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
The figure shows the xy-plane with the quadrants labeled. The graph of a linear function h (not shown), where \( y=h(x) \), is a line completely contained in only quadrants Ⅰ and Ⅱ of the xy-plane. Which of the following could define the function h?
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 \( y=f(x) \) in the xy-plane has a slope of 25 and passes through the point (0, 0). Which equation defines \( f \)?
原库参考解析
直线方程・斜率与截距挑战
Line j is defined by \( y=4x+9 \). Line k is parallel to line j in the xy-plane. What is the slope of line k ?
原库参考解析
直线方程・斜率与截距挑战
For a linear function \( f \), the table below shows three values of x and their corresponding values of \( f(x) \). Which of the following could define \( f \)?
原库参考解析
直线方程・斜率与截距进阶
For a linear relationship between x and y, the table gives three values of x and their corresponding values of y, where t is a constant. Which equation represents this relationship?
原库参考解析
直线方程・斜率与截距进阶
Line j is shown in the xy-plane. Line k (not shown) is parallel to line j. What is the slope of line k?
原库参考解析
直线方程・斜率与截距挑战
In the linear function g, \( g(14)=28 \) and \( g(12)=35 \) . Which equation defines g?
原库参考解析
直线方程・斜率与截距挑战
The function \( f \) is defined by \( f(x)=\frac{2}{15}x \). 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・斜率与截距挑战
Line q in the xy-plane has a slope of \( \frac{1}{3} \) and passes through the point (0, 70). Which equation defines line q?
原库参考解析
直线方程・斜率与截距进阶
The function \( f \) is defined by \( f(x)=x-16 \). In the xy-plane, the x-intercept of the graph of \( y=f(x) \) is \( (a,\ 0) \). What is the value of a?
原库参考解析
直线方程・斜率与截距进阶
An equation of the line in the xy-plane that contains the points (0, 4) and (2, 0) is \( ax+by=1 \), where a and b are constants. What is the value of a?
原库参考解析
直线方程・斜率与截距进阶
For the linear function \( f \), \( f(-5)=-7 \), and the slope of the graph of \( y=f(x) \) in the xy-plane is 3. Which equation defines \( f \)?
原库参考解析
