WEBVTT

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让我们来学习解绝对值方程式

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先来复习下

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当计算一个数值的绝对值时

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以-1为例 来计算绝对值

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你需要做的是

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计算出1与0之间的距离

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对-1来说 下面以数轴示例

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这个没画好

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这是数轴 这是0

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这是-1

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-1与0之间的距离是1

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那-1的绝对值就是1

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1和0之间的距离也是1

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所以1的绝对值同样是1

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也就是说 绝对值为与0之间的距离

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来换一种更简单的方式来理解

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绝对值其实就是取数值的正值

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-7346的绝对值是7346.

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理解这点以后 下面

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开始学习解绝对值方程式

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来看这个方程式

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x-5的绝对值等于10

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可以这样解读这个等式

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也希望能够这样理解 这就是说

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x与5之间的距离为10

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与5之间的距离是10的数值有几个

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大家应该已经想到解法了

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不过下面要学习更系统的解法

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这个方程式有两种情况

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一种是x-5等于10

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如果假定绝对值里面的值为10

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取绝对值后

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会得到10

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或者假设x-5等于-10

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如果x-5等于-10 取绝对值后

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也是10

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所以第二种情况是 x-5等于-10

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两种情况同时满足这个方程式

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先解这个

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等式两边加5

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得到x=15

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再解这个 等式两边加5

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x等于-5

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得出计算结果

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有两个x值满足这个方程式

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x可以是15

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15-5等于10 取绝对值后

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得到10 或者x为-5

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-5减5等于-10

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取绝对值后 还是10

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注意 这两个值

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与5之间的距离都是10

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再解一个

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再来一个

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现在有

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x+2的绝对值等于6

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这说明什么呢

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也就是说 要么x+2

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也就是绝对值符号里的这个值 它等于6

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或者绝对值符号里的这个值

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x+2 它等于-6

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假设等于-6时

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取绝对值后 得到6

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那么 假设x+2等于-6

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然后 等式两边减2

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得到 x等于4

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再看这个方程式 两边减2

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得到x等于-8

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这就是方程式的两种解法

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为了加深理解

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对于绝对值 可以把它看成是距离

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这样重写这个绝对值方程式

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x减-2的绝对值等于6

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其实就是说

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哪些值与-2之间的距离刚好是6呢

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还记得吗 在这里讲过

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哪些值与5之间的距离是10

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不管被5减的值是几

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它们与5之间的距离肯定是10

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这边呢

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哪些值与-2之间的距离是6

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那就是4 或-8

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自己试着验证下结果

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再做一个

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再来一个 这次用紫色

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这有4x的绝对值

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来计算一个有难度的

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4x减1

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4x-1的绝对值 等于--

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这里让它等于19

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这样 跟之前的解法一样

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一种情况是4x减1等于19

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另一种情况是4x减1等于-19

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因为当取绝对值以后

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都会得到19

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或者4x减1等于-19

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现在只需要解这两个方程式

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等式两边加1

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两个同时做

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两边加1后 得到4x等于20

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这两边加1后

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得到4x等于-18

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等式两边除4 得到x等于5

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这个等式两边除4 得到x等于-18/4

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也等于-9/2

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这两个x值同时满足这个绝对值方程式

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试试看

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-9/2乘4

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等于-18

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-18再减1就是-19

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取绝对值后 得到19

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等于5时 4乘5是20

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减1等于19

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再取绝对值以后

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还是等于19

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下面来画绝对值方程式的图形 增加趣味性

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现在有

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y等于x+3的绝对值

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这是个函数 或者说图形

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并包含绝对值在里面

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这要考虑两种情况

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一种情况是

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绝对值符号里的这个值为正

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也就是x+3

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写这边 x+3>0

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另一种情况是x+3<0

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当x+3>0时

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这个图形 或这条直线 这里可能还不能称为一条直线

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这个函数 就等同于y=x+3

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如果这边这个值是大于0的话

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那么绝对值符号就形同虚设

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那么这个就完全等同于

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y=x+3

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那么什么情况下x+3>0

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不等式两边减3

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得到x>-3

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那么当x>-3时

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这个图形与y=x+3是等同的

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当x+3<0时

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在这种情况下

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绝对值符号里的值为负

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在这种情况下等式就变成

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y等于-（x+3）

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为什么呢

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仔细听 现在这里是负值 如果

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x+3是负值

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在这个前提下 如果它是负值的话

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那么当取负值的绝对值后

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它肯定是得到正值

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就相当于把它与-1相乘

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如果知道是取负值的绝对值

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就相当于用这个值乘-1

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因为绝对值都是正值

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也就是这里所讨论的情况

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x+3小于0

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两边减3 当x小于-3

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当x小于-3

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那么当x<-3时

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图形应该是这样

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当x>-3

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图形是这样

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下面一起来看

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完整的图形

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先画数轴

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这是x轴 这是y轴

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先把这个函数乘出来 这样可以得到mx

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加b的函数格式

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这个等于-x-3

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先看看这种函数图形

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的一般形式

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-x-3

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y轴截距是-3 所以1 2 3

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-x那就应该是向下倾斜 也就是

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向下斜率为1

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看起来就是这样

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x轴的截距应该是x等于

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当假设y等于0时 这边x

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等于-3

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穿过这条线

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这个点

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这个图形 如果没有这里这个限制条件

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应该是这样

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如果没有区间限制条件

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x轴的这个区间限制条件

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那么这个图形 应该是怎样的

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来看看

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y轴截距是3

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像这样

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x轴的截距呢

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当y等于0时 x等于-3

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还是一样穿过这个点

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斜率为1

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那就应该是这样的

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这就是这个函数的图形

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现在 可以得出结论 绝对值

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函数为紫色直线

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当x<-3时

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当x<-3 这里是x等于

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-3 当x<-3

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它就是这条紫线

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就是这条

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上面是当x<-3时的图形

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但是如果x>-3时 就应该是

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这条绿线了

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就是这条

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图形看起来很像奇怪的V

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当x>-3时 应该是正值

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所以这个图形 斜率为正

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但当x<-3时 本质上

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其实是取这个函数的负值 如果想要

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这么理解的话 所以这里是负斜率

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如果看到这种V型函数 或是

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V型图 那就表明它是

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绝对值函数