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
For the linear function \( f \), \( f(1)=654 \) and \( f(2)=648 \). Which of the following defines \( f \)?
原库参考解析
直线方程・斜率与截距进阶
In the xy-plane, line k passes through the points (3, 0) and (0, 5). Which of the following is an equation of line k?
原库参考解析
直线方程・斜率与截距进阶
The graph of the linear function \( f \) is shown, where \( y=f(x) \). What is the slope of the graph of \( f \)?
原库参考解析
直线方程・斜率与截距基础
In the xy-plane, what is the slope of the line that passes through the points (0, 0) and (13, 14)?
原库参考解析
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?
原库参考解析
直线方程・斜率与截距进阶
The function \( f \) is defined by \( f(x)=3x-\frac{1}{4} \) . What is the y-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
The graph of \( 2x+y=11 \) in the xy-plane is a line. What is the slope of the line?
原库参考解析
直线方程・斜率与截距进阶
Line t in the xy-plane has a slope of \( \frac{1}{8} \) and passes through the point (32, -7). Which equation defines line t?
原库参考解析
直线方程・斜率与截距挑战
A linear function g is defined by \( g(x)=ax+b \), where a and b are constants. If \( g(-2)=-7 \) and \( g(1)=5 \), what is the value of a?
原库参考解析
直线方程・斜率与截距挑战
Line s is defined by \( x-4y=24 \). Line t is perpendicular to line s in the xy-plane and passes through the points \( (k,\ 0) \) and \( (60,−k) \). What is the value of k?
原库参考解析
直线方程・斜率与截距进阶
The table shows four values of x and their corresponding values of y. There is a linear relationship between x and y. Which of the following equations represents this relationship?
原库参考解析
直线方程・斜率与截距挑战
In the xy-plane, line h passes through the point (0, 0) and is perpendicular to the line represented by the equation \( -3x+y=20 \). If line h also passes through the point (18, t), what is the value of t?
原库参考解析
