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Describe and correct the error a student made in graphing g(x) = f(-x) as a reflection across the y-axis of the graph of f(x) = |x+2|+1

User Alex Lungu
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Final answer:

The error in graphing g(x) = f(-x) as a reflection of f(x) = |x+2|+1 across the y-axis is due to the absolute value function, |x+2|, remaining unchanged when x is replaced by -x. Therefore, the graph of g(x) is identical to the original graph of f(x), not merely a reflection.

Step-by-step explanation:

To describe and correct the error made in graphing g(x) = f(-x) as a reflection across the y-axis of the graph of f(x) = |x+2|+1, we must first understand how transformations affect the graph of a function. The original function, f(x), is a piecewise function that combines absolute value, linear, and constant transformations. In this case, when we graph f(x) = |x+2|+1, it will appear as a V-shaped graph with its vertex at (-2,1) and opening upwards.

To graph g(x) = f(-x), we reflect f(x) across the y-axis. This reflection implies that each point (x, y) on f(x) maps to (-x, y) on g(x). However, due to the absolute value, the graph of the reflected function will look identical to the original graph, as f(-x) = |-x+2|+1 simplifies to the same expression as f(x), since |-x+2| = |x-2|.

For example, consider the graph of f(x) when x=10. The function will map this to f(10) = |10+2|+1 = 13. Reflecting over the y-axis, g(-10) = f(-(-10)) = f(10) = 13, which confirms that the y-value remains consistent for the same absolute x-value in both f(x) and g(x).

User Narendra Verma
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