- Define the given functions: \(f(x) = 72\sqrt{x}\) and \(g(x) = 9\sqrt{x}\).
- Test for a vertical transformation \(g(x) = c \cdot f(x)\), which yields \(c = \frac{1}{8}\), indicating a vertical compression by a factor of \(\frac{1}{8}\).
- Test for a horizontal transformation \(g(x) = f(k \cdot x)\), which yields \(k = \frac{1}{64}\), indicating a horizontal stretch, not a compression.
- Conclude that the single correct transformation is a \(\boxed{\text{vertical compression}}\).
Explanation
- Analyzing the Problem and Data
We are given two functions, \(f(x) = 72\sqrt{x}\) and \(g(x) = 9\sqrt{x}\). We want to find a single transformation that maps the graph of \(f(x)\) to the graph of \(g(x)\). A vertical transformation affects the output values (the \(y\)-coordinates), while a horizontal transformation affects the input values (the \(x\)-coordinates). We test both possibilities.
- Testing Vertical Transformation
First, let's test for a vertical transformation of the form \(g(x) = c \cdot f(x)\). Substituting the given expressions, we have:
\[9\sqrt{x} = c \cdot (72\sqrt{x})\]
Dividing both sides by \(\sqrt{x}\) (assuming \(x > 0\)), we get:
\[9 = 72c\]
\[c = \frac{9}{72} = \frac{1}{8}\]
Since \(0 < c < 1\), a vertical transformation with \(c = \frac{1}{8}\) corresponds to a vertical compression by a factor of \(\frac{1}{8}\).
- Testing Horizontal Transformation
Next, let's check for a horizontal transformation of the form \(g(x) = f(k \cdot x)\). Substituting the given expressions, we have:
\[9\sqrt{x} = 72\sqrt{kx}\]
\[9\sqrt{x} = 72\sqrt{k}\sqrt{x}\]
Dividing by \(\sqrt{x}\), we get:
\[9 = 72\sqrt{k}\]
\[\sqrt{k} = \frac{1}{8}\]
\[k = \left(\frac{1}{8}\right)^2 = \frac{1}{64}\]
A horizontal transformation \(f(kx)\) is a compression if \(k > 1\) and a stretch if \(0 < k < 1\). Since \(k = \frac{1}{64}\) and \(0 < \frac{1}{64} < 1\), this transformation represents a horizontal stretch, not a horizontal compression.
- Conclusion
Comparing our results with the provided options:
- Vertical translation: \(g(x) = f(x) + k\) - Not applicable as we found a multiplicative factor.
- Vertical compression: \(g(x) = c \cdot f(x)\) with \(0 < c < 1\) - This matches our finding of \(c = \frac{1}{8}\).
- Horizontal translation: \(g(x) = f(x - h)\) - Not applicable.
- Horizontal compression: \(g(x) = f(k \cdot x)\) with \(k > 1\) - Our result \(k = \frac{1}{64}\) does not satisfy this condition.
Therefore, the correct single transformation is a vertical compression. The final answer is \(\boxed{\text{vertical compression}}\).
Examples
In architecture and design, scaling is frequently used to adjust the size of models or building plans while keeping their relative proportions. For example, if an original design drawing uses a function \(y = 72\sqrt{x}\) to represent a specific curve's profile, and you need to scale the height of this profile down to \(y = 9\sqrt{x}\) for a smaller layout, you are effectively applying a vertical compression. By multiplying the entire function output by \(\frac{1}{8}\), you ensure the curve maintains its shape but fits within the smaller design space, which is a practical application of vertical transformations.