# Average Rate of Change Foundations

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261005-advance-20261006-r03-rate-of-change-foundations · 2026-10-05 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-13, -9) and (-12, -9), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation0/1

The output difference is 0 and the input difference is 1, giving 0/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0001/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 2. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-12, -7) and (-14, -5), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-1/1

The output difference is 2 and the input difference is -2, giving -1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0002/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 3. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-11, -5) and (-8, -2), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation1/1

The output difference is 3 and the input difference is 3, giving 1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0003/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 4. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-10, -3) and (-14, 1), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-1/1

The output difference is 4 and the input difference is -4, giving -1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0004/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 5. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-9, -1) and (-4, 4), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation1/1

The output difference is 5 and the input difference is 5, giving 1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0005/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 6. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-8, 1) and (-14, 7), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-1/1

The output difference is 6 and the input difference is -6, giving -1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0006/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 7. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-7, 3) and (0, 10), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation1/1

The output difference is 7 and the input difference is 7, giving 1/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0007/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 8. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-6, 5) and (-7, 13), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-8/1

The output difference is 8 and the input difference is -1, giving -8/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0008/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 9. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-5, 7) and (-3, 16), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation9/2

The output difference is 9 and the input difference is 2, giving 9/2. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0009/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 10. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-4, 9) and (-7, 19), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-10/3

The output difference is 10 and the input difference is -3, giving -10/3. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0010/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 11. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-3, 11) and (1, 22), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation11/4

The output difference is 11 and the input difference is 4, giving 11/4. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0011/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 12. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-2, 13) and (-7, 25), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-12/5

The output difference is 12 and the input difference is -5, giving -12/5. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0012/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 13. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (-1, 15) and (5, 28), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation13/6

The output difference is 13 and the input difference is 6, giving 13/6. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0013/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 14. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (0, 17) and (-7, 31), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-2/1

The output difference is 14 and the input difference is -7, giving -2/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0014/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 15. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (1, 19) and (2, 34), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation15/1

The output difference is 15 and the input difference is 1, giving 15/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0015/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 16. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (2, 21) and (0, 37), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-8/1

The output difference is 16 and the input difference is -2, giving -8/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0016/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 17. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (3, 23) and (6, 40), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation17/3

The output difference is 17 and the input difference is 3, giving 17/3. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0017/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 18. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (4, 25) and (0, 43), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-9/2

The output difference is 18 and the input difference is -4, giving -9/2. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0018/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 19. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (5, 27) and (10, 46), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation19/5

The output difference is 19 and the input difference is 5, giving 19/5. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0019/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 20. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (6, 29) and (0, 49), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-10/3

The output difference is 20 and the input difference is -6, giving -10/3. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0020/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 21. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (7, 31) and (14, 52), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation3/1

The output difference is 21 and the input difference is 7, giving 3/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0021/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 22. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (8, 33) and (7, 55), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-22/1

The output difference is 22 and the input difference is -1, giving -22/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0022/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 23. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (9, 35) and (11, 58), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation23/2

The output difference is 23 and the input difference is 2, giving 23/2. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0023/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 24. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (10, 37) and (7, 61), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-8/1

The output difference is 24 and the input difference is -3, giving -8/1. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0024/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 25. Average Rate between Explicit Function Endpoints

For the same specified function, the permitted endpoints are (11, 39) and (15, 64), with distinct inputs. Find the average rate (y1−y0)/(x1−x0). Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation25/4

The output difference is 25 and the input difference is 4, giving 25/4. Reverse both differences together, not just one. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0025/)

Average rate of change between distinct inputs · qf-rate-endpoint-quotient · candidate





## 26. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=0 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationNo

The input difference is 0, so the eligibility answer is No. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0026/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 27. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-1 and x1=-2. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -1, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0027/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 28. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-2 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 2, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0028/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 29. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-3 and x1=-6. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -3, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0029/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 30. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-4 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 4, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0030/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 31. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-5 and x1=-10. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -5, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0031/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 32. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-6 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 6, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0032/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 33. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-7 and x1=-14. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -7, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0033/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 34. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-8 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 8, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0034/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 35. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-9 and x1=-18. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -9, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0035/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 36. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-10 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 10, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0036/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 37. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-11 and x1=-22. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -11, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0037/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 38. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-12 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 12, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0038/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 39. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-13 and x1=-26. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -13, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0039/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 40. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-14 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 14, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0040/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 41. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-15 and x1=-30. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -15, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0041/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 42. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-16 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 16, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0042/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 43. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-17 and x1=-34. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -17, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0043/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 44. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-18 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 18, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0044/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 45. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-19 and x1=-38. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -19, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0045/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 46. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-20 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 20, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0046/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 47. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-21 and x1=-42. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -21, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0047/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 48. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-22 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 22, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0048/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 49. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-23 and x1=-46. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is -23, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0049/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 50. Nonzero-Denominator Eligibility for an Average Rate

The proposed permitted endpoint inputs are x0=-24 and x1=0. Is their difference a valid nonzero denominator for an average rate of change?

View answer and explanationYes

The input difference is 24, so the eligibility answer is Yes. Equal inputs fail this teaching condition; no value is assigned to their quotient here. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0050/)

Average rate of change between distinct inputs · qf-rate-denominator-validity · candidate





## 51. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-12)x+(-7). For distinct inputs 0 and 1, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-12/1

The common constant cancels in the endpoint difference, leaving slope m=-12. The exact average rate is -12/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0051/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 52. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-11)x+(-4). For distinct inputs 0 and -2, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-11/1

The common constant cancels in the endpoint difference, leaving slope m=-11. The exact average rate is -11/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0052/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 53. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-10)x+(-1). For distinct inputs 0 and 3, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-10/1

The common constant cancels in the endpoint difference, leaving slope m=-10. The exact average rate is -10/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0053/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 54. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-9)x+(2). For distinct inputs 0 and -4, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-9/1

The common constant cancels in the endpoint difference, leaving slope m=-9. The exact average rate is -9/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0054/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 55. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-8)x+(5). For distinct inputs 0 and 5, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-8/1

The common constant cancels in the endpoint difference, leaving slope m=-8. The exact average rate is -8/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0055/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 56. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-7)x+(8). For distinct inputs 0 and -6, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-7/1

The common constant cancels in the endpoint difference, leaving slope m=-7. The exact average rate is -7/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0056/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 57. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-6)x+(11). For distinct inputs 0 and 7, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-6/1

The common constant cancels in the endpoint difference, leaving slope m=-6. The exact average rate is -6/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0057/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 58. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-5)x+(14). For distinct inputs 0 and -8, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-5/1

The common constant cancels in the endpoint difference, leaving slope m=-5. The exact average rate is -5/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0058/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 59. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-4)x+(17). For distinct inputs 0 and 9, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-4/1

The common constant cancels in the endpoint difference, leaving slope m=-4. The exact average rate is -4/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0059/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 60. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-3)x+(20). For distinct inputs 0 and -10, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-3/1

The common constant cancels in the endpoint difference, leaving slope m=-3. The exact average rate is -3/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0060/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 61. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-2)x+(23). For distinct inputs 0 and 11, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-2/1

The common constant cancels in the endpoint difference, leaving slope m=-2. The exact average rate is -2/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0061/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 62. Average Rate of a Real Affine Function

On the full real domain, f(x)=(-1)x+(26). For distinct inputs 0 and -12, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-1/1

The common constant cancels in the endpoint difference, leaving slope m=-1. The exact average rate is -1/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0062/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 63. Average Rate of a Real Affine Function

On the full real domain, f(x)=(0)x+(29). For distinct inputs 0 and 13, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation0/1

The common constant cancels in the endpoint difference, leaving slope m=0. The exact average rate is 0/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0063/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 64. Average Rate of a Real Affine Function

On the full real domain, f(x)=(1)x+(32). For distinct inputs 0 and -14, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation1/1

The common constant cancels in the endpoint difference, leaving slope m=1. The exact average rate is 1/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0064/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 65. Average Rate of a Real Affine Function

On the full real domain, f(x)=(2)x+(35). For distinct inputs 0 and 15, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation2/1

The common constant cancels in the endpoint difference, leaving slope m=2. The exact average rate is 2/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0065/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 66. Average Rate of a Real Affine Function

On the full real domain, f(x)=(3)x+(38). For distinct inputs 0 and -16, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation3/1

The common constant cancels in the endpoint difference, leaving slope m=3. The exact average rate is 3/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0066/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 67. Average Rate of a Real Affine Function

On the full real domain, f(x)=(4)x+(41). For distinct inputs 0 and 17, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation4/1

The common constant cancels in the endpoint difference, leaving slope m=4. The exact average rate is 4/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0067/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 68. Average Rate of a Real Affine Function

On the full real domain, f(x)=(5)x+(44). For distinct inputs 0 and -18, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation5/1

The common constant cancels in the endpoint difference, leaving slope m=5. The exact average rate is 5/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0068/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 69. Average Rate of a Real Affine Function

On the full real domain, f(x)=(6)x+(47). For distinct inputs 0 and 19, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation6/1

The common constant cancels in the endpoint difference, leaving slope m=6. The exact average rate is 6/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0069/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 70. Average Rate of a Real Affine Function

On the full real domain, f(x)=(7)x+(50). For distinct inputs 0 and -20, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation7/1

The common constant cancels in the endpoint difference, leaving slope m=7. The exact average rate is 7/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0070/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 71. Average Rate of a Real Affine Function

On the full real domain, f(x)=(8)x+(53). For distinct inputs 0 and 21, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation8/1

The common constant cancels in the endpoint difference, leaving slope m=8. The exact average rate is 8/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0071/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 72. Average Rate of a Real Affine Function

On the full real domain, f(x)=(9)x+(56). For distinct inputs 0 and -22, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation9/1

The common constant cancels in the endpoint difference, leaving slope m=9. The exact average rate is 9/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0072/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 73. Average Rate of a Real Affine Function

On the full real domain, f(x)=(10)x+(59). For distinct inputs 0 and 23, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation10/1

The common constant cancels in the endpoint difference, leaving slope m=10. The exact average rate is 10/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0073/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 74. Average Rate of a Real Affine Function

On the full real domain, f(x)=(11)x+(62). For distinct inputs 0 and -24, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation11/1

The common constant cancels in the endpoint difference, leaving slope m=11. The exact average rate is 11/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0074/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 75. Average Rate of a Real Affine Function

On the full real domain, f(x)=(12)x+(65). For distinct inputs 0 and 25, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation12/1

The common constant cancels in the endpoint difference, leaving slope m=12. The exact average rate is 12/1, including zero slope. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0075/)

Average rate of change between distinct inputs · qf-rate-affine-average · candidate





## 76. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(1)(x−(0))²+(0), with a≠0. For distinct inputs -1 and 1, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation0/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 0/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0076/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 77. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-2)(x−(-11))²+(-11), with a≠0. For distinct inputs -9 and -12, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-2/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -2/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0077/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 78. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(3)(x−(-10))²+(-10), with a≠0. For distinct inputs -11 and -7, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation6/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 6/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0078/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 79. Average Rate of a Vertex-Form Quadratic

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.

View answer and explanation-12/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -12/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0079/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 80. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(5)(x−(-8))²+(-8), with a≠0. For distinct inputs -9 and -3, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation20/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 20/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0080/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 81. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-6)(x−(-7))²+(-7), with a≠0. For distinct inputs -1 and -8, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-30/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -30/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0081/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 82. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(7)(x−(-6))²+(-6), with a≠0. For distinct inputs -7 and 1, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation42/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 42/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0082/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 83. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-8)(x−(-5))²+(-5), with a≠0. For distinct inputs 3 and -6, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-56/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -56/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0083/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 84. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(9)(x−(-4))²+(-4), with a≠0. For distinct inputs -5 and 5, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation72/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 72/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0084/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 85. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-10)(x−(-3))²+(-3), with a≠0. For distinct inputs 7 and -4, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-90/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -90/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0085/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 86. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(11)(x−(-2))²+(-2), with a≠0. For distinct inputs -3 and 9, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation110/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 110/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0086/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 87. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-12)(x−(-1))²+(-1), with a≠0. For distinct inputs 11 and -2, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-132/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -132/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0087/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 88. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(13)(x−(0))²+(0), with a≠0. For distinct inputs -1 and 13, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation156/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 156/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0088/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 89. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-14)(x−(1))²+(1), with a≠0. For distinct inputs 15 and 0, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-182/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -182/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0089/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 90. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(15)(x−(2))²+(2), with a≠0. For distinct inputs 1 and 17, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation210/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 210/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0090/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 91. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-16)(x−(3))²+(3), with a≠0. For distinct inputs 19 and 2, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-240/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -240/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0091/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 92. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(17)(x−(4))²+(4), with a≠0. For distinct inputs 3 and 21, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation272/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 272/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0092/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 93. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-18)(x−(5))²+(5), with a≠0. For distinct inputs 23 and 4, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-306/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -306/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0093/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 94. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(19)(x−(6))²+(6), with a≠0. For distinct inputs 5 and 25, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation342/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 342/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0094/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 95. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-20)(x−(7))²+(7), with a≠0. For distinct inputs 27 and 6, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-380/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -380/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0095/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 96. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(21)(x−(8))²+(8), with a≠0. For distinct inputs 7 and 29, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation420/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 420/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0096/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 97. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-22)(x−(9))²+(9), with a≠0. For distinct inputs 31 and 8, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-462/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -462/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0097/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 98. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(23)(x−(10))²+(10), with a≠0. For distinct inputs 9 and 33, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation506/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 506/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0098/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 99. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(-24)(x−(11))²+(11), with a≠0. For distinct inputs 35 and 10, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-552/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence -552/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0099/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate





## 100. Average Rate of a Vertex-Form Quadratic

On the full real domain, let q:R→R be q(x)=(25)(x−(12))²+(12), with a≠0. For distinct inputs 11 and 37, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation600/1

The constant k cancels, and factoring the square difference gives a(x0+x1−2h), hence 600/1. Symmetric distinct inputs can give zero without making q constant. This is an endpoint average rate, not a derivative.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261005-advance-20261006-r03-rate-of-change-0100/)

Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r03-rate-of-change-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261005-advance-20261006-r03-rate-of-change-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261005-advance-20261006-r03-rate-of-change-foundations/validation-report.json)