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 \( g \), the graph of \( y=g(x) \) in the xy-plane has a slope of 5 and passes through the point (2, 37). Which equation defines \( g \)?
原库参考解析
直线方程・斜率与截距挑战
Line \( k \) is defined by \( y=4x+\frac{1}{2} \). 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・斜率与截距进阶
Line k is shown in the xy-plane. Line j (not shown) is perpendicular to line k and passes through the point (7, 15). What is the y-coordinate of the y-intercept of line j?
原库参考解析
直线方程・斜率与截距进阶
For the linear function r, the table shows three values of x and their corresponding values of r(x). Function r is defined by \( r(x)=ax+b \), where a and b are constants. What is the value of \( a+b \)?
原库参考解析
直线方程・斜率与截距挑战
For the linear function g, the graph of \( y=g(x) \) in the xy-plane has a slope of 3 and passes through the point (2, 28). Which equation defines g?
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 \( y=42x+6 \). If line s also passes through the point (4, d), what is the value of d?
原库参考解析
直线方程・斜率与截距进阶
The graph shows the linear relationship between x and y. Which table gives three values of x and their corresponding values of y for this relationship?
原库参考解析
直线方程・斜率与截距挑战
What is an equation of the graph shown?
原库参考解析
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 . Which equation represents this relationship?
原库参考解析
直线方程・斜率与截距挑战
The function \( f \) is defined by \( f(x)=98x \). What is the slope of the graph of \( y=f(x) \) in the xy-plane?
原库参考解析
直线方程・斜率与截距进阶
The graph of the linear function \( f \) is shown, where \( y=f(x) \). What is the y-intercept of the graph of \( f \)?
原库参考解析
直线方程・斜率与截距挑战
For the linear function p, \( p(c)=-2 \) , where c is a constant, \( p(5)=34 \), and the slope of the graph of \( y=p(x) \) in the xy-plane is 6. For the linear function t, \( t(c)=-4 \) and \( t(6)=52 \). What is the slope of the graph of \( y=t(x) \) in the xy-plane?
原库参考解析
