这部分需要会什么？
- 由两点或图象求斜率
- 识别斜截式、点斜式和一般式
- 解释截距并判断平行与垂直
核心概念与公式
两点 \((x_1,y_1),(x_2,y_2)\) 且 \(x_1\ne x_2\) 时，斜率 \(m=(y_2-y_1)/(x_2-x_1)\)。它表示横坐标每增加 1，纵坐标的变化量。直线常写成 \(y=mx+b\) 或 \(y-y_1=m(x-x_1)\)。
\(y\) 截距为 \((0,b)\)；求 \(x\) 截距需令 \(y=0\)。不同非竖直平行线斜率相同；两条非竖直直线垂直时 \(m_1m_2=-1\)。竖直线 \(x=c\) 的斜率不存在，它与水平线垂直。
做题时怎样使用
- 有两点先求斜率；有一点和斜率优先用点斜式
- 需要截距时分别令另一个坐标为零
- 把所得直线代回给定点，检查坐标和单位
容易混淆的地方
斜率分子与分母必须采用相同的点的顺序。截距是坐标轴上的位置；\(y\) 截距的纵坐标才是式中的 \(b\)。
用一个例题串起方法
知识点教学例题一条直线经过 \((2,7)\) 与 \((5,16)\)。求它的方程及 \(y\) 截距。
展开解题思路 ＋
\(m=(16-7)/(5-2)=3\)。用点斜式得 \(y-7=3(x-2)\)，即 \(y=3x+1\)。因此 \(y\) 截距为 \((0,1)\)。
本知识点题库
先独立作答，再查看答案与解析。
直线方程・斜率与截距进阶
Line t in the xy-plane has a slope of \( \frac{1}{3} \) and passes through the point (15, −6). Which equation defines line t?
参考答案
直线方程・斜率与截距进阶
For the linear function \( f \), the graph of \( y=f(x) \) in the xy-plane has a slope of 0 and passes through the point (0, 36). Which equation defines \( f \)?
参考答案
直线方程・斜率与截距挑战
For a linear relationship between x and y, the table gives three values of x and their corresponding values of y in terms of a constant r. What is the slope of the line in the xy-plane that represents this relationship?
参考答案
直线方程・斜率与截距挑战
The table shows three values of x and their corresponding values of y. There is a linear relationship between x and y. In the xy-plane, what is the x-intercept of the graph that represents this relationship?
参考答案
直线方程・斜率与截距挑战
A line passes through the points (5, 7) and (13, 35) in the xy-plane. What is the slope of the line?
参考答案
直线方程・斜率与截距进阶
Line p is defined by the equation \( y=\frac{7}{5}x-\frac{49}{5} \) and line r is defined by the equation \( ax+by=1 \), where a and b are constants. In the xy-plane, line r is perpendicular to line p and passes through the point (12, 7). What is the value of b?
参考答案
直线方程・斜率与截距进阶
Line h is defined by \( \frac{1}{2}x+\frac{1}{9}y−54=0 \). Line j is perpendicular to line h in the xy-plane. What is the slope of line j?
参考答案
直线方程・斜率与截距挑战
Line t in the xy-plane has a slope of \( \frac{1}{9} \) and passes through the point (45, -7). Which equation defines line t?
参考答案
直线方程・斜率与截距进阶
In the xy-plane, line k has a slope of 4 and a y-intercept of \( \left(0,-28\right) \) . What is the x-coordinate of the x-intercept of line k?
参考答案
直线方程・斜率与截距挑战
The table shows two values of x and their corresponding values of y. In the xy-plane, the graph of the linear equation representing this relationship passes through the point \( \left(\frac{1}{6},\ a\right) \) . What is the value of a?
参考答案
直线方程・斜率与截距挑战
Line j in the xy-plane passes through the points (7, 0) and (0, 3). Line k passes through the origin and is perpendicular to line j. What is the x-coordinate of the point where lines j and k intersect?
参考答案
直线方程・斜率与截距挑战
The table gives the average time t, in minutes, it takes Mariana to travel a certain distance d, in miles. Which equation could represent this linear relationship?
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