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
The cost y, in dollars, for a manufacturer to make x rings is represented by the line shown.
What is the cost, in dollars, for the manufacturer to make 50 rings?
原库参考解析
直线方程・斜率与截距进阶
The function \( f \) is defined by \( f\left(x\right)=2x-\frac{1}{4} \). What is the y-intercept of the graph of \( y=f\left(x\right) \) in the xy-plane?
原库参考解析
直线方程・斜率与截距挑战
For the linear function p, \( p(c)=−5, \) where c is a constant, \( p\left(3\right)=23 \) and the slope of the graph of \( y=p\left(x\right) \) in the xy-plane is 7. For the linear function t, \( t\left(c\right)=-6 \) and \( t\left(4\right)=34 \) . What is the slope of the graph of \( y=t\left(x\right) \) in the xy-plane?
原库参考解析
直线方程・斜率与截距挑战
The table shows three values of x and their corresponding values of y, where s is a constant. 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
\( f\left(x\right)=29 \)
For the given linear function \( f \) , which table gives three values of x and their corresponding values of \( f\left(x\right) \)?
原库参考解析
直线方程・斜率与截距进阶
In the xy-plane, line k and line l are perpendicular and intersect at the point (2, 8). If line k is defined by the equation \( y=mx+b \), where m and b are constants and \( m>1, \)which of the following points lies on line l?
原库参考解析
直线方程・斜率与截距挑战
For the linear function p, \( p\left(c\right)=-6 \), where c is a constant, \( p\left(5\right)=42 \) , 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 \( t\left(6\right)=56 \) . What is the slope of the graph of \( y=t\left(x\right) \) in the xy-plane?
原库参考解析
直线方程・斜率与截距挑战
Line k is defined by \( y=8x+7 \). Line j is parallel to line k in the xy-plane and passes through the point (0, 15). Which equation defines line j?
原库参考解析
直线方程・斜率与截距进阶
The point \( \left(v,-5\right) \) lies on the line shown. What is the value of v?
原库参考解析
直线方程・斜率与截距进阶
\( f\left(x\right)=22 \)
For the given linear function \( f \), which table gives three values of \( x \) and their corresponding values of \( f\left(x\right) \) ?
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
The function \( f \) is defined by \( f\left(x\right)=6x+14 \). What is the y-coordinate of the y-intercept of the graph of \( y=f\left(x\right) \) in the xy-plane?
原库参考解析
