To analyze the transformation of the function \( g(x) = 3f(x - 1) \) based on the given function \( f(x) = x \), let's break down the components step by step:
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Starting function: The original function is \( f(x) = x \), which is a linear function that graphs as a straight line through the origin with a slope of 1.
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Horizontal translation: The expression \( f(x - 1) \) indicates a horizontal translation. Specifically, replacing \( x \) with \( x - 1 \) shifts the graph of \( f(x) \) to the right by 1 unit. Therefore, the graph of \( f(x - 1) \) will pass through the point \((1, 1)\) instead of \((0, 0)\).
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Vertical stretch: The coefficient outside the function, which is 3 in \( g(x) = 3f(x - 1) \), represents a vertical stretch. Since 3 is greater than 1, this means the output values of the function will be multiplied by 3. Thus, every point on the graph of \( f(x - 1) \) will be stretched vertically by a factor of 3.
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Equation after transformations: The resulting equation will be \( g(x) = 3(x - 1) \). We can simplify this to \( g(x) = 3x - 3 \). This is a linear function with a slope of 3 and a y-intercept of -3.
Summary of transformations:
- The graph of \( f(x) = x \) is shifted 1 unit to the right due to \( (x - 1) \).
- Then, the graph is stretched vertically by a factor of 3 due to the multiplication by 3.
Based on this analysis, the correct description of the change in the graph is:
- There is a translation to the right on the x-axis by 1 unit, and there is a vertical stretch by a factor of 3.
This corresponds to none of the options you listed exactly, as the first option mentioned leftward translation, which is incorrect. The correct interpretation includes a rightward shift and a vertical stretch.
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