Lines, curves and coordinates · 直线、曲线与坐标
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| gradient/ˈɡreɪdɪənt/ | 斜率 | xié lǜ |
| y-intercept/waɪ ˌɪntəˈsept/ | y 轴截距 | y zhóu jié jù |
| parallel/ˈpærəlel/ | 平行 | píng xíng |
| perpendicular/ˌpɜːpənˈdɪkjʊlə/ | 垂直 | chuí zhí |
| midpoint/ˈmɪdpɔɪnt/ | 中点 | zhōng diǎn |
| distance/ˈdɪstəns/ | 距离 | jù lí |
| roots/ruːts/ | 根 | gēn |
| parabola/pəˈræbələ/ | 抛物线 | pāo wù xiàn |
| vertex/ˈvɜːteks/ | 顶点 | dǐng diǎn |
| graphically/ˈɡræfɪkli/ | 用图像求解 | yòng tú xiàng qiú jiě |
Connect a rule with a graph
- Coordinates let you test where a line or curve passes. A picture suggests relationships; algebra checks exact values.
- Label axes and relevant points when sketching. A graph's scale affects what can be read accurately from it.
将规则与图像连接
- 坐标让您测试直线或曲线经过的位置。图片暗示了关系；代数检查精确值。
- 绘制草图时标注轴和相关点。图像的刻度影响能否从中准确读取内容。
State which lines have a gradient
- In $y=mx+c$, m is the gradient 斜率 and c the y-intercept y 轴截距. For a nonvertical line, $m=(y_2-y_1)/(x_2-x_1)$.
- Distinct nonvertical parallel 平行 lines have equal gradients; nonvertical perpendicular 垂直 lines have gradient product negative 1. Vertical lines have undefined gradient and are perpendicular to horizontal lines.
说明哪些线具有斜率
- 在 $y=mx+c$ 中，m是斜率 斜率，c是y轴截距 y轴截距。对于非垂直线，$m=(y_2-y_1)/(x_2-x_1)$。
- 不同的非垂直平行 平行线具有相等的斜率；非垂直垂直 垂直线的斜率乘积为负1。垂直线斜率未定义，且与水平线垂直。
Change the gradient and the intercept · 改变斜率和截距
$y = mx + c$
See what m and c each control before you calculate them. · 在计算 m 和 c 之前，先明确它们各自控制什么。
Find the gradient of the line through (2, 1) and (6, 9). · 求经过 (2, 1) 和 (6, 9) 两点的直线的斜率。
Rise over run: (9 − 1) ÷ (6 − 2) = 8 ÷ 4 = 2. · 上升比水平距离：(9 − 1) ÷ (6 − 2) = 8 ÷ 4 = 2。
That line is y = 2x + c. Find c. · 该直线方程为 y = 2x + c。求 c 的值。
Substitute (2, 1): 1 = 4 + c, so c = −3. Check with (6, 9): 2(6) − 3 = 9 ✓. · 代入 (2, 1)：1 = 4 + c，得 c = −3。用 (6, 9) 验证：2(6) − 3 = 9 ✓。
Which gradient is perpendicular to a line of gradient 4? · 哪个斜率与斜率为 4 的直线垂直？
The gradients must multiply to −1, and 4 × (−1/4) = −1. · 两斜率之积必须为 −1，且 4 × (−1/4) = −1。
Measure segments using coordinates
- The midpoint 中点 averages each coordinate: $M=((x_1+x_2)/2,(y_1+y_2)/2)$.
- The distance 距离 is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. A perpendicular bisector passes through the midpoint at a right angle to the segment.
使用坐标测量线段
- 中点 中点平均每个坐标：$M=((x_1+x_2)/2,(y_1+y_2)/2)$。
- 距离 距离是 $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$。垂直平分线穿过中点并与线段成直角。
Find the distance between (1, 2) and (4, 6). · 求 (1, 2) 与 (4, 6) 之间的距离。
Differences 3 and 4, so d = √(9 + 16) = √25 = 5 — the 3-4-5 triangle. · 差值分别为 3 和 4，故 d = √(9 + 16) = √25 = 5 —— 这是 3-4-5 直角三角形。
Roots can cross or touch
- A quadratic graph is a parabola 抛物线; its vertex 顶点 is the turning point. Real roots 根 are x-values where it meets the x-axis, including a repeated root that touches.
- To solve graphically 用图像求解, read intersections of the relevant graphs. Equate their expressions to check exact coordinates and distinguish a strict inequality from equality at an endpoint.
根可以交叉或接触
- 二次图像是抛物线 抛物线；其顶点 顶点是转折点。实根 根是它与x轴相交处的x值，包括接触点的重根。
- 要用图像求解 用图像求解，读取相关图像的交点。令它们的表达式相等以检查精确坐标，并区分端点处的严格不等式与等式。
What are the roots of a quadratic, geometrically? · 二次方程的根在几何上代表什么？
A real root satisfies y = 0. The graph can cross the x-axis or touch it at a repeated root. · 实根满足 y = 0。图像可以在重根处穿过或触碰x轴。
Put the steps of finding a line through two points in order. · 将求过两点直线方程的步骤排序。
Checking with the unused point is the step that catches an arithmetic slip. · 用未使用的点验证是发现算术错误的关键步骤。
Find a line from two points. For $(2,1)$ and $(6,9)$, $m=(9-1)/(6-2)=2$. In $y=mx+c$, $c=y-mx=1-2(2)=-3$. Thus $y=2x-3$; substituting the other point gives 9 as required.
从两点求一条直线。 对于 $(2,1)$ 和 $(6,9)$，$m=(9-1)/(6-2)=2$。在 $y=mx+c$ 中，$c=y-mx=1-2(2)=-3$。因此 $y=2x-3$；代入另一点得到9，符合要求。
Find both coordinates of an intersection. The equations $y=x+1$ and $y=x^2-3x+1$ give $x(x-4)=0$. Substitution gives $(0,1)$ and $(4,5)$, not just two x-values. Sheet 1.3 also asks for a labelled quadratic sketch.
求交点的两个坐标。 方程 $y=x+1$ 和 $y=x^2-3x+1$ 给出 $x(x-4)=0$。代入得到 $(0,1)$ 和 $(4,5)$，而不仅仅是两个x值。表1.3还要求绘制带标签的二次图像。
Equal gradients alone do not distinguish identical from distinct parallel lines. The negative-reciprocal test does not cover a vertical line. Check the equation form before applying a rule.
仅凭斜率相等不能区分相同与不同的平行线。负倒数测试不适用于垂直线。应用规则前请检查方程形式。