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
What is the slope of the graph of \( y=\frac{1}{4}(27x+12)+7x \) 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 s passes through the point \( (0,\ 0) \) and is parallel to the line represented by the equation \( y=36x+6 \). If line s also passes through the point \( (9,\ d) \), what is the value of d?
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
Line k is defined by \( y=-\frac{22}{3}x+2. \) 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・斜率与截距挑战
For the linear function g, \( g(4)=10 \) and \( g(x+1)-g(x)=6 \). Which equation defines g?
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
Line \( p \) is defined by \( kx+7y=16. \) Line p passes through the points \( (r,\ 0) \) and \( (0,\ t) \) in the xy-plane, where \( k \), \( r \), and \( t \) are nonzero constants and \( r \) is not equal to \( t \). Which expression represents the value of \( k \)?
原库参考解析
直线方程・斜率与截距进阶
For the linear function \( g \), the table shows four values of \( x \) and their corresponding values of \( g\left(x\right) \). The function can be written as \( g\left(x\right)=mx+b \), where \( m \) and \( b \) are constants. What is the value of \( b \)?
原库参考解析
直线方程・斜率与截距挑战
In the xy-plane, lines k and j are perpendicular and intersect at the point(4, 23). 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・斜率与截距进阶
In the linear function f, f(0)=3 and f(5)=3. Which equation defines f?
原库参考解析
直线方程・斜率与截距进阶
The graph of -6x+4.4y=-660 in the xy-plane has intercepts at (a,0) and (0,b). What is the value of ab?
原库参考解析
直线方程・斜率与截距挑战
In the xy-plane, line r passes through the points (2,16) and (7, 17). Line s passes through the point (1, 2) and is perpendicular to line r. An equation of line s is ax+7y=c, where a and c are constants. What is the value of c?
原库参考解析
