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R0055-28484 · q-20261005-advance-20261006-r02-quadratic-function-0084
On the full real domain, let q:R→R be q(x)=(9)(x−(-4))²+(-2), with a≠0. Reflect the graph point (0,q(0)) across x=-4. Give the reflected (input,output) pair.
G001 — On the full real domain, let q:R→R be q(x)=(9)(x−(-4))²+(-2), with a≠0. Reflect the graph point (0,q(0)) across x=-4. Give the reflected (input,output) pair.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [qf-quadratic-function-axis-mirror, quadratic_mirror]
After
derived = [[reflected_input_unchanged_output_graph_POINT]]
Conditions: none
List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged.
Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001
Before
Legacy mirror ORIGINAL x minus h := sub(0, -4)
After
Legacy mirror ORIGINAL x minus h = 4
Conditions: none
Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Legacy mirror ORIGINAL squared displacement := mul(4, 4)
After
Legacy mirror ORIGINAL squared displacement = 16
Conditions: none
Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Legacy mirror ORIGINAL a times displacement square := mul(9, 16)
After
Legacy mirror ORIGINAL a times displacement square = 144
Conditions: none
Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Legacy mirror ORIGINAL graph output plus k := sum(144, -2)
After
Legacy mirror ORIGINAL graph output plus k = 142
Conditions: none
Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Legacy mirror ORIGINAL twice h := mul(2, -4)
After
Legacy mirror ORIGINAL twice h = -8
Conditions: none
Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Legacy mirror ORIGINAL reflected input := sub(-8, 0)
After
Legacy mirror ORIGINAL reflected input = -8
Conditions: none
Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Legacy mirror reflected displacement := sub(-8, -4)
After
Legacy mirror reflected displacement = -4
Conditions: none
Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
source roles = [9, -4, -2, 0, -8]
After
derived = [4, -4, 142, 2h-(2h-x)=x]
Conditions: none
x_prime=2h-x gives x_prime-h=-(x-h),so the square and q value are unchanged.Return the FULL graph point,not the axis or vertex.A point at x=h reflects to itself;derived coordinates/values are not SAFE-clipped.
Rule: Legacy_mirror_FULLreal_vertex_quadratic_involution_unchanged_output · Uses: G001, S008
S009 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.S008 · premise_and_result_audit — Recompute Legacy mirror reflected displacement with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Legacy mirror ORIGINAL reflected input with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Legacy mirror ORIGINAL twice h with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Legacy mirror ORIGINAL graph output plus k with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Legacy mirror ORIGINAL a times displacement square with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Legacy mirror ORIGINAL squared displacement with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Legacy mirror ORIGINAL x minus h with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.(-8, 142)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 958ec9ff98088d7e4e2e16ff5131f97931b90f3db8cb91b93b32d9e513b320b6candidate record SHA-256: de833b22d35ad6e0a969762afea094886b6e35c4f5dc0942e76d469b7045467asemantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4eLean: NOT_RUN · human review: NOT_PERFORMED