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 p is defined by \( x-18y=18 \). Line w is parallel to line p in the xy-plane. What is the slope of line w?
原库参考解析
直线方程・斜率与截距挑战
The graph shows a linear relationship between x and y. Which equation represents this relationship, where R is a positive constant?
原库参考解析
直线方程・斜率与截距挑战
Line t in the xy-plane has a slope of \( \frac{1}{4} \) and passes through the point (20, -9). Which equation defines line t?
原库参考解析
直线方程・斜率与截距进阶
What is the x-intercept of the graph shown?
原库参考解析
直线方程・斜率与截距挑战
Line p in the xy-plane contains the points (11,0) and (11,11). Which equation defines line p?
原库参考解析
直线方程・斜率与截距挑战
The line shown can be represented by the equation \( ax+by=32 \), where a and b are constants. Which of the following is true about a and b?
原库参考解析
直线方程・斜率与截距挑战
Line t in the xy-plane has a slope of \( \frac{1}{9} \) and passes through the point (45, -8). Which equation defines line t?
原库参考解析
直线方程・斜率与截距进阶
The graph of \( 4x+y=19 \) in the xy-plane is a line. What is the slope of the line?
参考答案
直线方程・斜率与截距进阶
A linear function g is defined by \( g(x)=ax+b \) where a and b are constants. If \( g(-4)=-6 \) and \( g(3)=8 \), what is the value of a?
参考答案
直线方程・斜率与截距进阶
Line q in the xy-plane passes through the points (0, 20) and (1, 44). Which equation defines line q?
参考答案
直线方程・斜率与截距挑战
\( 9x+10y=24 \)
If the graph of the given equation is shifted up 5 units in the xy-plane, what is the y-coordinate of y-intercept of the resulting graph?
参考答案
直线方程・斜率与截距挑战
For a linear relationship between x and y the table gives three values of x and their corresponding values of y in terms of a constant r . What is the slope of the line in the xy-plane that represents this relationship?
参考答案
