2.1 找出线的斜度和平方

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• 选择节查找线条的曲率和公式

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  查找线条的曲率和公式

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  The grade, or , of a road is measured in a percentage. For example, if a road has a downgrade of 7%, this means, that over every 100 horizontal feet, the road will slope down 7 feet vertically .
  ::道路的等级或程度以百分比衡量。 比如,如果道路降级为7%,这意味着每100个水平英尺,公路将垂直向下倾斜7英尺。

  

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  If a highway has a downgrade of 12% for 3 miles (5280 feet in a mile), how much will the road drop? What is the slope of this stretch of highway?
  ::如果一条高速公路在3英里(每英里5280英尺)上降12%,公路会跌落多少?这段公路的斜坡是多少?

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  Finding the Slope and Equation of a Line
  ::查找线条的曲率和公式

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  The slope of a line determines how steep or flat it is. When we place a line in the coordinate plane , we can measure the slope, or steepness, of a line. Recall the parts of the coordinate plane, also called a $\begin{align*}x-y\end{align*}$ plane and the Cartesian plane , after the mathematician Descartes.
  ::线的斜坡决定它有多陡峭或平坦。 当我们在坐标平面上设置一条线时, 我们可以测量线的斜度或陡度。 回顾坐标平面的各个部分, 也称为X- y 平面和笛卡尔平面, 以数学运货师为后。

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  To plot a point, order matters. First, every point is written $\begin{align*}(x, y),\end{align*}$ where $\begin{align*}x\end{align*}$ is the movement in the $\begin{align*}x-\end{align*}$ direction and $\begin{align*}y\end{align*}$ is the movement in the $\begin{align*}y-\end{align*}$ direction. If $\begin{align*}x\end{align*}$ is negative, the point will be in the $\begin{align*}2^{nd}\end{align*}$ or $\begin{align*}3^{rd}\end{align*}$ quadrants . If $\begin{align*}y\end{align*}$ is negative, the point will be in the $\begin{align*}3^{rd}\end{align*}$ or $\begin{align*}4^{th}\end{align*}$ quadrants. The quadrants are always labeled in a counter-clockwise direction and using Roman numerals.
  ::要绘制一个点, 命令很重要 。 首先, 每个点都是书面的 (x, y) , 其中 x 是 x - 方向的移动, y 是 y - 方向的移动 。 如果 x 是负的, 点将是 2 或 3 之四 。 如果 y 是 负的, 点将是 3 或 4 之四 。 四重点总是以反时针方向贴上标签, 并使用罗马数字 。

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  The point in the $\begin{align*}4^{th}\end{align*}$ quadrant would be (9, -5).
  ::第四象限的点将是( 9, 5 ) 。

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  To find the slope of a line or between two points, first, we start with right triangles. Let’s take the two points (9, 6) and (3, 4). Plotting them on a $\begin{align*}x-y\end{align*}$ plane, we have:
  ::为了找到一条线的斜坡或两点之间的斜坡, 首先, 我们从右三角开始。 让我们选择两点( 9, 6 ) 和 ( 3, 4 ) 。 在 x - y 平面上绘制它们, 我们有 :

  

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  To turn this segment into a right triangle, draw a vertical line down from the higher point, and a horizontal line from the lower point, towards the vertical line. Where the two lines intersect is the third vertex of the slope triangle.
  ::要将此段转换为右三角形, 请从高点向下绘制一条垂直线, 从下点向下绘制一条水平线, 到垂直线。 在两条线相交之处, 两条线是斜度三角形的第三个顶点 。

  

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  Now, count the vertical and horizontal units along the horizontal and vertical sides ( $\begin{align*}{\color{red}{\mathbf{red}}}\end{align*}$ dotted lines).
  ::现在,沿着水平和垂直两侧(红色虚线)计算垂直和水平单位。

  

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  The slope is a fraction with the vertical distance over the horizontal distance, also called the “ rise over run .” Because the vertical distance goes down, we say that it is -2. The horizontal distance moves towards the negative direction (the left), so we would say that it is -6. So, for slope between these two points, the slope would be $\begin{align*}\frac{-2}{-6}\end{align*}$ or $\begin{align*}\frac{1}{3}\end{align*}$ .
  ::斜坡是水平距离垂直距离的一个小块,水平距离上也称为“环向上升 ” 。 因为垂直距离是 -2. 水平距离向负方向(左)移动,所以我们说它就是 -6. 因此,对于这两点之间的斜坡来说,斜坡将是-2-6或13。

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  Note : You can also draw the right triangle above the line segment.
  ::注意:您也可以在线段上绘制右三角形。

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  Now, let's find the slope of the following lines.
  ::现在,让我们找到以下线条的斜坡。

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  1. Use a slope triangle to find the slope of the line below.
     ::使用斜度三角形查找下线的斜度。

  

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  Notice the two points that are drawn on the line. These are given to help you find the slope. Draw a triangle between these points and find the slope.
  ::注意线上绘制的两点。 这些是用来帮助您找到斜坡的。 在这些点之间绘制三角形并找到斜坡 。

  

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  From the slope triangle above, we see that the slope is $\begin{align*}\frac{-4}{4} = -1\end{align*}$ .
  ::从上面的斜坡三角形,我们看到斜坡是 - 441。

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  Whenever a slope reduces to a whole number, the “run” will always be positive 1. Also, notice that this line points in the opposite direction as the line segment above. We say this line has a negative slope because the slope is a negative number and points from the $\begin{align*}2^{nd}\end{align*}$ to $\begin{align*}4^{th}\end{align*}$ quadrants. A line with positive slope will point in the opposite direction and point between the $\begin{align*}1^{st}\end{align*}$ and $\begin{align*}3^{rd}\end{align*}$ quadrants.
  ::1. 另外,请注意,这条线点与上面的线段方向相反。我们说,这条线的斜坡是负的,因为斜坡是2号至4号方位的负数和点。正斜坡的直线将指向1号和3号方位之间的相反方向和点。

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  2. If we go back to our previous example with points (9, 6) and (3, 4), we can find the vertical distance and horizontal distance another way.
     ::如果我们回到我们以前的例子,加上点(9,6)和点(3,4),我们可以找到另一条垂直距离和水平距离。

  

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  From the picture, we see that the vertical distance is the same as the difference between the $\begin{align*}y-\end{align*}$ values and the horizontal distance is the difference between the $\begin{align*}x-\end{align*}$ values. Therefore , the slope is $\begin{align*}\frac{6-4}{9-3}\end{align*}$ . We can extend this idea to any two points, $\begin{align*}(x_1, y_1)\end{align*}$ and $\begin{align*}(x_2, y_2)\end{align*}$ .
  ::从图中可以看出,垂直距离与 y- 值和水平距离之间的差别是相同的。 因此,斜坡是 6-49-3 。 我们可以将这个概念扩大到任何两个点, (x1,y1) 和 (x2,y2) 。

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  Slope Formula : For two points $\begin{align*}(x_1, y_1)\end{align*}$ and $\begin{align*}(x_2, y_2),\end{align*}$ the slope between them is $\begin{align*}\frac{y_2 - y_1}{x_2 - x_1}\end{align*}$ . The symbol for slope is $\begin{align*}m\end{align*}$ .
  ::斜坡公式: 对于两个点(x1,y1)和(x2,y2),两点之间的斜坡是 y2-y1,x2-x1. 斜坡的符号是 m。

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  It does not matter which point you choose as $\begin{align*}(x_1, y_1)\end{align*}$ or $\begin{align*}(x_2, y_2)\end{align*}$ .
  ::您选择哪个点为 (x1,y1) 或 (x2,y2) 并不重要 。

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  Let's find the slope of the following lines using the Slope Formula.
  ::让我们用“斜坡公式”找到以下线条的斜坡。

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  1. Find the slope between (-4, 1) and (6, -5).
     ::在(4、1)和(6、5)之间找到斜坡。

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  Set $\begin{align*}(x_1, y_1) = (-4, 1)\end{align*}$ and $\begin{align*}(x_2, y_2) = (6, -5)\end{align*}$ .
  ::设置 (x1,y1) = (- 4, 1) 和 (x2,y2) = (6, 5) 。

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  $$\begin{align*}m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 -(-4)}{-5 - 1} = \frac{10}{-6} = -\frac{5}{3}\end{align*}$$
  ::my2-y1x2-x1=6-(-4)-5-1=10-653

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  2. Find the slope between (9, -1) and (2, -1).
     ::查找在(9,-1)和(2,-1)之间的斜坡。

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  Set $\begin{align*}(x_1, y_1) = (9, -1)\end{align*}$ and $\begin{align*}(x_2, y_2) = (2, -1)\end{align*}$ .
  ::设置 (x1,y1) =(9,-1) 和(x2,y2) =(2,-1)。

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  $$\begin{align*}m = \frac{-1 - (-1)}{2 - 9} = \frac{0}{-7} = 0\end{align*}$$
  ::m1-(- 1)2-9=0-7=0

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  Here, we have zero slope . Plotting these two points we have a horizontal line. This is because the $\begin{align*}y-\end{align*}$ values are the same. Anytime the $\begin{align*}y-\end{align*}$ values are the same we will have a horizontal line and the slope will be zero.
  ::在这里,我们有零斜度。 绘制这两个点的横线。 这是因为 y - 值是相同的。 只要 y - 值是相同的, 我们就会有一个水平线, 而斜度是零 。

  

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find how much the road will drop and the slope of the stretch of highway. 
  ::早些时候,有人要求你找到 道路将下降多少 和高速公路的斜坡。

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  The road slopes down 12 feet over every 100 feet.
  ::路坡每100英尺下下12英尺。

  

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  Let's set up a ratio to find out how much the road slopes in 3 miles, or $\begin{align*}3 \cdot 5280 = 15,840\end{align*}$ feet.
  ::让我们设定一个比率 来找出三英里内 公路坡有多高 也就是3 5 5 280=15 840英尺

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  $$\begin{align*}\frac{12}{100}&= \frac{x}{15,840} \\ 15840 \cdot \frac{12}{100} &= x \\ x &= 1900.8 \end{align*}$$
  ::12100=x15,8401584012100=xx=1900.8

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  The road drops 1900.8 feet over the 3 miles. The slope of the road is $\begin{align*}\frac{12}{100}\end{align*}$ or $\begin{align*}\frac{3}{25}\end{align*}$ when the fraction is reduced.
  ::公路在3英里处跌落1900.8英尺,路坡为12100或325英尺,当分数减少时。

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  Example 2
  ::例2

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  Use a slope triangle to find the slope of the line below.
  ::使用斜度三角形查找下线的斜度。

  

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  Counting the squares, the vertical distance is down 6, or -6, and the horizontal distance is to the right 8, or +8. The slope is then $\begin{align*}\frac{-6}{8}\end{align*}$ or $\begin{align*}-\frac{2}{3}\end{align*}$ .
  ::计算方形时,垂直距离向下为6或-6,水平距离向右为8或+8。 斜度为-68或-23。

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  Example 3
  ::例3

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  Find the slope between (2, 7) and (-3, -3).
  ::查找(2、7)和(3、3和3)之间的斜坡。

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  Use the Slope Formula. Set $\begin{align*}(x_1, y_1) = (2, 7)\end{align*}$ and $\begin{align*}(x_2, y_2) = (-3, -3)\end{align*}$ .
  ::使用斜坡公式。 设置 (x1,y1) = (2, 7) 和 (x2,y2) = (- 3, 3) 。

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  $$\begin{align*}m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 -7}{-3 -2} = \frac{-10}{-5} = 2\end{align*}$$
  ::my2 - y1x2 - x1 - x1 - 3 - 7 - 3 - 2 - 2 - 10 - 5=2

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  Example 4
  ::例4

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  Find the slope between (-4, 5) and (-4, -1).
  ::在(4、5)和(4、1)之间找到斜坡。

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   Again, use the Slope Formula. Set $\begin{align*}(x_1, y_1) = (-4, 5)\end{align*}$ and $\begin{align*}(x_2, y_2) = (-4, -1)\end{align*}$ .
  ::再次使用斜坡公式。 设定 (x1, y1) = (- 4, 5) 和 (x2, y2) = (- 4, -1) 。

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  $$\begin{align*}m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 -5}{-4 -(-4)} = \frac{-6}{0}\end{align*}$$
  ::my2-y1x2-x1_x1___1-5-4-(-4)_(-4)_________________________________________________________________________________________________________________

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  You cannot divide by zero. Therefore, this slope is undefined . If you were to plot these points, you would find they form a vertical line. All vertical lines have an undefined slope .
  ::您不能除以 0 。 因此, 此斜坡是未定义的 。 如果您要绘制这些点, 您就会发现它们形成一条垂直线 。 所有垂直线都有一个未定义的斜坡 。

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  Important Note : Always reduce your slope fractions. Also, if the numerator or denominator of a slope is negative, then the slope is negative. If they are both negative, then we have a negative number divided by a negative number, which is positive, thus a positive slope.
  ::重要注意 : 总是减少您的斜坡分数。 另外, 如果斜坡的分子或分母为负数, 那么斜坡为负数。 如果两者均为负数, 那么我们就会发现负数除以负数, 负数是正数, 因此是正数 。

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  Review
  ::回顾

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  Find the slope of each line by using slope triangles.
  ::使用斜度三角形查找每条线的斜度。

  1. 
  2. 
  3. 
  4. 
  5. 
  6. 

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  Find the slope between each pair of points using the Slope Formula.
  ::使用“斜坡公式”在每对点之间查找斜度。

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  7. (-5, 6) and (-3, 0)
     :5-5,6)和(3,0)
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  8. (1, -1) and (6, -1)
     :1,-1)和(6,-1)
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  9. (3, 2) and (-9, -2)
     :3,2和3,3,2)和(9,9,2)
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  10. (8, -4) and (8, 1)
      :8,4和8,1)
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  11. (10, 2) and (4, 3)
      :10、2和4、3)
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  12. (-3, -7) and (-6, -3)
      :3-3,7)和(6,6,3)
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  13. (4, -5) and (0, -13)
      :4、5和5)和(0、13)
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  14. (4, -15) and (-6, -11)
      :4,15,4,4,15)和(6,11)
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  15. (12, 7) and (10, -1)
      :12,7)和(10,-1)
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  16. Challenge The slope between two points $\begin{align*}(a, b)\end{align*}$ and (1, -2) is $\begin{align*}\frac{1}{2}\end{align*}$ . Find $\begin{align*}a\end{align*}$ and $\begin{align*}b\end{align*}$ .
      ::两点(a、b)和1点(1、2点)之间的斜坡是12点。

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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 。