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・斜率与截距基础
\( f\left(x\right)=4x+b \)
For the linear function \( f \), b is a constant and \( f\left(7\right)=28 \) . What is the value of b?
原库参考解析
直线方程・斜率与截距基础
The function \( f \) is defined by \( f\left(x\right)=\frac{1}{10}x-2 \) . 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・斜率与截距基础
The graph shows the linear relationship between x and y. Which table gives three values of x and their corresponding values of y for this relationship?
原库参考解析
直线方程・斜率与截距挑战
If the graph of \( 27x+33y=297 \) is shifted down 5 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
The function \( f \) is defined by \( f\left(x\right)=\frac{1}{10}x-2 \) . 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
The graph shows the relationship between the number of shares of stock from Company A, x, and the number of shares of stock from Company B, y, that Simone can purchase. Which equation could represent this relationship?
原库参考解析
直线方程・斜率与截距基础
\( h\left(x\right)=x+b \)
For the linear function ℎ, b is a constant and ℎ(0) = 45. What is the value of b?
原库参考解析
直线方程・斜率与截距基础
What is the equation of the line that passes through the point (0, 5) and is parallel to the graph of \( y=7x+4 \) in the xy-plane?
原库参考解析
直线方程・斜率与截距挑战
The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is \( \left(k-5,\ b\right) \), where k and b are constants. What is the value of b?
原库参考解析
直线方程・斜率与截距挑战
\( 5x+7y=1 \)
\( ax+by=1 \)
In the given pair of equations, a and b are constants. The graph of this pair of equations in the xy-plane is a pair of perpendicular lines. Which of the following pairs of equations also represents a pair of perpendicular lines?
原库参考解析
直线方程・斜率与截距挑战
The function ℎ is defined by \( h\left(x\right)=4x+28 \). The graph of \( y=h\left(x\right) \) in the xy-plane has an x-intercept at \( \left(a,\ 0\right) \) and a y-intercept at \( \left(0,\ b\right) \) , where a and b are constants. What is the value of \( a+b \) ?
原库参考解析
直线方程・斜率与截距挑战
ax + by = b
In the equation above, a and b are constants and 0< a < b. Which of the following could represent the graph of the equation in the xy-plane?
原库参考解析
