# Question 28476: Graph Point Reflected across the Axis [Return to catalogue page 143](https://ocme.evemisslab.com/en/questions/page/143/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261005-advance-20261006-r02-quadratic-function-foundations/#q-20261005-advance-20261006-r02-quadratic-function-0076) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0055-28476 · q-20261005-advance-20261006-r02-quadratic-function-0076 ## Original question and givens On the full real domain, let q:R→R be q(x)=(1)(x−(0))²+(0), with a≠0. Reflect the graph point (0,q(0)) across x=0. Give the reflected (input,output) pair. - G001 — On the full real domain, let q:R→R be q(x)=(1)(x−(0))²+(0), with a≠0. Reflect the graph point (0,q(0)) across x=0. Give the reflected (input,output) pair. ## Definition and scope 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. - [mko-quadratic-function](https://ocme.evemisslab.com/en/mko/mko-quadratic-function/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Legacy mirror ORIGINAL x minus h := sub(0, 0) After Legacy mirror ORIGINAL x minus h = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Legacy mirror ORIGINAL squared displacement := mul(0, 0) After Legacy mirror ORIGINAL squared displacement = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Legacy mirror ORIGINAL a times displacement square := mul(1, 0) After Legacy mirror ORIGINAL a times displacement square = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Legacy mirror ORIGINAL graph output plus k := sum(0, 0) After Legacy mirror ORIGINAL graph output plus k = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Legacy mirror ORIGINAL twice h := mul(2, 0) After Legacy mirror ORIGINAL twice h = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Legacy mirror ORIGINAL reflected input := sub(0, 0) After Legacy mirror ORIGINAL reflected input = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Legacy mirror reflected displacement := sub(0, 0) After Legacy mirror reflected displacement = 0 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore source roles = [1, 0, 0, 0, 0] After derived = [0, 0, 0, 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. Step identifiers and dependenciesRule: Legacy_mirror_FULLreal_vertex_quadratic_involution_unchanged_output · Uses: G001, S008 ## Reverse checks - 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. ## Result (0, 0) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 99b44b21080151dfeb5b99bb6167d00a0c6d9fff3afc8ee1ee1e20d666122489 - candidate record SHA-256: fe7e03dbb84db394e0a529877d5c6659e6e917f3cdcab8a47ab6df62a962c85a - semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683 - question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r02-quadratic-function-0076.json)