5.8两点直线的斜率

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  两点直线的斜率

  

  [Figure 1]

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  Hadiya has a very active dog that she would like to train for agility. Hadiya's dad is helping her make a ramp off their back deck so Hadiya and Max can practice. Measuring from the ground at the base of the deck, the ramp will be placed out 4 feet, the horizontal  distance , and reach the 7-foot-high deck, the  vertical  distance. Using the base of the deck as point  (0,0), solve for the   of the agility ramp.
  ::Hadiya有一条非常活跃的狗,她想训练它来适应敏捷。Hadiya的爸爸正在帮她从他们的后甲板上做一个斜坡,这样Hadiya和Max就可以练习。从甲板底部的地面测量,斜坡将放置在4英尺之外,是水平距离,到达7英尺高的甲板,是垂直距离。用甲板底部作为角点(0,0),解决易敏坡。

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  In this concept, you will learn to solve for the slope of a line using two points on a line, or  coordinates .
  ::在此概念中,您将学会用一条线上的两点或坐标来解析一条线的斜坡。

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  Solving for Slope of a Line Using Coordinates
  ::使用坐标解解析行的曲线

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  Coordinates  are values on a line that show a point's exact position. They are indicated by  " data-term="Parentheses" role="term" tabindex="0"> parentheses , (x,y). The point (7,-4) means that the x value of a particular point is 7 units to the right of point (0,0) and the y value of the same point is 4 units below point (0,0).
  ::坐标是显示点准确位置的线上的值。它们用括号(x,y)表示。点(7,4)意味着一个特定点的x值是点(0,0)右侧的7个单位,而同一点的Y值是点(0,0)下方的4个单位。

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  If the graph is given, the slope of a line can be found by choosing two points on the line and counting the units  vertically  and  horizontally  to find the change in y over the change in x,  $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\end{array}$ .
  ::如果给定了图形, 可以通过在线上选择两点并垂直和水平计算单位来找到线的斜度, 以在 x, y x 的更改中查找 y 的变化 。

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    $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{y_2-y_1}{x_2-x_1}\end{array}$
  ::y x = y 2 - y 1 x 2 - x 1

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    $\begin{array}{r}{y}_2-y_1\end{array}$ represents the vertical change, or  difference , between two points on a line.
  ::y 2 - y 1 表示线上两点之间的垂直变化或差异。

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    $\begin{array}{r}{x}_2-x_1\end{array}$ represents the horizontal change, or difference, between two points on a line.
  ::x 2 - x 1 表示线上两点之间的水平变化或差异。

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  Since slope is a ratio, improper fractions are not to be converted to mixed numbers.
  ::由于斜坡是一个比率,不适当的分数不能转换成混合数字。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Hadiya and her agile dog, Max.
  ::之前,你被问及Hadiya和她的敏捷狗Max的问题。

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  Hadiya's dad is making an agility ramp that rises 7 feet and  stretches  out 4 feet from a point (0,0) at the base of the deck in their backyard. What is the slope of the ramp?
  ::Hadiya的爸爸正在建造一个闪亮的斜坡,从他们的后院的甲板底部(0,0)从一个点(0)伸展到4英尺。斜坡的斜坡是多少?

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  First, write the  equation .
  ::首先,写出方程式

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  slope =  $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\end{array}$
  ::斜坡 = y x

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  Next,  substitute  in the given information.
  ::其次,在给定信息中替换。

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  The change in y,  $\begin{array}{r}\mathrm{\Delta }y\end{array}$  , goes from 0 to 7 feet, a change of +7.
  ::y, y, y, 的改变从0到7英尺, 变化为+7 。

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  The change in x,   $\begin{array}{r}\mathrm{\Delta }x\end{array}$ , goes from 0 to 4 feet. Assume to the right and a change of +4.
  ::x, x x, 从0英尺到4英尺。 假定右边, 并更改+4 。

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  Then, slope =  $\begin{array}{r}\frac{7}{4}\end{array}$ .
  ::然后,斜坡=7 4

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  The answer is  $\begin{array}{r}\frac{7}{4}\end{array}$  .
  ::答案是7 4。

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  Example 2
  ::例2

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  Find the slope of line  $\begin{array}{r}CD\end{array}$  below.
  ::找到下面CD线的斜坡。

  

  [Figure 2]

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  First, write the  formula  for the slope.
  ::首先,写下斜坡的公式。

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      $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{y_2-y_1}{x_2-x_1}\end{array}$  
  
  ::y x = y 2 - y 1 x 2 - x 1

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  Next, fill in the values that are known from the coordinates given, using C as Point 1 and D as Point 2.
  ::下一步,填写从给定坐标中已知的值,以C点1和D点2作为C点1和D点2。

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    $\begin{array}{r}\frac{y_{}-y_1}{x_2-x_1}=\end{array}$ $\begin{array}{r}\frac{9-3}{9-7}\end{array}$
  ::y - y 1 x 2 - x 1 = 9 - 3 9 - 7 y 1 x 2 - x 1 = 9 - 3 9 - 7

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  Then, simplify.
  ::那么,简化。

    $\begin{array}{r}\frac{6}{2}=3\end{array}$

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  The answer is that the slope of line CD equals 3. The slope is positive.
  ::答案是光线 CD 的斜坡等于 3. 斜坡是正的。

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  Example 3
  ::例3

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  Find the slope of line  $\begin{array}{r}FG\end{array}$  below.
  ::找到下面FG线的斜坡

  

  [Figure 3]

   

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  First, write the formula.
  ::首先,写公式。

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     $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{y_2-y_1}{x_2-x_1}\end{array}$
  ::y x = y 2 - y 1 x 2 - x 1

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  Next, fill in the values that are known from the coordinates given using F as Point 1 and G as Point 2.
  ::下一步,填入从给出的坐标上已知的数值,使用F点1作为F点,G点2作为G点。

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    $\begin{array}{r}\frac{7-3}{3-6}\end{array}$   $\begin{array}{r}\frac{y_{}-y_1}{x_2-x_1}=\end{array}$
  ::7 - 3 3 - 6 y - 6 y y 1 x 2 - x 1 = 7 - 3 3 - 6 y y y 1 x 2 - x 1

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  Then, simplify. Remember that slopes are not given as mixed numbers.
  ::那么,简化。记住,斜坡不是以混合数字表示的。

    $\begin{array}{r}\frac{4}{-3}=-\frac{4}{3}\end{array}$

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  $\begin{array}{r}-\frac{4}{3}\end{array}$ The answer is that the slope of line FG equals 
  ::- 4 3 答案是,FG线的斜度等于

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  The slope is negative
  ::斜坡为负负

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  Example 4
  ::例4

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  Draw a line that goes through a point at (–4, –1) and has a slope of  $\begin{array}{r}\frac{3}{7}\end{array}$ . What are the coordinates of the second point?
  ::绘制一条穿过( - 4 - 1) 点的线, 斜度为 3 7 。 第二点的坐标是 多少 ?

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  First, on a  coordinate plane , plot the given point at (-4, -1)
  ::首先,在坐标平面上,在( 4 - 1) 处绘制给定点( 4 - 1) 。

  

  [Figure 4]

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  Next, remember the formula for the slope of a line.
  ::下一位, 请记住一条线的斜坡的公式 。

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  slope =    $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\end{array}$     
  ::斜坡 = y x

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  Then substitute the values that are given.
  ::然后替换给定的值。

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    $\begin{array}{r}\frac{3}{7}=\end{array}$   $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\end{array}$
  ::3 7 = y x

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  This means that the change in y,  $\begin{array}{r}\mathrm{\Delta }y\end{array}$ , is equal to +3.
  ::这意味着y, y, 的更改等于+3 。

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  The change in x,  $\begin{array}{r}\mathrm{\Delta }x\end{array}$ , is equal to +7.
  ::x, x, 等于+7 的更改。

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  Next, starting at the given point (-4, -1), move up +3 units on the  y-axis  and right +7 units on the  x-axis .
  ::下一步,从给定点( 4 - 1) 开始, 在 y 轴上移动+3 单位, 在 x 轴上移动 + 3 单位, 右 + 7 单位 。

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  Then, mark the point, read its location, and draw a line through both points.
  ::然后,标记点, 读读它的位置, 并通过两个点绘制一条线 。

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  The answer is (3, 2).
  ::答案是(3,2)

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  Example 5
  ::例5

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  Draw a line that goes passes through (–5, 4) and has a slope of  $\begin{array}{r}-\frac{2}{3}\end{array}$ . Give the coordinates of a second point.
  ::绘制一条经过( -5, 4) 且斜度为-2 3 的线条。 给第二个点的坐标 。

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  First, on a coordinate plane, plot the given point at (-5, 4).
  ::首先,在坐标飞机上,在(5,4)处绘制给定点(5,4)。

  

  [Figure 5]

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  Next, remember the formula for the slope of a line.
  ::下一位, 请记住一条线的斜坡的公式 。

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  slope =    $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\end{array}$
  ::斜坡 = y x

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  Then substitute the values that are given.
  ::然后替换给定的值。

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     $\begin{array}{r}-\frac{2}{3}\end{array}$    =  $\begin{array}{r}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\end{array}$
  ::- 2 3 = y x

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  Since the slope is negative, the minus sign can be assigned to either value.
  ::由于斜坡为负,减号可划为任一值。

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  The change in y,  $\begin{array}{r}\mathrm{\Delta }y\end{array}$ , is equal to -2.
  ::y, y, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, y, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, e, 等等等等等等等等等等于 - - - -2, 等 -2, 等 -2, 等等之 2。。) 。

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  The change in x,  $\begin{array}{r}\mathrm{\Delta }x\end{array}$ , is equal to +3.
  ::x, x, x, 等于+3 的更改。

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  Next, starting at the given point (-5, 4), move down 2 units on the y-axis and right 3 units on the x-axis.Then, mark the point, read its location, and draw a line through both points.
  ::接下来,从给定点( 5, 4) 开始, 在 y 轴上向下移动 2 个单位, 在 x 轴上向右移动 3 个单位。 然后, 标记点, 读读它的位置, 并通过两个点绘制一条线 。

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  The answer is (-2, 2).
  ::答案是(-2, 2) 。

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  There are an infinite number of points along this line, one of which could be identified by solving the above with the negative assigned to the change in x rather than the change in y.
  ::沿着这条线有无限数量的点数,其中之一可以通过解决上述问题而确定,对x的改变而不是y的改变是负数。

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  Review
  ::回顾

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  Find the slope of each line shown.
  ::查找显示的每条线的斜坡 。

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  1. 

     [Figure 6]
     
     ::[图6]
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  2. 

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     ::[图7]
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  3. 

     [Figure 8]
     
     ::[表8]
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  4. 

     [Figure 9]
     
     ::[图9]
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  5. 

     [Figure 10]
     
     ::[图10]
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  6. 

     [Figure 11]
     
     ::[表11]
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  7. 

     [Figure 12]
     
     ::[图12]
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  8. 

     [Figure 13]
     
     ::[图13]
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  9. On the coordinate grid below, draw a line that passes through (–3, 2) and has a slope of  $\begin{array}{r}\frac{1}{2}\end{array}$ .
     ::在下面的协调网格上,划一条横穿(-3,2)的线,斜坡为12。

  

  [Figure 14]

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  10. Is this slope positive or negative?
      ::这个斜坡是正的还是负的?
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  11. On the coordinate grid below, draw a line that passes through (-2, 5) and has a slope of -4.
      ::在下面的坐标网格上,划一条横穿(-2,5)的线,斜坡为-4。

  

  [Figure 15]

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  12. Is this slope positive or negative?
      ::这个斜坡是正的还是负的?

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  Review (Answers)
  ::回顾(答复)

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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 。

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  Resources
  ::资源