Polynomial Affine Input Shift Root Coordinate Tool

Description

Transform a root coordinate under affine input substitution.

Polynomial transformations change coefficient scales or root coordinates under an explicit substitution. Translation, reflection and dilation use different signs, while a single complex phase does not apply a complete root-of-unity filter. The original function’s domain still matters after rational cancellation.

Transform a root coordinate under affine input substitution.

When to use this tool

Use Polynomial Affine Input Shift Root Coordinate when checking the stated polynomial transformation or complex-coordinate term.

How to use the tool and how it works

Enter root, scale, shift. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read root. Use the output convention below.

Transform a root coordinate under affine input substitution. Require scale != 0; output root = (originalRoot-shift)/scale.^[1]

Input field	Meaning and units	Accepted input
root	Original root; returned coordinate is (root-shift)/scale for p(scale*x+shift).	Required. Finite number.
scale	Scale	Required. Finite number.
shift	Shift	Required. Finite number.

Output field	Interpretation	Returned shape
root	Original root; returned coordinate is (root-shift)/scale for p(scale*x+shift).	Required. Finite number.

json

{
  "input": {
    "root": 7,
    "scale": 2,
    "shift": 3
  },
  "output": {
    "root": 2
  }
}

Limitations and assumptions

This assumes the transformed function is p(scale*x+shift); the original root is supplied, not solved or validated.
Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.

Alternative or Complementary methods

Use the full transformed expression and explicit variable substitution for complete symbolic equivalence, filtering or root analysis.

References

Calculation and input contract: algebra/polynomial-affine-input-shift-root-coordinate — tool.io

Similar or alternative tools

Polynomial Affine Substitution Leading Coefficient
Return leading coefficient under affine input substitution.
Polynomial Coefficient Binomial Shift Term
Return one binomial-shift coefficient contribution.