- Analyze the functional forms \(f(x) = -2x - 4\) and \(g(x) = -10x - 4\).
- Test vertical transformations \(g(x) = c \cdot f(x)\), which fail to equate coefficients.
- Test horizontal transformations \(g(x) = f(k \cdot x)\), where we find \(k = 5\).
- Conclude that the transformation is a horizontal shrink.
- Final answer: \(\boxed{a}\).
Explanation
- Problem Analysis
We are given the functions \(f(x) = -2x - 4\) and \(g(x) = -10x - 4\). We want to determine which transformation converts the graph of \(f(x)\) into the graph of \(g(x)\). A general function transformation can involve either vertical or horizontal adjustments. A vertical transformation involves multiplying the output by a factor \(c\), i.e., \(g(x) = c \cdot f(x)\). A horizontal transformation involves multiplying the input by a factor \(k\), i.e., \(g(x) = f(k \cdot x)\).
- Testing Vertical Transformation
First, let's examine if a vertical transformation \(g(x) = c \cdot f(x)\) exists.
Substituting \(f(x)\), we get:
\[g(x) = c \cdot (-2x - 4) = -2cx - 4c\]
Comparing this to \(g(x) = -10x - 4\), we require:
\(-2c = -10\) and \(-4c = -4\).
From \(-2c = -10\), we find \(c = 5\).
From \(-4c = -4\), we find \(c = 1\).
Since these two conditions yield different values for \(c\), a vertical transformation alone cannot convert \(f(x)\) into \(g(x)\).
- Testing Horizontal Transformation
Next, let's examine if a horizontal transformation \(g(x) = f(k \cdot x)\) exists.
Substituting \(k \cdot x\) into \(f(x)\), we get:
\[g(x) = f(k \cdot x) = -2(k \cdot x) - 4 = -2kx - 4\]
Comparing this to \(g(x) = -10x - 4\), we require:
\(-2k = -10\) and \(-4 = -4\).
From \(-2k = -10\), we find \(k = 5\).
Since \(k=5\) satisfies the equation and the constant term \(-4\) matches, a horizontal transformation exists. Replacing \(x\) with \(5x\) represents a horizontal shrink by a factor of \(\frac{1}{5}\). Thus, the transformation is a horizontal shrink.
- Conclusion
Based on our analysis, the transformation that maps \(f(x) = -2x - 4\) to \(g(x) = -10x - 4\) is a horizontal shrink, corresponding to option a. The final answer is \(\boxed{a}\).
Examples
In engineering, a horizontal shrink is useful when you need to condense the time axis of a signal or waveform. For example, if you have a sound wave represented by a mathematical function \(f(t)\), applying a horizontal shrink (e.g., \(g(t) = f(2t)\)) would result in the sound playing twice as fast, effectively shifting its pitch higher. This is a common application of function transformations in audio processing and signal analysis.