The graph of the function y = f(x) is shown in the xy-plane above. Which of the following is the graph of f(x + 2)?
The graph of f(x + 2) shifts the graph of f(x) to the left by 2 units.
When a function is transformed by adding a constant to the variable x, it results in a horizontal shift. Specifically, f(x + 2) translates the graph of f(x) leftward by 2 units, altering the x-coordinates of all points accordingly while preserving their corresponding y-values.
This graph does not reflect a leftward shift of the original function. Instead, it appears to be a vertical shift or a transformation that does not correspond to the required change for f(x + 2). The characteristics of the original graph are not maintained.
This graph shows an unaltered version of the original function and does not demonstrate any horizontal shift. Therefore, it cannot represent f(x + 2), as the absence of the necessary transformation fails to align with the function's definition.
Similar to the previous options, this graph does not exhibit the leftward shift needed for f(x + 2). Instead, it looks like the original function has been modified in a way that does not correspond to a simple horizontal transformation, making it incorrect.
This graph correctly illustrates a leftward shift of the original function by 2 units, accurately representing f(x + 2). Each point on the graph has moved left, maintaining the same height (y-value) as per the transformation rules of function graphs.
This graph indicates a rightward shift instead of a leftward one, which is the opposite of what is required for f(x + 2). The points have all moved to the right, misrepresenting the transformation that should have taken place.
The function transformation from f(x) to f(x + 2) necessitates a leftward shift of the graph by 2 units. Among the provided options, only option D accurately reflects this transformation, making it the correct choice. Understanding how shifts affect the graph of a function is essential in analyzing and visualizing function behavior across transformations.
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