The graph of g(x) is steeper or less steep than the graph of f(x).
Understand the Problem
The question is asking about how the transformation of the function f(x) affects the slope of the graph of g(x) compared to f(x). Specifically, it wants to know whether g(x) is steeper or less steep than f(x) based on the given transformation.
Answer
The graph of \( g(x) \) is less steep than the graph of \( f(x) \).
Answer for screen readers
The graph of ( g(x) ) is less steep than the graph of ( f(x) ).
Steps to Solve
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Understand the Transformation
The given function is ( f(x) = x ) and it is transformed to create ( g(x) = \frac{1}{2} f(x) ). Therefore, we rewrite it as: $$ g(x) = \frac{1}{2} x $$
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Identify the Slope of ( f(x) )
The slope of the function ( f(x) = x ) is 1, which means it rises 1 unit for every 1 unit it runs horizontally.
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Identify the Slope of ( g(x) )
The slope of ( g(x) = \frac{1}{2} x ) is ( \frac{1}{2} ). This means that for every 1 unit it runs horizontally, it only rises ( \frac{1}{2} ) unit.
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Compare the Slopes
Since the slope of ( g(x) ) is less than that of ( f(x) ) (where ( \frac{1}{2} < 1 )), we can conclude that the graph of ( g(x) ) is less steep than the graph of ( f(x) ).
The graph of ( g(x) ) is less steep than the graph of ( f(x) ).
More Information
When the function is multiplied by a factor less than 1, its slope decreases, leading to a less steep graph. Conversely, multiplying by a factor greater than 1 would increase the slope and make the graph steeper.
Tips
- Confusing the effect of multiplication on the slope with addition or other transformations. Remember, multiplying by a positive fraction will decrease the steepness.
- Misinterpreting the graphical effect; ensure you visualize how the coefficient affects the slope.
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