Porisma

Worked solution

Fold negative outputs above the x-axis

Use a graphing calculator to sketch (1) y=x2−1y=x^2-1 and (2) y=∣x2−1∣y=|x^2-1|. Copy the curves. What do you observe about the curves?

CJC · Transformations · Output modulus activity

Sketch both curves

Answer

The roots stay (−1,0),(1,0)(-1,0),(1,0); (0,−1)(0,-1) becomes (0,1)(0,1).

Retain the original curve above or on the x-axis. Reflect only the part below the x-axis in that axis: for this parabola it is the arc −1<x<1-1<x<1. Both source copies show the original lower arc and the reflected upper arc.

Expand diagram

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Original x²−1 and output modulus |x²−1|, roots −1 and 1, original minimum (0,−1), reflected point (0,1). The outer upward branches stay unchanged.
Original x²−1 and output modulus |x²−1|, roots −1 and 1, original minimum (0,−1), reflected point (0,1). The outer upward branches stay unchanged.
Original x²−1 and output modulus |x²−1|, roots −1 and 1, original minimum (0,−1), reflected point (0,1). The outer upward branches stay unchanged.
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State the output-modulus rule

Answer

∣f(x)∣≥0|f(x)|\ge0 wherever f is defined.

The two source steps are: retain the portion with y≥0y\ge0, then reflect the portion with y<0y<0 in the x-axis. Independently, this is ∣f(x)∣=f(x)|f(x)|=f(x) when f(x)≥0f(x)\ge0 and ∣f(x)∣=−f(x)|f(x)|=-f(x) when f(x)<0f(x)<0. It does not reflect the entire graph or restrict the input to positive x.

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