To understand the transformation of the function from \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we can analyze the components of the function:
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Shifting: The term \( (x - 3) \) indicates a horizontal shift. Specifically, replacing \( x \) with \( x - 3 \) shifts the graph to the right by 3 units.
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Width: The transformation does not include a vertical stretch or compression (which would involve multiplying by a factor other than 1), nor is there a horizontal stretching or compression (would be represented by a factor inside the parentheses with \( x \)). The simplest quadratic function, \( x^2 \), has a standard width. Since no such adjustments are made, the shape remains the same; only the position changes.
Thus, the correct description of the transformation from \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \) is:
Shifts 3 units to the right but have the same shape.