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第 10 週 · 二次函數與圖像 · 第 4 節

圖解法解二次方程 · 根 = 拋物線與 x 軸的交點

把「畫圖」和「解方程」連起來 —— 解 ax² + bx + c = 0，就是找拋物線 y = ax² + bx + c 跟 x 軸的交點。配滿圖例，閉環判別式。

▶ 第一關： 根 = 拋物線與 x 軸的交點
xyroot 1root 2roots = x-intercepts (where y = 0)y = ax²+bx+c
y = x² − 5x + 6, set y = 0xyx = 2x = 3(2,0)(3,0)(x−2)(x−3)=0 ⇒ roots 2 and 3
read the roots off the graphxy(0, 0)(7, 0)y = x² − 7xcuts x-axis at x = 0 and x = 7
▶ 第二關： 交點個數 ↔ 判別式 Δ（閉環 W9D3）
Δ > 02 pointscross twiceΔ = 01 pointjust touchesΔ < 00 pointsnever touches
the translation bridgesign of Δb²−4acreal rootshow manyx-axis cutshow manyΔ > 0 → 2 → 2 Δ = 0 → 1 → 1 Δ < 0 → 0 → 0all three columns ALWAYS match
▶ 第三關： 圖像驗代數 + 圖解 f(x) = k
graph: read it23=algebra: compute itx² − 5x + 6 = 0(x − 2)(x − 3) = 0x = 2 or x = 3same roots — two ways, one answer
solve x² = 9 by sliding a line y = 9xyy = 9x = −3x = 3y = x²intersections give the solutions x = ±3
⚔ BOSS： 判交點數 · 讀根 · 對應 Δ（含 a < 0）
a < 0: opens down, still cuts x-axisxyrootrooty = −x²+2x+3Δ > 0 ⇒ 2 cuts, even opening downward