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 table shows three 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
For the linear function \( f \), the graph of \( y=f\left(x\right) \) in the xy-plane has a slope of 28 and passes through the point (0, 0). The function \( f \) is defined by \( f\left(x\right)=ax \), where a is a constant. What is the value of a?
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 \( y=\frac{1}{2}\left(15x+12\right)+3x \) in the xy-plane?
原库参考解析
直线方程・斜率与截距进阶
The function \( f \) is defined by \( f(x)=12x+18. \) 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
直线方程・斜率与截距进阶
Line k is defined by \( y=7x+2 \). Line j is parallel to line k in the xy-plane and passes through the point (0, 3). Which equation defines line j?
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 \( y=\frac{1}{5}\left(19x+19\right)+7x \) in the xy-plane?
原库参考解析
直线方程・斜率与截距进阶
Line q in the xy-plane has a slope of \( \frac{1}{7} \) and passes through the point (0, 60). Which equation defines line q?
原库参考解析
直线方程・斜率与截距进阶
Line r is defined by the equation 4x-9y= 5. Line s is parallel to line r in the xy-plane. What is the slope of line s?
原库参考解析
直线方程・斜率与截距进阶
The table shows two values of x and their corresponding values of y. The graph of the linear equation representing this relationship passes through the point (\( \frac{1}{3} \), a), What is the value of a?
原库参考解析
直线方程・斜率与截距挑战
g(x)=5(16x -17)
What is the y-coordinate of the y-intercept of the graph of y = g(x) - 5 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 equation 6x+ By = 42 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
In the xy-piane, line l passes through the point (0, 0) and is parallel to the line represented by the equation y= 6x + 3. If line l also passes through the point (4, d), what is the value of d?
原库参考解析
