- Analyze the functions \(f(x) = 10x - 2\) and \(g(x) = 10x + 7\).
- Model a vertical translation as \(g(x) = f(x) + k\).
- Solve for \(k\) by setting \(10x + 7 = (10x - 2) + k\), yielding \(k = 9\).
- Conclude that the transformation is a vertical translation 9 units up, so the correct option is: \(\boxed{\text{translation 9 units up}}\).
Explanation
- Analysis of Transformation Types
To determine the transformation that maps the graph of \(f(x) = 10x - 2\) to the graph of \(g(x) = 10x + 7\), we analyze how a transformation affects a function's equation. A vertical translation of a function \(f(x)\) by \(k\) units is given by \(g(x) = f(x) + k\). If \(k > 0\), the graph shifts upward; if \(k < 0\), it shifts downward. A horizontal translation by \(h\) units is given by \(g(x) = f(x - h)\). If \(h > 0\), the graph shifts to the right; if \(h < 0\), it shifts to the left.
- Calculating the Vertical Translation
We test for a vertical translation by setting \(g(x) = f(x) + k\). Substituting the given expressions, we have:
\[10x + 7 = (10x - 2) + k\]
Expanding and solving for \(k\):
\[10x + 7 = 10x - 2 + k\]
\[7 = -2 + k\]
\[k = 7 + 2 = 9\]
Since \(k = 9\), this indicates a vertical translation of 9 units in the positive direction (upward).
- Verifying and Concluding the Transformation
We check for horizontal translations to see if any other provided options are valid. Setting \(g(x) = f(x + h)\), we have:
\[10x + 7 = 10(x + h) - 2\]
\[10x + 7 = 10x + 10h - 2\]
\[7 = 10h - 2\]
\[9 = 10h\]
\[h = 0.9\]
This would correspond to a horizontal shift of 0.9 units to the left, which does not match any of the provided 9-unit shift options. Thus, the correct transformation is a vertical translation of 9 units up.
Examples
Imagine you are designing a game where an object moves along a path defined by a linear function. If you want to move the entire path vertically upward to increase the difficulty or change the scenery, you are applying a vertical translation. For example, shifting a simple platform from the ground level to a height of 9 meters involves adding 9 to the height function, mirroring the transformation we calculated here. Understanding these shifts allows for precise control over the positioning and movement of objects in graphical software and game development.