6.9 绝对值平方图

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  绝对值平方图

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  What if you were given an absolute value function like $\begin{array}{r}y=|x-8|\end{array}$ ? How could you graph this function? After completing this Concept, you'll be able to make a table of values to graph absolute value functions like this one.
  ::如果给您给您一个绝对值函数, 如 yx- 8? 您如何绘制此函数 ? 完成此概念后, 您将能够绘制一个数值表来绘制此绝对值函数 。

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  Watch This
  ::观察这个

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  Guidance
  ::指导指南指南指导指南

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  Now let’s look at how to graph absolute value functions.
  ::现在让我们来看看如何绘制绝对值函数的图表。

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  Example A
  ::例A

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  Consider the function $\begin{array}{r}y=|x-1|\end{array}$ . We can graph this function by making a table of values:
  ::考虑函数 yx- 1\\\\\\\\\\\\\\\\\\\\\\\\\ x\\\\\\\\\\\\\\\\\\\ x\\\\\\\\\\\ x\\\\\\\\\\\\\\\\\ x\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\ x\\\\\\\ x\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\我们我们可以我们可以通过我们可以我们可以通过我们可以通过我们可以图这个函数这个函数,我们可以可以图形这个函数 可以通过这个函数 通过图这个函数 可以通过这个函数 可以通过一个表格 可以通过一个表格表表表表 可以通过一个表格列表的表格的表格列表的表格的表格的表格的表格 的表格。这个函数。这个函数。这个函数。

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y=|x-1|\end{array}$
  -2	$\begin{array}{r}y=|-2-1|=|-3|=3\end{array}$
  -1	$\begin{array}{r}y=|-1-1|=|-2|=2\end{array}$
  0	$\begin{array}{r}y=|0-1|=|-1|=1\end{array}$
  1	$\begin{array}{r}y=|1-1|=|0|=0\end{array}$
  2	$\begin{array}{r}y=|2-1|=|1|=1\end{array}$
  3	$\begin{array}{r}y=|3-1|=|2|=2\end{array}$
  4	$\begin{array}{r}y=|4-1|=|3|=3\end{array}$

  

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  You can see that the graph of an absolute value function makes a big “V”. It consists of two line rays (or line segments), one with positive slope and one with negative slope , joined at the vertex or cusp .
  ::可以看到绝对值函数的图形产生一个大的“V”。 它由两线射线(或线段段)组成,一线射线为正斜度,一线为负斜度,在顶部或顶部加入。

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  We’ve already seen that to solve an absolute value equation we need to consider two options:
  ::为了解决一个绝对价值方程式, 我们需要考虑两种选择:

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  1. The expression inside the absolute value is not negative.
     ::绝对值中的表达式不是负值。
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  2. The expression inside the absolute value is negative.
     ::绝对值内的表达式为负值。

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  Combining these two options gives us the two parts of the graph.
  ::结合这两个选项后,我们可以看到图表的两部分。

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  For instance, in the above example, the expression inside the absolute value sign is $\begin{array}{r}x-1\end{array}$ . By definition, this expression is nonnegative when $\begin{array}{r}x-1\ge 0\end{array}$ , which is to say when $\begin{array}{r}x\ge 1\end{array}$ . When the expression inside the absolute value sign is nonnegative, we can just drop the absolute value sign. So for all values of $\begin{array}{r}x\end{array}$ greater than or equal to 1, the equation is just $\begin{array}{r}y=x-1\end{array}$ .
  ::例如,在上述例子中,绝对值符号内的表达式为x-1。根据定义,这种表达式在 x-10 时是非负的,也就是说在 x=1 时是非负的。当绝对值符号内的表达式是非负的时,我们只需降低绝对值符号。因此,对于X大于或等于1的所有值,等式只是y=x-1。

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  On the other hand, when $\begin{array}{r}x-1<0\end{array}$ — in other words, when $\begin{array}{r}x<1\end{array}$ — the expression inside the absolute value sign is negative. That means we have to drop the absolute value sign but also multiply the expression by -1. So for all values of $\begin{array}{r}x\end{array}$ less than 1, the equation is $\begin{array}{r}y=-(x-1)\end{array}$ , or $\begin{array}{r}y=-x+1\end{array}$ .
  ::另一方面,当 x - 1 < 0 - 换句话说, 当 x < 1 - 时, 绝对值符号内的表达为负值时。 这意味着我们必须降低绝对值符号, 但也将表达数乘以 - 1 。 因此, 对于所有不小于 1 的 x 1 值来说, 等式是 y (x-1) 或 yx+ 1 。

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  These are both graphs of straight lines, as shown above. They meet at the point where $\begin{array}{r}x-1=0\end{array}$ — that is, at $\begin{array}{r}x=1\end{array}$ .
  ::这些是上面所示的两条直线的图形。 它们相交于 x-1=0 点, 即 x=1 点。

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  We can graph absolute value functions by breaking them down algebraically as we just did, or we can graph them using a table of values. However, when the absolute value equation is linear, the easiest way to graph it is to combine those two techniques, as follows:
  ::我们可以像我们刚才那样,通过将绝对值函数分解为代数来绘制绝对值函数,或者用一个数值表来绘制这些函数。然而,当绝对值方程式是线性时,最简单的图表方法是将这两种技术结合起来,具体如下:

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  1. Find the vertex of the graph by setting the expression inside the absolute value equal to zero and solving for $\begin{array}{r}x\end{array}$ .
     ::通过在绝对值内设定等于零的表达式并解析 x 来查找图形的顶点。
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  2. Make a table of values that includes the vertex, a value smaller than the vertex, and a value larger than the vertex. Calculate the corresponding values of $\begin{array}{r}y\end{array}$ using the equation of the function.
     ::制作包含顶点、小于顶点的值和大于顶点的值的表格。使用函数的方程式计算 y 的相应值。
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  3. Plot the points and connect them with two straight lines that meet at the vertex.
     ::用在顶端相会的两条直线绘制点并连接它们。

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  Example B
  ::例例BB

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  Graph the absolute value function $\begin{array}{r}y=|x+5|\end{array}$ .
  ::绘制绝对值函数 yx+5。

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  Solution
  ::解决方案

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  Step 1 : Find the vertex by solving $\begin{array}{r}x+5=0\end{array}$ . The vertex is at $\begin{array}{r}x=-5\end{array}$ .
  ::第一步 1: 通过解析 x+5=0 来查找顶点。 顶点在 x% 5 处 。

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  Step 2 : Make a table of values:
  ::第2步:编制一个数值表:

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y=|x+5|\end{array}$
  -8	$\begin{array}{r}y=|-8+5|=|-3|=3\end{array}$
  -5	$\begin{array}{r}y=|-5+5|=|0|=0\end{array}$
  -2	$\begin{array}{r}y=|-2+5|=|3|=3\end{array}$

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  Step 3 : Plot the points and draw two straight lines that meet at the vertex:
  ::第3步:绘制点和绘制在顶端相交的两条直线:

  

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  Example C
  ::例例C

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  Graph the absolute value function: $\begin{array}{r}y=|3x-12|\end{array}$
  ::绝对值函数: y3x- 12

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  Solution
  ::解决方案

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  Step 1 : Find the vertex by solving $\begin{array}{r}3x-12=0\end{array}$ . The vertex is at $\begin{array}{r}x=4\end{array}$ .
  ::第1步:通过解析 3x- 12=0 查找顶点。 顶点在 x=4 。

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  Step 2 : Make a table of values:
  ::第2步:编制一个数值表:

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y=|3x-12|\end{array}$
  0	$\begin{array}{r}y=|3(0)-12|=|-12|=12\end{array}$
  4	$\begin{array}{r}y=|3(4)-12|=|0|=0\end{array}$
  8	$\begin{array}{r}y=|3(8)-12|=|12|=12\end{array}$

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  Step 3 : Plot the points and draw two straight lines that meet at the vertex.
  ::第3步:绘制点和绘制在顶端相交的两条直线。

  

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  Watch this video for help with the Examples above.
  ::观看此视频, 帮助了解上面的例子 。

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  Guided Practice
  ::实践指南 实践指南 实践指南 实践指南 实践指南 实践指南 实践指南

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  Graph the absolute value function: $\begin{array}{r}y=3|x-4|\end{array}$
  ::绝对值函数: y= 3\\\\ x- 4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Solution
  ::解决方案

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  Step 1 : Find the vertex by solving $\begin{array}{r}x-4=0\end{array}$ . The vertex is at $\begin{array}{r}x=4\end{array}$ .
  ::第1步:通过解析 x-4=0 查找顶点。顶点为 x=4 。

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  Step 2 : Make a table of values:
  ::第2步:编制一个数值表:

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y=3|x-4|\end{array}$
  0	$\begin{array}{r}y=3|0-4|=3|-4|=3\cdot 4=12\end{array}$
  4	$\begin{array}{r}y=3|4-4|=3|0|=3\cdot 0=0\end{array}$
  8	$\begin{array}{r}y=3|8-4|=3|4|=3\cdot 4=12\end{array}$

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  Notice this is the same table as Example C. The function $\begin{array}{r}y=3|x-4|\end{array}$ is equivalent to the function $\begin{array}{r}y=|3x-12|.\end{array}$ This is because positive numbers can be factored out, or distributed into the absolute value function.
  ::注意此表与例C相同。 函数 y= 3x- 4 等同函数 y3x- 12。 这是因为正数可以乘以参数, 或者分布到绝对值函数 。

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  Step 3 : Plot the points and draw two straight lines that meet at the vertex.
  ::第3步:绘制点和绘制在顶端相交的两条直线。

  

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  Explore More
  ::探索更多

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  Graph the absolute value functions.
  ::图形显示绝对值函数。

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  1. $\begin{array}{r}y=|x+3|\end{array}$
     ::yx+3
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  2. $\begin{array}{r}y=|x-6|\end{array}$
     ::~ ~ ~ ~ - ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
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  3. $\begin{array}{r}y=|4x+2|\end{array}$
     ::y4x+2
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  4. $\begin{array}{r}y=|5-6x|\end{array}$
     ::y5 - 6x
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  5. $\begin{array}{r}y=|2x-1|\end{array}$
     ::y2x-1__________________________________________________________________________________________________________________________________________________________________________________________
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  6. $\begin{array}{r}y=3|2x-7|\end{array}$
     ::y=32x-7
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  7. $\begin{array}{r}y=0.05|x-1.25|\end{array}$
     ::y=0.05x-1.25
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  8. $\begin{array}{r}y=\frac{1}{2}|x+10|\end{array}$
     ::y=12x+10
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  9. $\begin{array}{r}y=\left|\frac{x}{3}-4\right|\end{array}$
     ::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
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  10. $\begin{array}{r}y=-2\left|\frac{x}{2}-5\right|\end{array}$
      ::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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  Answers for Explore More Problems
  ::探索更多问题的答案

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。