To analyze the transformation from \( f(x) = 4x - 5 \) to \( g(x) = 42x - 5 \), we can look at how the function's coefficient of \( x \) changes.
- In the original function \( f(x) = 4x - 5 \), the coefficient of \( x \) is 4.
- In the transformed function \( g(x) = 42x - 5 \), the coefficient of \( x \) is 42.
To see how the graph has transformed, we can find the factor by which the coefficient has changed. We can do this by taking the ratio of the new coefficient to the old coefficient:
\[ k = \frac{42}{4} = 10.5 \]
Stretch or Compression
Since the coefficient of \( x \) has increased from 4 to 42, it indicates that the function is being stretched vertically. A higher coefficient means that for each unit increase in \( x \), \( g(x) \) increases more than \( f(x) \).
Conclusion
So we can summarize the answer as:
- The value of \( k \) is \( 10.5 \).
- The graph shows a stretch.
None of the responses you provided match the calculation of \( k = 10.5 \), so make sure to confirm this against the options given in your setup if applicable. If you meant to refer to \( 2 \) specifically in the context or options, that likely does not directly apply to the transformation between \( 4 \) and \( 42 \).
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