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 x-intercept of the graph of \( px+39y=1 \) in the xy-plane, where p is a positive constant?
参考答案
直线方程・斜率与截距基础
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?
参考答案
直线方程・斜率与截距基础
What is the y-coordinate of the y-intercept of the graph shown?
参考答案
直线方程・斜率与截距进阶
Line k is defined by \( y=-\frac{10}{3}x+3 \). Line j is perpendicular to line k in the xy-plane. What is the slope of line j?
参考答案
直线方程・斜率与截距挑战
There is a linear relationship between the amount of salt x, in grams, and the amount of water y, in liters, in a certain body of seawater. The table shows this relationship for three samples of this seawater.
Which equation represents this relationship?
Which equation represents this relationship?
参考答案
直线方程・斜率与截距进阶
For the linear function \( f \), the table shows three values of x and their corresponding values of \( f(x) \). Which equation defines \( f \)?
参考答案
直线方程・斜率与截距进阶
Line k is defined by \( y=2x+17 \). Line j is perpendicular to line k in the xy-plane. What is the slope of line j?
参考答案
直线方程・斜率与截距基础
The graph of \( 5x+y=12 \) in the xy-plane is a line. What is the slope of the line?
参考答案
直线方程・斜率与截距挑战
Line j in the xy-plane passes through the points \( \left(\frac{47}{k},-\frac{2}{k}+\frac{4}{47}\right) \) and \( \left(\frac{94}{k},-\frac{4}{k}+\frac{4}{47}\right) \), where k is a constant. What is the slope of line j?
参考答案
直线方程・斜率与截距进阶
\( f\left(x\right)=-10x+b \)
For the linear function\( f \), b is a constant and\( f\left(\frac{2}{5}\right)=-53 \). What is the value of b?
原库参考解析
直线方程・斜率与截距进阶
What is the y-intercept of the graph shown?
原库参考解析
直线方程・斜率与截距进阶
Line r is defined by the equation \( 6x-5y=6 \). Line s is parallel to line r in the xy-plane. What is the slope of line s?
原库参考解析
