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3 Responses
Think “flip vs fold”: y = |f(x)| flips any part of the graph below the x-axis upward (zeros stay put), while y = f(|x|) folds the right-hand side (x≥0) across the y-axis and throws away the original left side, making an even graph (e.g., for f(x)=x−2, |f(x)|=|x−2| has vertex (2,0) but f(|x|)=|x|−2 has vertex (0,−2)). I finally stopped mixing them up after literally folding a scrap of paper over the y-axis in class and realizing I’d been “flipping when I should’ve been folding.”
Output-abs vs input-abs: y = |f(x)| takes your original graph and flips any part below the x-axis upward (zeros stay put), while y = f(|x|) is made by taking the right-hand half of f and mirroring it across the y-axis-i.e., f(|x|)=f(x) for x≥0 and f(-x) for x<0. Think of |f(x)| as “bouncing off the floor,” and f(|x|) as “fold the paper along the y-axis and trace the right side onto the left.”
Think of these as two different “folds” of the graph-one vertical, one horizontal-and that’s the whole vibe! For y = |f(x)|, you take the original graph of y = f(x) and flip any part that’s below the x-axis straight up across the x-axis: points with negative y-values get reflected to positive, x-intercepts stay put, the domain doesn’t change, and everything ends up with y ≥ 0. For y = f(|x|), you’re feeding in only nonnegative inputs, so the right-hand side (x ≥ 0) of the original graph gets duplicated to the left: the result is even, and anything that used to happen on the left side is ignored and replaced by a mirror of the right side (although if f was already symmetric, it kind of looks like you “kept” the old left side too). Simple example: let f(x) = x − 2. Then |f(x)| = |x − 2|, a V-shape with its point at (2, 0): at x = −3 you get |−5| = 5. But f(|x|) = |x| − 2, another V with its point at (0, −2): at x = −3 you get 3 − 2 = 1. So to sketch quickly: for |f(x)|, draw f and fold the negative parts upward; for f(|x|), sketch f on x ≥ 0 and mirror that across the y-axis to fill in x ≤ 0.