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・斜率与截距挑战
| x | f(x) |
| 1 | -64 |
| 2 | 0 |
| 3 | 64 |
For the linear function f, the table shows three values of x and their corresponding values of f (x). Function f is defined by f(x) = ax＋ b,where a and b are constants.What is the value of a - b?
原库参考解析
直线方程・斜率与截距进阶
Line p is defined by 4y + 8x = 6. Line r is perpendicular to line p in the xy-plane. What is the slope of line r?
原库参考解析
直线方程・斜率与截距基础
| x | f(x) |
| 0 | 29 |
| 1 | 32 |
| 2 | 35 |
For the linear function f, the table shows three values of x and their corresponding values of f(x). Which equation defines f(x)?
原库参考解析
直线方程・斜率与截距基础
The function f is defined by f(x) = 7x -84.What is the x-intercept of the graph of y = f(x) in the xy-plane?
原库参考解析
直线方程・斜率与截距基础
What is the y-intercept of the graph shown?
原库参考解析
直线方程・斜率与截距挑战
| x | y |
| k | 13 |
| k + 7 | -15 |
The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is (k - 5,b), where k and b are constants. What is the value of b?
原库参考解析
直线方程・斜率与截距进阶
In the xy-plane, the graph of the linear function f contains the points (0, 2) and (8, 34). Which equation defines f, where y= f(x)?
原库参考解析
直线方程・斜率与截距进阶
The function h is defined by h(x)= 4x + 28. The graph of y= h(x) in the xy-plane has an x-intercept at (a, 0) and a y-intercept at (0, b), where a and b are constants. What is the value of a + b?
原库参考解析
直线方程・斜率与截距进阶
A line in the xy-plane has a slope of
原库参考解析
直线方程・斜率与截距基础
The graph shows the linear relationship between x and y. Which table gives three values of x and their corresponding values of y for this relationship?
原库参考解析
直线方程・斜率与截距基础
What is the y-intercept of the graph shown?
原库参考解析
直线方程・斜率与截距进阶
| x | 10 | 15 | 20 | 25 |
| f(x) | 82 | 137 | 192 | 247 |
The table shows four values of x and their corresponding values of f(x). There is a linear relationship between x and f(x) that is defined by the equation f(x) = mx - 28, where m is a constant. What is the value of m?
原库参考解析
