Porisma

Worked solution

Take the modulus of a cubic input

Sketch y=(∣x∣+1)(∣x∣−2)(∣x∣−3)y=(|x|+1)(|x|-2)(|x|-3) from y=(x+1)(x−2)(x−3)y=(x+1)(x-2)(x-3).

CJC · Transformations · Example 15

Retain and mirror the right-hand side

Answer

The new roots are x=±2,±3x=\pm2,\pm3; the y-intercept remains (0,6)(0,6).

Discard the original x<0x<0 portion, including its root −1-1. Keep the portion with x≥0x\ge0, then reflect it in the y-axis. Negative outputs between 2 and 3 stay negative and also appear between −3 and −2; input modulus does not make all outputs non-negative.

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Preserve every printed extremum label

Answer

The right-hand extrema have matching mirror images on the left.

The source labels (0.131,6.06)(0.131,6.06) and (2.53,−0.879)(2.53,-0.879), with mirror labels (−0.131,6.06)(-0.131,6.06) and (−2.53,−0.879)(-2.53,-0.879). These are retained printed approximate coordinates. Independently, f(0)=6f(0)=6 and the one-sided slopes change sign at 0, giving a corner minimum there; the nearby maxima are slightly higher.

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Even input-modulus cubic with roots −3,−2,2,3 and y-intercept 6. Source approximate extrema (±0.131,6.06) and (±2.53,−0.879); original negative-input cubic shown dashed and discarded, including its root −1.
Even input-modulus cubic with roots −3,−2,2,3 and y-intercept 6. Source approximate extrema (±0.131,6.06) and (±2.53,−0.879); original negative-input cubic shown dashed and discarded, including its root −1.
Even input-modulus cubic with roots −3,−2,2,3 and y-intercept 6. Source approximate extrema (±0.131,6.06) and (±2.53,−0.879); original negative-input cubic shown dashed and discarded, including its root −1.
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