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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-r02-quadratic-function-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.
On the full real domain, let q:R→R be q(x)=(1)(x−(0))²+(0), with a≠0. Evaluate q(0) exactly.
0
Subtract h, square the displacement, multiply by a, and add k: q(0)=0. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(-11))²+(-9), with a≠0. Evaluate q(-13) exactly.
-17
Subtract h, square the displacement, multiply by a, and add k: q(-13)=-17. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(-10))²+(-8), with a≠0. Evaluate q(-7) exactly.
19
Subtract h, square the displacement, multiply by a, and add k: q(-7)=19. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(-9))²+(-7), with a≠0. Evaluate q(-13) exactly.
-71
Subtract h, square the displacement, multiply by a, and add k: q(-13)=-71. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(5)(x−(-8))²+(-6), with a≠0. Evaluate q(-3) exactly.
119
Subtract h, square the displacement, multiply by a, and add k: q(-3)=119. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-6)(x−(-7))²+(-5), with a≠0. Evaluate q(-8) exactly.
-11
Subtract h, square the displacement, multiply by a, and add k: q(-8)=-11. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(7)(x−(-6))²+(-4), with a≠0. Evaluate q(-4) exactly.
24
Subtract h, square the displacement, multiply by a, and add k: q(-4)=24. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-8)(x−(-5))²+(-3), with a≠0. Evaluate q(-8) exactly.
-75
Subtract h, square the displacement, multiply by a, and add k: q(-8)=-75. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(9)(x−(-4))²+(-2), with a≠0. Evaluate q(0) exactly.
142
Subtract h, square the displacement, multiply by a, and add k: q(0)=142. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-10)(x−(-3))²+(-1), with a≠0. Evaluate q(-8) exactly.
-251
Subtract h, square the displacement, multiply by a, and add k: q(-8)=-251. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(11)(x−(-2))²+(0), with a≠0. Evaluate q(-1) exactly.
11
Subtract h, square the displacement, multiply by a, and add k: q(-1)=11. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-12)(x−(-1))²+(1), with a≠0. Evaluate q(-3) exactly.
-47
Subtract h, square the displacement, multiply by a, and add k: q(-3)=-47. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(13)(x−(0))²+(2), with a≠0. Evaluate q(3) exactly.
119
Subtract h, square the displacement, multiply by a, and add k: q(3)=119. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-14)(x−(1))²+(3), with a≠0. Evaluate q(-3) exactly.
-221
Subtract h, square the displacement, multiply by a, and add k: q(-3)=-221. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(15)(x−(2))²+(4), with a≠0. Evaluate q(7) exactly.
379
Subtract h, square the displacement, multiply by a, and add k: q(7)=379. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-16)(x−(3))²+(5), with a≠0. Evaluate q(2) exactly.
-11
Subtract h, square the displacement, multiply by a, and add k: q(2)=-11. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(17)(x−(4))²+(6), with a≠0. Evaluate q(6) exactly.
74
Subtract h, square the displacement, multiply by a, and add k: q(6)=74. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-18)(x−(5))²+(7), with a≠0. Evaluate q(2) exactly.
-155
Subtract h, square the displacement, multiply by a, and add k: q(2)=-155. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(19)(x−(6))²+(8), with a≠0. Evaluate q(10) exactly.
312
Subtract h, square the displacement, multiply by a, and add k: q(10)=312. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-20)(x−(7))²+(9), with a≠0. Evaluate q(2) exactly.
-491
Subtract h, square the displacement, multiply by a, and add k: q(2)=-491. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(21)(x−(8))²+(10), with a≠0. Evaluate q(9) exactly.
31
Subtract h, square the displacement, multiply by a, and add k: q(9)=31. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-22)(x−(9))²+(11), with a≠0. Evaluate q(7) exactly.
-77
Subtract h, square the displacement, multiply by a, and add k: q(7)=-77. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(23)(x−(10))²+(12), with a≠0. Evaluate q(13) exactly.
219
Subtract h, square the displacement, multiply by a, and add k: q(13)=219. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(-24)(x−(11))²+(13), with a≠0. Evaluate q(7) exactly.
-371
Subtract h, square the displacement, multiply by a, and add k: q(7)=-371. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(25)(x−(12))²+(14), with a≠0. Evaluate q(17) exactly.
639
Subtract h, square the displacement, multiply by a, and add k: q(17)=639. Retain the coefficient sign; at x=h the square is zero.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-evaluation · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(-12))²+(2), with a≠0. Give the vertex as an ordered coordinate pair.
(-12, 2)
The vertex is (h,k)=(-12, 2), not merely the scalar extremum value k. Its axis is x=-12.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(-11))²+(-1), with a≠0. Give the vertex as an ordered coordinate pair.
(-11, -1)
The vertex is (h,k)=(-11, -1), not merely the scalar extremum value k. Its axis is x=-11.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(-10))²+(-2), with a≠0. Give the vertex as an ordered coordinate pair.
(-10, -2)
The vertex is (h,k)=(-10, -2), not merely the scalar extremum value k. Its axis is x=-10.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(-9))²+(5), with a≠0. Give the vertex as an ordered coordinate pair.
(-9, 5)
The vertex is (h,k)=(-9, 5), not merely the scalar extremum value k. Its axis is x=-9.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(-8))²+(-4), with a≠0. Give the vertex as an ordered coordinate pair.
(-8, -4)
The vertex is (h,k)=(-8, -4), not merely the scalar extremum value k. Its axis is x=-8.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(-7))²+(-5), with a≠0. Give the vertex as an ordered coordinate pair.
(-7, -5)
The vertex is (h,k)=(-7, -5), not merely the scalar extremum value k. Its axis is x=-7.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(-6))²+(8), with a≠0. Give the vertex as an ordered coordinate pair.
(-6, 8)
The vertex is (h,k)=(-6, 8), not merely the scalar extremum value k. Its axis is x=-6.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(-5))²+(-7), with a≠0. Give the vertex as an ordered coordinate pair.
(-5, -7)
The vertex is (h,k)=(-5, -7), not merely the scalar extremum value k. Its axis is x=-5.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(-4))²+(-8), with a≠0. Give the vertex as an ordered coordinate pair.
(-4, -8)
The vertex is (h,k)=(-4, -8), not merely the scalar extremum value k. Its axis is x=-4.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(-3))²+(11), with a≠0. Give the vertex as an ordered coordinate pair.
(-3, 11)
The vertex is (h,k)=(-3, 11), not merely the scalar extremum value k. Its axis is x=-3.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(-2))²+(-10), with a≠0. Give the vertex as an ordered coordinate pair.
(-2, -10)
The vertex is (h,k)=(-2, -10), not merely the scalar extremum value k. Its axis is x=-2.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(-1))²+(-11), with a≠0. Give the vertex as an ordered coordinate pair.
(-1, -11)
The vertex is (h,k)=(-1, -11), not merely the scalar extremum value k. Its axis is x=-1.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(0))²+(14), with a≠0. Give the vertex as an ordered coordinate pair.
(0, 14)
The vertex is (h,k)=(0, 14), not merely the scalar extremum value k. Its axis is x=0.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(1))²+(-13), with a≠0. Give the vertex as an ordered coordinate pair.
(1, -13)
The vertex is (h,k)=(1, -13), not merely the scalar extremum value k. Its axis is x=1.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(2))²+(-14), with a≠0. Give the vertex as an ordered coordinate pair.
(2, -14)
The vertex is (h,k)=(2, -14), not merely the scalar extremum value k. Its axis is x=2.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(3))²+(17), with a≠0. Give the vertex as an ordered coordinate pair.
(3, 17)
The vertex is (h,k)=(3, 17), not merely the scalar extremum value k. Its axis is x=3.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(4))²+(-16), with a≠0. Give the vertex as an ordered coordinate pair.
(4, -16)
The vertex is (h,k)=(4, -16), not merely the scalar extremum value k. Its axis is x=4.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(5))²+(-17), with a≠0. Give the vertex as an ordered coordinate pair.
(5, -17)
The vertex is (h,k)=(5, -17), not merely the scalar extremum value k. Its axis is x=5.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(6))²+(20), with a≠0. Give the vertex as an ordered coordinate pair.
(6, 20)
The vertex is (h,k)=(6, 20), not merely the scalar extremum value k. Its axis is x=6.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(7))²+(-19), with a≠0. Give the vertex as an ordered coordinate pair.
(7, -19)
The vertex is (h,k)=(7, -19), not merely the scalar extremum value k. Its axis is x=7.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(8))²+(-20), with a≠0. Give the vertex as an ordered coordinate pair.
(8, -20)
The vertex is (h,k)=(8, -20), not merely the scalar extremum value k. Its axis is x=8.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(9))²+(23), with a≠0. Give the vertex as an ordered coordinate pair.
(9, 23)
The vertex is (h,k)=(9, 23), not merely the scalar extremum value k. Its axis is x=9.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(10))²+(-22), with a≠0. Give the vertex as an ordered coordinate pair.
(10, -22)
The vertex is (h,k)=(10, -22), not merely the scalar extremum value k. Its axis is x=10.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(11))²+(-23), with a≠0. Give the vertex as an ordered coordinate pair.
(11, -23)
The vertex is (h,k)=(11, -23), not merely the scalar extremum value k. Its axis is x=11.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(12))²+(26), with a≠0. Give the vertex as an ordered coordinate pair.
(12, 26)
The vertex is (h,k)=(12, 26), not merely the scalar extremum value k. Its axis is x=12.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-vertex · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(-7))²+(-12), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=-12
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=-12; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(-6))²+(-11), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=-11
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=-11; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(-5))²+(-10), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=-10
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=-10; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(-4))²+(-9), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=-9
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=-9; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(-3))²+(-8), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=-8
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=-8; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(-2))²+(-7), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=-7
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=-7; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(-1))²+(-6), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=-6
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=-6; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(0))²+(-5), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=-5
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=-5; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(1))²+(-4), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=-4
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=-4; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(2))²+(-3), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=-3
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=-3; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(3))²+(-2), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=-2
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=-2; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(4))²+(-1), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=-1
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=-1; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(5))²+(0), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=0
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=0; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(6))²+(1), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=1
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=1; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(7))²+(2), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=2
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=2; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(8))²+(3), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=3
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=3; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(9))²+(4), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=4
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=4; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(10))²+(5), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=5
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=5; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(11))²+(6), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=6
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=6; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(12))²+(7), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=7
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=7; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(13))²+(8), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=8
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=8; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(14))²+(9), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=9
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=9; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(15))²+(10), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=10
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=10; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(16))²+(11), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global maximum=11
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global maximum=11; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(17))²+(12), with a≠0. Give the global minimum or global maximum value, according to the coefficient sign. Do not substitute a restricted interval.
global minimum=12
Because (x−h)²≥0, a>0 gives a global minimum and a<0 a global maximum, attained at x=h. Here global minimum=12; the function is unbounded in the other direction.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-extremum · candidate
On the full real domain, let q:R→R be q(x)=(1)(x−(0))²+(0), with a≠0. Reflect the graph point (0,q(0)) across x=0. Give the reflected (input,output) pair.
(0, 0)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (0, 0); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-2)(x−(-11))²+(-9), with a≠0. Reflect the graph point (-13,q(-13)) across x=-11. Give the reflected (input,output) pair.
(-9, -17)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-9, -17); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(3)(x−(-10))²+(-8), with a≠0. Reflect the graph point (-7,q(-7)) across x=-10. Give the reflected (input,output) pair.
(-13, 19)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-13, 19); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-4)(x−(-9))²+(-7), with a≠0. Reflect the graph point (-13,q(-13)) across x=-9. Give the reflected (input,output) pair.
(-5, -71)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-5, -71); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(5)(x−(-8))²+(-6), with a≠0. Reflect the graph point (-3,q(-3)) across x=-8. Give the reflected (input,output) pair.
(-13, 119)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-13, 119); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-6)(x−(-7))²+(-5), with a≠0. Reflect the graph point (-8,q(-8)) across x=-7. Give the reflected (input,output) pair.
(-6, -11)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-6, -11); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(7)(x−(-6))²+(-4), with a≠0. Reflect the graph point (-4,q(-4)) across x=-6. Give the reflected (input,output) pair.
(-8, 24)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-8, 24); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-8)(x−(-5))²+(-3), with a≠0. Reflect the graph point (-8,q(-8)) across x=-5. Give the reflected (input,output) pair.
(-2, -75)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-2, -75); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(9)(x−(-4))²+(-2), with a≠0. Reflect the graph point (0,q(0)) across x=-4. Give the reflected (input,output) pair.
(-8, 142)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-8, 142); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-10)(x−(-3))²+(-1), with a≠0. Reflect the graph point (-8,q(-8)) across x=-3. Give the reflected (input,output) pair.
(2, -251)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (2, -251); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(11)(x−(-2))²+(0), with a≠0. Reflect the graph point (-1,q(-1)) across x=-2. Give the reflected (input,output) pair.
(-3, 11)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-3, 11); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-12)(x−(-1))²+(1), with a≠0. Reflect the graph point (-3,q(-3)) across x=-1. Give the reflected (input,output) pair.
(1, -47)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (1, -47); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(13)(x−(0))²+(2), with a≠0. Reflect the graph point (3,q(3)) across x=0. Give the reflected (input,output) pair.
(-3, 119)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-3, 119); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-14)(x−(1))²+(3), with a≠0. Reflect the graph point (-3,q(-3)) across x=1. Give the reflected (input,output) pair.
(5, -221)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (5, -221); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(15)(x−(2))²+(4), with a≠0. Reflect the graph point (7,q(7)) across x=2. Give the reflected (input,output) pair.
(-3, 379)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (-3, 379); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-16)(x−(3))²+(5), with a≠0. Reflect the graph point (2,q(2)) across x=3. Give the reflected (input,output) pair.
(4, -11)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (4, -11); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(17)(x−(4))²+(6), with a≠0. Reflect the graph point (6,q(6)) across x=4. Give the reflected (input,output) pair.
(2, 74)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (2, 74); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-18)(x−(5))²+(7), with a≠0. Reflect the graph point (2,q(2)) across x=5. Give the reflected (input,output) pair.
(8, -155)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (8, -155); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(19)(x−(6))²+(8), with a≠0. Reflect the graph point (10,q(10)) across x=6. Give the reflected (input,output) pair.
(2, 312)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (2, 312); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-20)(x−(7))²+(9), with a≠0. Reflect the graph point (2,q(2)) across x=7. Give the reflected (input,output) pair.
(12, -491)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (12, -491); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(21)(x−(8))²+(10), with a≠0. Reflect the graph point (9,q(9)) across x=8. Give the reflected (input,output) pair.
(7, 31)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (7, 31); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-22)(x−(9))²+(11), with a≠0. Reflect the graph point (7,q(7)) across x=9. Give the reflected (input,output) pair.
(11, -77)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (11, -77); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(23)(x−(10))²+(12), with a≠0. Reflect the graph point (13,q(13)) across x=10. Give the reflected (input,output) pair.
(7, 219)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (7, 219); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(-24)(x−(11))²+(13), with a≠0. Reflect the graph point (7,q(7)) across x=11. Give the reflected (input,output) pair.
(15, -371)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (15, -371); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
On the full real domain, let q:R→R be q(x)=(25)(x−(12))²+(14), with a≠0. Reflect the graph point (17,q(17)) across x=12. Give the reflected (input,output) pair.
(7, 639)
The reflected input is 2h−x and its squared displacement is unchanged. The reflected point is (7, 639); a point on the axis is its own reflection.
Reviewed step-by-step derivation
Quadratic functions in vertex form · qf-quadratic-function-axis-mirror · candidate
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