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
For the linear function \( f \), the graph of \( y=f(x) \) in the xy-plane has a slope of 2 and has a y-intercept at (0, -5). Which equation defines \( f \)?
原库参考解析
直线方程・斜率与截距进阶
What is the slope of the graph of \( y=\frac{5x}{13}-23 \) in the xy-plane?
原库参考解析
直线方程・斜率与截距基础
A line in the xy-plane has a slope of \( -\frac{1}{2} \) and passes through the point (0, 3). Which equation represents this line?
原库参考解析
直线方程・斜率与截距基础
The graph of the linear function \( f \) is shown, where \( y=f\left(x\right) \). What is the x-intercept of the graph of \( f \)?
原库参考解析
直线方程・斜率与截距挑战
Line h is defined by \( \frac{1}{5}x+\frac{1}{7}y-70=0 \) . 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・斜率与截距挑战
\( \frac{3}{5}x+\frac{3}{4}y=7 \)
Which table gives three values of x and their corresponding values of y for the given equation?
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・斜率与截距进阶
Line k is defined by y=\( \frac{1}{4}x \) +1. Line j is parallel to line k in the xy-plane. What is the slope of j ?
原库参考解析
直线方程・斜率与截距挑战
The table gives the number of hours, ℎ, of labor and a plumber’s total charge f (ℎ), in dollars, for two different jobs.
There is a linear relationship between ℎ and f (ℎ). Which equation represents this relationship?
原库参考解析
直线方程・斜率与截距挑战
The graph shows the momentum y, in newton-seconds, of an object x seconds after the object started moving, for 0≤ x ≤8. What is the average rate of change, in newton-seconds per second, in the momentum of the object from x =2 to x =6?
原库参考解析
直线方程・斜率与截距挑战
The y-intercept of the graph of 12x +2y =18 in the xy-plane is (0, y). What is the value of y ?
原库参考解析
直线方程・斜率与截距挑战
f(x) =39
For the given linear function f, which table gives three values of x and their corresponding values of f(x) ?
原库参考解析
