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 j is defined by \( 4x+5y=55 \). Line k is parallel to line j in the xy-plane. An equation of line k is \( 24x+ry=15 \) , where r is a constant. If line k passes through the point \( \left(0,\ b\right) \) , what is the value of b?
原库参考解析
直线方程・斜率与截距进阶
For the linear function \( f \), the graph of \( y=f\left(x\right) \) in the xy-plane has a slope of \( −54 \) andpasses through the point \( (0,0). \) Which equation 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 has a slope of 5 and a y-intercept of \( \left(0,-35\right) \). What is the x-coordinate of the x-intercept of line k?
原库参考解析
直线方程・斜率与截距挑战
Which equation defines the linear function f ?
原库参考解析
直线方程・斜率与截距进阶
For the linear function h, h(0) = 0 and h(8) = 7. Which equation defines h?
原库参考解析
直线方程・斜率与截距挑战
A line in the xy-plane passes through the points (\( 0,\ -2 \)) and (\( 1 \), \( 7 \)). Which of the following is an equation of this line?
原库参考解析
直线方程・斜率与截距挑战
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・斜率与截距进阶
The graph of the linear equation \( 4x+By=36 \)is shown, where B is a constant. What is the value of B?
原库参考解析
直线方程・斜率与截距进阶
Line r in the xy-plane has a slope of 8 and passes through the point (0, 11). Which equation defines line r?
原库参考解析
直线方程・斜率与截距进阶
\( h\left(x\right)=x+b \)
For the linear function h, b is a constant and h(0) = 21. What is the value of b?
原库参考解析
直线方程・斜率与截距挑战
The graph of the linear function \( f \) is shown, where \( y=f\left(x\right) \). Which equation defines \( f \) ?
原库参考解析
直线方程・斜率与截距进阶
For the linear function p, \( p\left(c\right)=-6 \) , where c is a constant,\( p\left(3\right)=26 \) , and the slope of the graph of \( y=p\left(x\right) \) in the xy-plane is 8. For the linear function t, \( t\left(c\right)=-7 \) and \( t\left(4\right)=38 \). What is the slope of the graph of \( y=t\left(x\right) \) in the xy-plane?
原库参考解析
