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在这个视频中 我要做几个

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把直线方程化成斜截式的例子

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就算是复习

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这就是说把直线方程化成

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y=mx+b的形式 m是斜率 b是y截距

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我们做几个这样的问题

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题目说有一条斜率是-5的直线

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所以m=-5 它的y截距是6

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所以b=6

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非常简单

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直线的方程是y=-5x+6

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不赖嘛 做下一题

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直线的斜率是-1 经过点(4/5,0)

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题目给了斜率 斜率是-1

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所以m=-1

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但是我们不能确定y截距是多少

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我们知道直线方程的形式是

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y等于斜率 -1x加b b是y截距

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现在 我们可以利用坐标的信息

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直线经过这个点

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我们可以用这个信息来解出b

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直线经过点也就是说

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当x=4/5

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y=0一定满足这个方程

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代入 当x=4/5 y=0

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所以0=-1×4/5+b

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向下滚动一点

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我看一下 0=-4/5+b

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方程两边加4/5

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这是4/5

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这边也要加4/5

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我这么做的原因是要消掉这个

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就得到b=4/5

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这样我们就得到了直线的方程

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y等于-1乘x

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也可以写成-x 加b 4/5 就像这样

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现在看这一题

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直线穿过点(2,6)和(5,0)

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题目没有明确给我们斜率和y截距

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但是我们可以通过坐标把这些求出来

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所以首先要做的就是算出斜率

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我们知道斜率m等于y变化量除x变化量

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也就等于 y变化量是多少？

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从这一点开始

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这是(6,0) 这样做

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这是6 标上颜色 减0

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6-0 这就是y变化量

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x变化量是2-5

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标上颜色是为了给你们展示

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当先用这个y 用上面这个6

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那我就要先用这个x

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我要告诉你们 这是坐标(2,6)

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这是坐标(5,0)

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2和5不能交换

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那样就会得到答案的相反数

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但在这里得到什么？

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这就是6-0=6 2-5=-3

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所以这就变成了-6/3

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也就是-2 这就是斜率

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到目前为止我们知道直线一定是

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y等于斜率 用橙色表示

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-2x加y截距

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现在就和上一题一样了

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我们可以用其中一个点求出b

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任意一点

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这两个点都在直线上

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所以这两点都满足方程

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我要用点(5,0)

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因为如果有个数是0通常比较好算

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运算比较简单

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所以把(5,0)代入

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当x=5 y=0

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当这是-2×5时 y=0

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当x=5 再加b

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所以得到0=-10+b

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如果方程两边加10

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如果两边加10 这两个约掉了

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就得到b=10+0=10

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就得到b=10

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现在我们求出了直线的方程

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方程是y 用个新颜色

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y等于-2x加b 就是加10 做完了

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再做一题

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好了 直线过点(3,5)

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点(3,0)

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就像上个问题

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从求斜率m开始

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斜率就是高度除跨度

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也就是y变化量除x变化量

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如果你们把这作为家庭作业

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就不用写这些

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我只是想确定你们理解

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这些是相同的

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y变化量除x变化量是多少？

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这等于 先从这边开始

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我想让你们知道可以随便找一个点

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像这样 0-5

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我先用这个坐标

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把这看做是结束点 0-5

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记得当我们刚开始学这个

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我总是把x当做分子

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错 应该用y做分子

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这是坐标的第二个数

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这应该是除-3-3

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这是坐标(-3,0)

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这是坐标(3,5) 减去它

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得到什么？

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这就等于 用个中性的颜色

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这就等于分子

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-5除-3-3是-6

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负号消掉了 就得到5/6

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我们已知直线方程的形式是

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我们把一个点的坐标代入求b

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开始

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我通常喜欢用含有0的点

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所以当x=-3 y=0 加b

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所有我做的就是用-3替换x 用0替换y

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我知道可以这么做是因为它在直线上

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它肯定满足直线方程

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我们来解出b

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就得到0等于 如果用-3除3

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这就成了1

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如果6除3 就得到2

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所以它变成了-5/2+b

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方程两边加5/2

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加5/2 加5/2

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我喜欢改变记号以便你们都能熟悉

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所以方程变成了5/2等于

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这是0 等于b b等于5/2

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所以直线的方程是y=5/6x+b

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b我们算出来是5/2 加5/2 做完了

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再做一题 这有个图

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我们来算出这个图的方程

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这实际上 在某种水平上 更简单一点

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斜率是多少？

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斜率就是y变化量除x变化量 我们看一下

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当x变化 当x变化了1

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这就是x变化量 x变化量是1

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我只是要把x变化1 增加1

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y变化量是多少？

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看起来y恰好变化了4

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看起来当△x=1时

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△y y变化量等于4

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所以y变化量除x变化量

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当x变化量是1 y变化量是4

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所以斜率是4 现在 y截距是多少？

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现在我们看一下图像

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看起来它与y轴交于y=-6

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或者点(0,-6)

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所以我们知道b=-6

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所以就求出了直线的方程

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直线方程是y等于

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斜率乘x加y截距 我可以这么写

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-6 就是加-6

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所以这就是直线的方程

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再做一题

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题目说f(1.5)=-3 f(-1)=2

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这是什么？

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所有这些就是以一种奇特的方式告诉你们

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当x=1.5 当把1.5代入函数

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函数值就是-3

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这就是说点(1.5,3)在直线上

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然后题目给了一个点 当x=-1

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这只是以一种特别的方式告诉你们

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这两个点都在直线上 没啥特别的

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我认为这个问题的关键就是

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让你们熟悉函数记号

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当你们看到这样的东西不会害怕

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如果你们让x=1.5 就得到-3

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如果设y=f(x) 这就是坐标

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所以这是y坐标

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当x=1.5 y应该等于-3

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不管怎样 我已经说过好多次了

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我们来算出直线斜率

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斜率就是y变化量除x变化量 等于

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2减去这个数 -3

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这就是y值 除 所有这些除

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-1减去这个数

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这样写 -1减这个数 减1.5

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我用这个颜色是为了给你们展示一下

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-1和2都来源于这个点

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这就是为啥这两个在前面 如果先用这些数

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我就要把这个x和y放到前面

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如果把2放前面 就要把-1放前面

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这就是为啥我用颜色区分

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所以这就等于2-(-3)

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也就是2+3 等于5

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-1-1.5是-2.5

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5/2.5等于2

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所以直线的斜率是-2

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实际上 顺便和你们说一下

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我用什么顺序都没关系

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如果先用这个坐标

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那么就要先用这个坐标

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换种方式

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如果写成-3-2除1.5-(-1)

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这就是减2除1.5-(-1)

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得到的结果是相同的 等于多少？

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-3-2=-5除1.5-(-1)

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这就是1.5+1 除2.5

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同样的 等于-2

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所以我刚给你们展示了

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哪个点作为开始点和结束点都是相同的

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只要上下一致

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如果这是开始的y 这就是开始的x

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如果这是结束的y 这就应该是结束的x

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但是不管怎样 我们知道斜率是-2

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所以我们知道方程是y等于

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-2x加y截距

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我们用其中一个坐标

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我用这个 因为我不想算小数

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我们知道y=2

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所以x=-1时 y=2

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当然还要加b

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所以2等于 -2乘-1等于2加b

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如果方程两边减2

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方程两边减去它

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左边就是0 等于b

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所以b是0

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所以直线方程是y=-2x

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实际上 如果你们想写成函数形式

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就应该是f(x)=-2x

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我假设y=f(x)

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但是这确实是个方程 题目没有提到y

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所以这里要写成f(x)=2x

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每个坐标都是x和f(x)

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所以你们可以把斜率的定义看成

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f(x)变化量除x变化量

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这些就是一个东西相同的看法