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R0056-28679 · q-20261005-advance-20261006-r03-rate-of-change-0079
On the full real domain, let q:R→R be q(x)=(-4)(x−(-9))²+(-9), with a≠0. For distinct inputs -5 and -10, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.
G001 — On the full real domain, let q:R→R be q(x)=(-4)(x−(-9))²+(-9), with a≠0. For distinct inputs -5 and -10, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.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-rate-quadratic-average, quadratic_average]
After
derived = [exact_quadratic_endpoint_average_not_derivative]
Conditions: none
Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions.
Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001
Before
Rate ORIGINAL x1 minus x0 := sub(-10, -5)
After
Rate ORIGINAL x1 minus x0 = -5
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Rate quadratic ORIGINAL displacement0 := sub(-5, -9)
After
Rate quadratic ORIGINAL displacement0 = 4
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Rate quadratic ORIGINAL square0 := mul(4, 4)
After
Rate quadratic ORIGINAL square0 = 16
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Rate quadratic ORIGINAL a times square0 := mul(-4, 16)
After
Rate quadratic ORIGINAL a times square0 = -64
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Rate quadratic ORIGINAL endpoint0 plus k := sum(-64, -9)
After
Rate quadratic ORIGINAL endpoint0 plus k = -73
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Rate quadratic ORIGINAL displacement1 := sub(-10, -9)
After
Rate quadratic ORIGINAL displacement1 = -1
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Rate quadratic ORIGINAL square1 := mul(-1, -1)
After
Rate quadratic ORIGINAL square1 = 1
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Rate quadratic ORIGINAL a times square1 := mul(-4, 1)
After
Rate quadratic ORIGINAL a times square1 = -4
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Rate quadratic ORIGINAL endpoint1 plus k := sum(-4, -9)
After
Rate quadratic ORIGINAL endpoint1 plus k = -13
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
source roles = [-4, -9, -9, -5, -10]
After
derived = [[-73, -13], dy=a*(x1-x0)*(x1+x0-2h)]
Conditions: none
Use the WHOLE vertex-form quadratic.The k terms cancel and the square difference factors as(x1-x0)(x1+x0-2h).Symmetric endpoints can give dy=0 although a is nonzero;the function need not be constant.
Rule: Rate_WHOLE_quadratic_difference_of_squares · Uses: G001, S010
Before
Rate ORIGINAL y1 minus y0 := sub(-13, -73)
After
Rate ORIGINAL y1 minus y0 = 60
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
Rate ORIGINAL dy over dx := div(60, -5)
After
Rate ORIGINAL dy over dx = -12
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
source roles = [-5, -10, -73, -13, -5, 60]
After
derived = [-12/1]
Conditions: none
Keep the SAME endpoint orientation:(y1-y0)/(x1-x0).Both differences may reverse together,not just one.The result may be negative or0;normalize a positive denominator and retain/1.This finite endpoint rate is not a general derivative certificate.
Rule: Rate_oriented_quotient_reverse_BOTH_differences · Uses: G001, S013
S014 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S013 · premise_and_result_audit — Recompute Rate ORIGINAL dy over dx with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute Rate ORIGINAL y1 minus y0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S010 · premise_and_result_audit — Recompute Rate quadratic ORIGINAL endpoint1 plus k with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute Rate quadratic ORIGINAL a times square1 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Rate quadratic ORIGINAL square1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Rate quadratic ORIGINAL displacement1 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Rate quadratic ORIGINAL endpoint0 plus k with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Rate quadratic ORIGINAL a times square0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Rate quadratic ORIGINAL square0 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Rate quadratic ORIGINAL displacement0 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Rate ORIGINAL x1 minus x0 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.-12/1
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: a08ac85bda369e0a36734352f8240fbd302d4cd036a812cea5aab0c4be7e9f87candidate record SHA-256: 429b6c1cc98d9ed314644949866719796b3b267e46d301e8a611986b402469a6semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759fLean: NOT_RUN · human review: NOT_PERFORMED