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
For a particular car, the linear function f gives the predicted power, in brake horsepower (bhp), for engine speeds between 1,000 revolutions per minute (rpm) and 6,000 rpm. According to this function, the car's predicted power is 228 bhp at an engine speed of 1,896 rpm and 600 bhp at an engine speed of 4,500 rpm. The equation \( f\left(x\right)=\frac{1}{7}\left(x-a\right)+228 \) defines f, where x is the engine speed, in rpm, and a is a constant. What is the value of a?
原库参考解析
直线方程・斜率与截距挑战
In the xy-plane, line l has a slope of 4 and an x-intercept of ( \( \frac{4}{3} \) , 0). What is the y-intercept of line l?
原库参考解析
直线方程・斜率与截距挑战
Line k is defined by y = 4x + 1. Line j is parallel to line k in the xy-plane and passes through the point (0, 5). Which equation defines line j?
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 42 and passes through the point ( 0, 0 ). The function f is defined by f (x) = 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 x-intercept of a line in the xy-plane that has a slope of\( -\frac{8}{5} \) and passes through the point (0, 16) ?
原库参考解析
直线方程・斜率与截距进阶
A line in the xy-plane has a slope of\( -\frac{1}{6} \) 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
For the linear function g , the graph of \( y=g\left(x\right) \) in the xy-plane has a slope of 2 and passes through the point (1, 17) . Which equation defines g?
原库参考解析
直线方程・斜率与截距进阶
Line t is defined by \( y=-\frac{1}{6}x+17 \). Line s is perpendicular to line t 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
If the graph of \( 12x+22y=132 \) is shifted down 4 units in the xy-plane, what is the y-intercept of the resulting graph?
原库参考解析
直线方程・斜率与截距挑战
Line k is shown in the xy-plane. Line j(not shown) is perpendicular to line k and passes through the point (19, 23). What is the y-coordinate of the y-intercept of line j?
原库参考解析
直线方程・斜率与截距挑战
For the linear function g, the graph of \( y=g\left(x\right) \) in the xy-plane has a slope of 5 and passes through the point (2, 47). Which equation defines g?
原库参考解析
直线方程・斜率与截距进阶
The graph of a line in the xy-plane passes through the point \( \left(1,\ 7\right) \) and crosses the x-axis at the point \( \left(3,\ 0\right) \). The line crosses the y-axis at the point \( \left(0,\ b\right) \). What is the value of b?
原库参考解析
