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 n is shown in the xy-plane. Line k (not shown) is perpendicular to line n. What is the slope of line k?
原库参考解析
直线方程・斜率与截距挑战
There is a linear relationship between x and y. The table shows three values of x and their corresponding values of y in terms of a constant n. What is the slope of the line that represents this relationship in the xy-plane?
原库参考解析
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 passes through the point (0, 2) and (3, 3). What is the slope of y = f(x) ?
原库参考解析
直线方程・斜率与截距进阶
Line n is shown in the xy-plane. Line k (not shown) is perpendicular to line n. What is the slope of line k?
原库参考解析
直线方程・斜率与截距挑战
There is a linear relationship between x and y. The table shows three values of x and their corresponding value of y in terms of a constant n. What is the slope of the line that represents this relationship in the xy-plane ?
原库参考解析
直线方程・斜率与截距进阶
In the xy-plane, line p has a slope of \( \frac{5}{3} \) and an x-intercept of(9,0). what is the y-coordinate of the y-intercept of line p?
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 and a > 0. In the xy-plane, the graph of y = f(x) has a y-intercept at (0, −26) and passes through the point (2, 9). What is the value of a + b?
原库参考解析
直线方程・斜率与截距挑战
Line j is shown in the xy-plane.Line k (not shown) is parallel to line j and passes through the point (0,5). Which equation defines line k?
原库参考解析
直线方程・斜率与截距挑战
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, 25). Which equation defines f?
原库参考解析
直线方程・斜率与截距进阶
The table shows three values of x and their corresponding values of y. There is a linear relationship between x and y. What is the y-coordinate of the y-intercept of the line in the xy-plane that represents this relationship?
原库参考解析
直线方程・斜率与截距挑战
In the xy-plane, lines k and j are perpendicular and intersect at the point (5,28). Line j passes through the origin, and line k passes through the point (p, 0), where p is a constant. What is the value of p?
原库参考解析
直线方程・斜率与截距挑战
The roofs on the buildings of a public garden are designed to allow water to drain into the garden. The table shows the relationship between the area x, in square feet, of a roof and the amount of water f (x), in gallons, drained from the roof into the garden over a certain period of time. Which equation could define f
原库参考解析
