determine which of the provided transformation options maps the graph of \(f(x)\) to the graph of \(g(x)\).
- Determining the Vertical Translation
We are looking for a transformation that maps \(f(x) = 7x + 6\) to \(g(x) = 7x - 4\). A vertical translation of a function \(f(x)\) by \(k\) units is represented by \(g(x) = f(x) + k\). Substituting the given expressions, we have:
\[7x - 4 = (7x + 6) + k\]
To find \(k\), we simplify the equation:
\[7x - 4 = 7x + 6 + k\]
Subtracting \(7x\) from both sides gives:
\[-4 = 6 + k\]
Solving for \(k\), we get:
\[k = -4 - 6 = -10\]
A value of \(k = -10\) corresponds to a vertical translation of 10 units downwards.
- Conclusion
Comparing our result of a vertical translation of 10 units down with the provided options:
Atranslation 10 units left
Btranslation 10 units down
Ctranslation 10 units up
Dtranslation 10 units right
Option b matches our result perfectly. A vertical translation of 10 units down shifts the entire graph of \(f(x) = 7x + 6\) to the position where it coincides with \(g(x) = 7x - 4\). The final answer is option b.
Examples
Imagine you are adjusting the height of a shelf on a wall. If the shelf is initially mounted at a height defined by a linear function \(f(x) = 7x + 6\), where \(x\) represents the distance from a corner and the output represents the height from the floor, moving the entire shelf down by 10 units means every point on the shelf now sits at a height determined by \(g(x) = (7x + 6) - 10 = 7x - 4\). This linear transformation is a direct application of moving objects in coordinate space, commonly used in architecture, game design, and physical modeling.