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
For the linear function g, the table shows four values of x and their corresponding values of \( g\left(x\right) \). The function can be written as \( g\left(x\right)=mx+b \), where m and b are constants. 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 69 and \( f\left(0\right)=3 \). Which equation defines \( f \)?
原库参考解析
直线方程・斜率与截距挑战
The function \( f \) is defined by \( f\left(x\right)=-9x+9 \). 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・斜率与截距挑战
The table shows three values of x and their corresponding values of y, where a and b are constants. There is a linear relationship between x and y. In the xy-plane, the y-intercept of the line representing this relationship is (\( 0 \), \( −16 \)). What is the value of a + b?
原库参考解析
直线方程・斜率与截距挑战
Line p in the xy-plane is defined by \( y=x+\frac{8}{3} \) and passes through the points (a, 0) and (0, b), where a and b are constants. What is the value of a?
原库参考解析
直线方程・斜率与截距挑战
In the xy-plane, line p has a slope of \( \frac{3}{4} \) and an x-intercept of (16, 0). What is the y-coordinate of the y-intercept of line p?
原库参考解析
直线方程・斜率与截距挑战
A line in the xy-plane has a slope of \( -\frac{1}{4} \)and passes through the point (0, 19). Which equation represents this line?
原库参考解析
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. What is the x-coordinate of the x-intercept of the graph that represents this linear relationship in the xy-plane?
原库参考解析
直线方程・斜率与截距挑战
Line s in the xy-plane is defined by \( 8x+2y=0 \). Line t is perpendicular to line s and passes through the points \( \left(6,\ 0\right) \) and \( \left(k,\ \frac{5}{4}\right) \). What is the value of k?
原库参考解析
直线方程・斜率与截距挑战
The graph of the linear equation \( 10x+By=50 \) 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 t in the xy-plane has a slope of\( -\frac{1}{3} \) and passes through the point (9, 4). Which equation defines line t?
原库参考解析
直线方程・斜率与截距进阶
In the xy-plane, the graph of \( y=f\left(x\right) \) is a line that passes through (0, 21) and has a slope of 4. Which equation defines \( f \)?
原库参考解析
