# Question 34531: Exact Evaluation of a Vertex-Form Quadratic [Return to catalogue page 173](https://ocme.evemisslab.com/en/questions/page/173/) [← 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-0006) 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. R0068-34531 · q-20261005-advance-20261006-r02-quadratic-function-0006 ## Original question and givens On the full real domain, let q:R→R be q(x)=(-6)(x−(-7))²+(-5), with a≠0. Evaluate q(-8) exactly. - G001 — On the full real domain, let q:R→R be q(x)=(-6)(x−(-7))²+(-5), with a≠0. Evaluate q(-8) exactly. ## 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-evaluation, [a, domain, h, k, operation, x]] After derived = [quadratic_evaluation, reals] Conditions: none Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact. Step identifiers and dependenciesRule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001 S002 · impliesBefore source roles = [-6, -7, -5] After derived = [q(x)=a*(x-h)^2+k, x in R] Conditions: none FIRST qualify nonzero a and the FULL real domain.The ORIGINAL function is vertex form,not a*x²+h*x+k.Keep h/k roles unchanged and do not silently replace the domain by a bounded interval. Step identifiers and dependenciesRule: Quadratic_FULL_real_vertex_form_nonzero_leading_coefficient · Uses: G001, S001 S003 · calculationBefore Quadratic ORIGINAL x minus h := sub(-8, -7) After Quadratic ORIGINAL x minus h = -1 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Quadratic ORIGINAL displacement squared := mul(-1, -1) After Quadratic ORIGINAL displacement squared = 1 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Quadratic ORIGINAL a times displacement squared := mul(-6, 1) After Quadratic ORIGINAL a times displacement squared = -6 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Quadratic ORIGINAL weighted square plus k := sum(-6, -5) After Quadratic ORIGINAL weighted square plus k = -11 Conditions: none Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [-6, -7, -5, -8] After derived = [-1, 1, -6, -11] Conditions: none Subtract h,square,multiply by SIGNED a,then add k.Do not lose a negative coefficient;at x=h the square is0.This is THIS input’s value,not a sampled global-extremum certificate. Step identifiers and dependenciesRule: Quadratic_EVALUATE_all_substitutions_and_signed_a · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S006 · premise_and_result_audit — Recompute Quadratic ORIGINAL weighted square plus k with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Quadratic ORIGINAL a times displacement squared with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Quadratic ORIGINAL displacement squared with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Quadratic ORIGINAL x minus h with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. - S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof. ## Result -11 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: 9c48c54d233bb1d82bd809bc991d2f0da1705991b9e999f9c0afb35a95c46fa1 - candidate record SHA-256: b6c39ea942a406c69ac1417c7e8f96d810ccbcfe00c8ae16f23e7b2c6975e7d2 - semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed - question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229 - 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-0006.json)