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
The table above shows some values of x and their corresponding values f(x) for the linear function f. 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・斜率与截距挑战
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・斜率与截距挑战
Oil and gas production in a certain area dropped from 4 million barrels in 2000 to 1.9 million barrels in 2013. Assuming that the oil and gas production decreased at a constant rate, which of the following linear functions f best models the production, in millions of barrels, t years after the year 2000?
原库参考解析
直线方程・斜率与截距挑战
The function f is defined by f(x) = ax + b , where a and b are constants. In the xy-plane, the graph of y = f(x) has an x-intercept at (2 , 0) and a y-intercept at (0 , -323). What is the value of b ?
原库参考解析
直线方程・斜率与截距挑战
In the xy-plane, the graph of y = x2 -9 intersects line p at (1 , a ) and (5 , b ), where a and b are constants. What is the slope of line p ?
原库参考解析
直线方程・斜率与截距进阶
j(x) = mx+144For the linear function j, m is a constant and j(12)=18. What is the value of j(10)?
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・斜率与截距基础
Line k is defined by y =3x +15. Line j is perpendicular to line k in the xy-plane. What is the slope of line j ?
原库参考解析
直线方程・斜率与截距基础
What is the y-intercept of the line graphed?
原库参考解析
直线方程・斜率与截距挑战
In the xy-plane, line k passes through the points (0, -5) and (1, -1). Which equation defines line k ?
原库参考解析
直线方程・斜率与截距挑战
When line n is graphed in the xy-plane, it has an x-intercept of (−4, 0) and a y-intercept of (0, \( \frac{86}{3} \) ). What is the slope of line n?
原库参考解析
直线方程・斜率与截距挑战
The table shows three values of x and their corresponding values of y, where n is a constant, for the linear relationship between x and y. What is the slope of the line that represents this relationship in the xy-plane?
原库参考解析
