1.
y = ax² + bx - 7 ...... [1]
圖形以 (-3, 2 ) 為頂點的二次函數:
y = a(x + 3)² + 2 ...... [2]
[2] 和 [1] 表示同一函數:
a(x + 3)² + 2 = ax² + bx - 7
ax² + 6ax + (9a + 2) = ax² + bx - 7
比較常數項:
9a + 2 = -7
9a = -9
a = -1
比較 x 項的係數:
6a = b
b = 6(-1)
b = -6
=====
2.
f(x) = ax² + bx + c
f(0) = 17
a(0)² + b(0) + c = 17
c = 17
f(-4) = f(3)
a(-4)² + b(-4) + c = a(3)² + b(3) + c
7a = 7b
a = b ...... [1]
f(3) = 5
ax² + bx + c = 5
a(3)² + b(3) + c = 5
9a + 3b + c = 5 ...... [2]
將 [1] 及 c = 17 代入 [2] 中:
9b + 3b + 17 = 5
12b = -12
b = -1
將 b = -1 代入 [2] 中:
a = -1
a + b + c = (-1) + (-1) + 17 = 15
=====
3.
二次函數: y = -x² + 2x + 8 ...... [1]
x 軸: y = 0 ...... [2]
[1] = [2]:
-x² + 2x + 8 = 0
x² - 2x - 8 = 0
(x + 2)(x - 4) = 0
x = -2 或 x = 4
所以 A、B兩點座標分別為 (-2, 0) 及 (4, 0)。
AB 線段長 = 4 - (-2) = 6
二次函數:
y = -x² + 2x + 8
y = -(x² - 2x) + 8
y = -(x² - 2x + 1) + 1 + 8
y = -(x - 1) + 9
頂點 C 的座標 = (1, 9)
ΔABC 面積 = (1/2) x 6 x 9 = 27
=====
4.
y = f(x) = x² + x - 6
當 x = 0 (y 軸):
y = (0)² + (0) - 6
y = -6
A 點的座標為 (0, -6)。
當 y = 0 (x 軸):
x² + x - 6 = 0
(x + 2)(x - 3) = 0
x = -2 或 x = 3
C、B 兩點的座標分別為 (-2, 0) 及 (3, 0)
ΔABC 面積 = (1/2) x [3 - (-2)] x 6 =15
=====
5.
設 A、B 的座標分別為 (a, 0) 及 (b, 0)。(設 a > b)
則 a 和 b 為 2x² + 12x + c = 0 之兩根。
兩根之和: a + b = -12/2= -6
兩根之積: ab = c/2
AB 線段長:
a - b = 4
(a - b)² = 16
(a + b)² - 4ab = 16
(-6)² - 4(c/2) = 16
c = 10
=====
6.
二次函數:
f(x) = ax² + bx + c ...... [1]
當 x = 2 時有極大值 9 的二次函數:
f(x) = a(x - 2)² + 9 ...... [2]
將 f(0) = 1 代入 [1] 中:
a(0)² + b(0) + c = 1
c = 1
將 f(0) = 1 代入 [2] 中:
a(0 - 2)² + 9 = 1
4a = -8
a = -2
所以 f(x) = -2x² + bx + 1
f(2) = 9 :
-2(2)² + b(2) + 1 = 9
2b = 16
b = 8
所以 f(x) = -2x² + 8x + 1
f(3) = -2(3)² + 8(3) + 1 = 7
=====
7.
頂點為 (-1, 3)。
設二次函數為 f(x) = a(x+ 1)² + 3
f(1) = 15
a(1 + 1)² + 3 = 15
4a = 12
a = 3
f(x) = 3(x + 1)² + 3
f(0) = 3(0 + 1)² + 3 = 6
參考資料: micatkie
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