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P2-5

直線圖像

Linear Graphs

PAPER 2 專題 5 / Topic 5

直線圖像 Linear Graphs

P2-5.1 直線方程基礎 Basics of Linear Equations

📐 直線方程形式 Forms of Linear Equations

中文	English	公式 Formula	用途 Usage
斜截式	Slope-Intercept Form	$y = mx + c$	讀斜率和 y-截距 Read slope & y-intercept
一般式	General Form	$Ax + By + C = 0$	標準答案形式 Standard answer form
點斜式	Point-Slope Form	$y - y_1 = m(x - x_1)$	已知一點和斜率 Given point & slope
兩點式	Two-Point Form	$\dfrac{y - y_1}{y_2 - y_1} = \dfrac{x - x_1}{x_2 - x_1}$	已知兩點 Given two points

P2-5.2 斜率與截距 Slope & Intercepts

📈 斜率公式與意義 Slope Formula & Meaning

斜率公式：

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\text{垂直變化}}{\text{水平變化}}$

斜率意義：

• $m > 0$：向右上升 ↗

• $m < 0$：向右下降 ↘

• $m = 0$：水平線

• $m$ 無定義：垂直線

Slope Formula:

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\text{rise}}{\text{run}}$

Slope Meaning:

• $m > 0$: Rising ↗

• $m < 0$: Falling ↘

• $m = 0$: Horizontal

• $m$ undefined: Vertical

📊 直線圖像類型 Types of Linear Graphs

$m > 0, c > 0$

向右上升，正截距
Rising, positive intercept

$m < 0, c > 0$

向右下降，正截距
Falling, positive intercept

$m > 0, c < 0$

向右上升，負截距
Rising, negative intercept

$m < 0, c < 0$

向右下降，負截距
Falling, negative intercept

P2-5.3 平行線與垂直線 Parallel & Perpendicular Lines

✏️ 平行線與垂直線條件 Conditions

中文	English	條件 Condition
平行線	Parallel Lines	$m_1 = m_2$ 斜率相等 Equal slopes
垂直線	Perpendicular Lines	$m_1 \times m_2 = -1$ 斜率乘積為 −1

⚡ 秒殺技巧 Quick Tips

1. 判斷升降：睇斜率 $m$ 正負

2. y-截距：代 $x = 0$ 入方程

3. x-截距：代 $y = 0$ 入方程

4. 平行：斜率相等

5. 垂直：斜率乘積 = −1

6. 過原點：$c = 0$（方程為 $y = mx$）

1. Rising/Falling: Check sign of $m$

2. y-intercept: Sub $x = 0$

3. x-intercept: Sub $y = 0$

4. Parallel: Equal slopes

5. Perpendicular: $m_1 \times m_2 = -1$

6. Through origin: $c = 0$ ($y = mx$)

P2-5.4 DSE 歷屆真題 Past Paper Questions

DSE 2023 Q25

題目：直線 $y = 2x - 3$ 的斜率和 y-截距分別是？

Question: What are the slope and y-intercept of the line $y = 2x - 3$?

📝 秒殺 Quick Solution

比較 $y = mx + c$：斜率 $m = \mathbf{2}$，y-截距 $c = \mathbf{-3}$

DSE 2022 Q26

題目：求過點 $(2, 5)$ 且斜率為 3 的直線方程。

Question: Find the equation of the line passing through $(2, 5)$ with slope 3.

📝 秒殺 Quick Solution

用點斜式：$y - 5 = 3(x - 2)$

$y - 5 = 3x - 6$

$y = 3x - 1$ 或 $\mathbf{3x - y - 1 = 0}$

DSE 2021 Q27

題目：直線 $L_1: 2x - y + 3 = 0$，求與 $L_1$ 平行的直線斜率。

Question: Line $L_1: 2x - y + 3 = 0$. Find the slope of lines parallel to $L_1$.

📝 秒殺 Quick Solution

改寫為斜截式：$y = 2x + 3$

斜率 $m = \mathbf{2}$

平行線斜率相等，所以答案是 $\mathbf{2}$

DSE 2020 Q28

題目：直線 $L$ 的斜率是 $\dfrac{1}{2}$，求與 $L$ 垂直的直線斜率。

Question: Line $L$ has slope $\dfrac{1}{2}$. Find the slope of lines perpendicular to $L$.

📝 秒殺 Quick Solution

垂直線：$m_1 \times m_2 = -1$

$\dfrac{1}{2} \times m_2 = -1$

$m_2 = \mathbf{-2}$

📝 練習題 Practice Questions

1. 求過 $(1, 4)$ 和 $(3, 10)$ 的直線斜率。
Find the slope of the line through $(1, 4)$ and $(3, 10)$.

2. 直線 $3x + 2y - 6 = 0$ 的 y-截距是？
What is the y-intercept of $3x + 2y - 6 = 0$?

3. 若直線斜率為 −4，求與之垂直的直線斜率。
If a line has slope −4, find the slope of perpendicular lines.

📋 答案 Answers

1. 3 2. 3 3. $\dfrac{1}{4}$

📚 MathsKiller 中英對照天書 | P2-5 直線圖像 Linear Graphs