Average Rate of Change Foundations
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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 explanation
0/1
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
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
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 explanation
1/1
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
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
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 explanation
1/1
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
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
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 explanation
1/1
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
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
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 explanation
9/2
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
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
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 explanation
11/4
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
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
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 explanation
13/6
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
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
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 explanation
15/1
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
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
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 explanation
17/3
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
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
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 explanation
19/5
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
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
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 explanation
3/1
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
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
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 explanation
23/2
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
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
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 explanation
25/4
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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 explanation
Yes
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
0/1
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
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 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
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 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
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 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
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 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
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 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
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 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
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 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
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 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
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 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
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 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
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 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
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 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
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 explanation
0/1
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
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
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 explanation
6/1
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
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
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 explanation
20/1
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
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
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 explanation
42/1
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
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
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 explanation
72/1
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
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
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 explanation
110/1
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
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
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 explanation
156/1
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
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
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 explanation
210/1
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
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
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 explanation
272/1
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
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
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 explanation
342/1
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
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
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 explanation
420/1
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
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
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 explanation
506/1
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
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
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 explanation
600/1
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
Average rate of change between distinct inputs · qf-rate-quadratic-average · candidate
Canonical question JSONL · English JSONL · Validation report