4.6 斜坡

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 斜斜度

  Collapse Expand

  斜斜度

  播放段落

  Slope 
  ::斜斜度

  播放段落

  Wheelchair ramps at building entrances must have a slope between $\begin{array}{r}\frac{1}{16}\end{array}$ and $\begin{array}{r}\frac{1}{20}\end{array}$ . If the entrance to a new office building is 28 inches off the ground, how long does the wheelchair ramp need to be?
  ::建筑物入口的轮椅坡道必须有一个116至120之间的斜坡,如果新办公楼的入口离地面28英寸,轮椅坡道需要多久?

  播放段落

  We come across many examples of slope in everyday life. For example, a slope is in the pitch of a roof, the grade or incline of a road, or the slant of a ladder leaning on a wall. In math, we use the word slope to define steepness in a particular way.
  ::我们在日常生活中遇到许多斜坡的例子。例如,斜坡位于屋顶、道路的等级或坡度或斜坡,或者斜坡位于墙上的梯子上。在数学中,我们用斜坡这个词来以某种特定的方式定义陡峭。

  播放段落

  $$\begin{array}{r}\text{Slope}=\frac{\text{distance moved vertically}}{\text{distance moved horizontally}}\end{array}$$

  
  ::Slope=远距离垂直移动 垂直移动 水平移动

  播放段落

  To make it easier to remember, we often word it like this:
  ::为了让人们更容易地记住,我们经常这样说:

  播放段落

  $$\begin{array}{r}\text{Slope}=\frac{\text{rise}}{\text{run}}\end{array}$$

  
  ::斜斜坡=中空

  

  播放段落

  In the picture above, the slope would be the ratio of the height of the hill to the horizontal length of the hill. In other words, it would be $\begin{array}{r}\frac{3}{4}\end{array}$ , or 0.75.
  ::在上图中,斜坡是山高与山水平长度之比。 换句话说,斜坡是34或0.75。

  播放段落

  If the car were driving to the right it would climb the hill - we say this is a positive slope . Any time you see the graph of a line that goes up as you move to the right, the slope is positive .
  ::如果车向右行驶,它会爬上山坡——我们说这是一个正斜坡。当你看到向右行向上行的图时,斜坡是正。

  播放段落

  If the car kept driving after it reached the top of the hill, it might go down the other side. If the car is driving to the right and descending , then we would say that the slope is negative .
  ::如果汽车在到达山顶后一直开到山顶,它可能会从另一侧下下去,如果汽车向右开往下,那么我们就会说斜坡是负的。

  

  播放段落

  Here’s where it gets tricky: If the car turned around instead and drove back down the left side of the hill, the slope of that side would still be positive. This is because the rise would be -3, but the run would be -4 (think of the $\begin{array}{r}x-\end{array}$ axis - if you move from right to left you are moving in the negative $\begin{array}{r}x-\end{array}$ direction). That means our slope ratio would be $\begin{array}{r}\frac{-3}{-4}\end{array}$ , and the negatives cancel out to leave 0.75, the same slope as before. In other words, the slope of a line is the same no matter which direction you travel along it.
  ::问题就在这里:如果汽车转过身往下向山坡的左侧倾斜,那一边的斜坡仍将是正的。 这是因为升幅会是 - 3, 但跑幅将是 - 4(如果从右向左移动,你会认为是 x - 轴 ) 。 这意味着我们的斜坡比率是 - 3 - 4 , 负数会取消,离开0.75, 与过去一样的斜坡。 换句话说,线的斜坡与你沿着这条斜坡前进的方向是相同的。

  播放段落

  Finding the Slope of a Line
  ::查找线条的曲线

  播放段落

  A simple way to find a value for the slope of a line is to draw a right triangle whose hypotenuse runs along the line. Then we just need to measure the distances on the triangle that correspond to the rise (the vertical dimension) and the run (the horizontal dimension).
  ::找到一条线的斜坡值的一个简单方法就是绘制一个右三角形,其下限沿线运行。 然后我们只需要测量三角形上与上升(垂直维度)和运行(水平维度)相对应的距离。

  播放段落

  Find the slopes for the three graphs shown.
  ::查找显示的三个图形的斜度。

  

  播放段落

  1. There are already right triangles drawn for each of the lines - in future problems you’ll do this part yourself. Note that it is easiest to make triangles whose vertices are lattice points (i.e. points whose coordinates are all integers).
  ::1. 每个线条都已经绘制了正确的三角形 -- -- 在未来的问题中,你会自己来做这部分,请注意,最容易的三角形的顶部是固定点(即坐标均为整数的点 ) 。

  播放段落

  a) The rise shown in this triangle is 4 units; the run is 2 units. The slope is $\begin{array}{r}\frac{4}{2}=2\end{array}$ .
  ::a) 这个三角形显示上升为4个单位;运行为2个单位。斜坡为42=2。

  播放段落

  b) The rise shown in this triangle is 4 units, and the run is also 4 units. The slope is $\begin{array}{r}\frac{4}{4}=1\end{array}$ .
  :b) 这个三角形显示上升为4个单位,运行为4个单位。斜坡为44=1。

  播放段落

  c) The rise shown in this triangle is 2 units, and the run is 4 units. The slope is $\begin{array}{r}\frac{2}{4}=\frac{1}{2}\end{array}$ .
  :c) 这个三角形显示上升为2个单位,运行为4个单位。斜坡为24=12。

  播放段落

  2. Find the slope of the line that passes through the points (1, 2) and (4, 7).
  ::2. 找出穿过点(1、2和4、7)的线坡。

  播放段落

  We already know how to graph a line if we’re given two points: we simply plot the points and connect them with a line. Here’s the graph:
  ::如果给出两点的话,我们已经知道如何绘制一条线的图解:我们只是绘制点并把它们与一条线连接起来。

  

  播放段落

  Since we already have coordinates for the vertices of our right triangle, we can quickly work out that the rise is $\begin{array}{r}7-2=5\end{array}$ and the run is $\begin{array}{r}4-1=3\end{array}$ (see diagram). So the slope is $\begin{array}{r}\frac{7-2}{4-1}=\frac{5}{3}\end{array}$ .
  ::由于我们已经有了我们右三角形的顶端坐标,我们可以迅速发现上升为7-2=5,运行为4-1=3(见图)。所以斜坡为7-24-1=53。

  播放段落

  If you look again at the calculations for the slope, you’ll notice that the 7 and 2 are the $\begin{array}{r}y-\end{array}$ coordinates of the two points and the 4 and 1 are the $\begin{array}{r}x-\end{array}$ coordinates. This suggests a pattern we can follow to get a general formula for the slope between two points $\begin{array}{r}(x_1,y_1)\end{array}$ and $\begin{array}{r}(x_2,y_2)\end{array}$ :
  ::如果您再次查看斜坡的计算, 您就会注意到 7 和 2 是两个点的 y - 坐标, 而 4 和 1 是 x - 坐标 。 这表明我们可以遵循一种模式来获得两个点( X1, y1) 和 (x2, y2) 之间斜坡的一般公式 :

  播放段落

  Slope between $\begin{array}{r}(x_1,y_1)\end{array}$ and $\begin{array}{r}(x_2,y_2)=\frac{y_2-y_1}{x_2-x_1}\end{array}$
  ::在 (x1,y1) 和 (x2,y2) =y2 -y1x2 - x1 之间

  播放段落

  or $\begin{array}{r}m=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}.\end{array}$
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}或者...

  播放段落

  In the second equation the letter $\begin{array}{r}m\end{array}$ denotes the slope (this is a mathematical convention you’ll see often) and the Greek letter delta $\begin{array}{r}(\mathrm{\Delta })\end{array}$ means change . So another way to express slope is change in $\begin{array}{r}y\end{array}$ divided by change in $\begin{array}{r}x\end{array}$ . In the next section, you’ll see that it doesn’t matter which point you choose as point 1 and which you choose as point 2.
  ::在第二个方程式中,字母m表示斜坡(这是一个数学惯例,你会经常看到),希腊字母三角洲()则表示变化。因此,表达斜坡的另一种方式是换换一,换一换一。在下一节中,你可以看到,选择哪个点是第1点,哪个点是第2点并不重要。

  播放段落

  Finding the Slopes of Horizontal and Vertical lines
  ::查找水平和垂直线条的曲线

  播放段落

  Determine the slopes of the two lines on the graph below.
  ::确定下图上两行的坡度。

  

  播放段落

  There are 2 lines on the graph: $\begin{array}{r}A(y=3)\end{array}$ and $\begin{array}{r}B(x=5)\end{array}$ .
  ::图中有两个行:A(y=3)和B(x=5)。

  播放段落

  Let’s pick 2 points on line $\begin{array}{r}A\end{array}$ —say, $\begin{array}{r}(x_1,y_1)=(-4,3)\end{array}$ and $\begin{array}{r}(x_2,y_2)=(5,3)\end{array}$ —and use our equation for slope:
  ::让我们在A-say线(x1,y1)=(-4,3)和(x2,y2)=(5,3)上选择两点,并用我们的方程式来表示斜坡:

  播放段落

  $$\begin{array}{r}m=\frac{y_2-y_1}{x_2-x_1}=\frac{(3)-(3)}{(5)-(-4)}=\frac{0}{9}=0.\end{array}$$

  
  ::my2-y1x2-x1=(3)-(3)(5)-(-4)=09=0。

  播放段落

  If you think about it, this makes sense - if $\begin{array}{r}y\end{array}$ doesn’t change as $\begin{array}{r}x\end{array}$ increases then there is no slope, or rather, the slope is zero. You can see that this must be true for all horizontal lines.
  ::如果您想一想,这样就有意义了 — — 如果您不随着x的增加而改变,那么就不会有斜坡,或者说,斜坡是零。您可以看到,所有水平线都必须如此。

  播放段落

  Horizontal lines ( $\begin{array}{r}y\end{array}$ = constant ) all have a slope of 0.
  ::水平线(y = 常数)都具有0的斜度。

  播放段落

  Now let’s consider line $\begin{array}{r}B\end{array}$ . If we pick the points $\begin{array}{r}(x_1,y_1)=(5,-3)\end{array}$ and $\begin{array}{r}(x_2,y_2)=(5,4)\end{array}$ , our slope equation is $\begin{array}{r}m=\frac{y_2-y_1}{x_2-x_1}=\frac{(4)-(-3)}{(5)-(5)}=\frac{7}{0}\end{array}$ . But dividing by zero isn’t allowed!
  ::现在让我们来考虑行B。如果我们选择点(x1,y1)=(5,-3)和(x2,y2)=(5,4),我们的斜度方程式是 m=y2 -y1x2-x1=(4)-(-3)(5)-(5)=70。但以零除法是不允许的!

  播放段落

  In math we often say that a term which involves division by zero is undefined . (Technically, the answer can also be said to be infinitely large—or infinitely small, depending on the problem.)
  ::在数学中,我们经常说,一个涉及零除法的术语是没有定义的。 (从技术上讲,答案也可以说是无限大——或者说无限小,视问题而定。 )

  播放段落

  Vertical lines $\begin{array}{r}(x=\end{array}$ constant ) all have an infinite (or undefined) slope.
  ::垂直线(x=contaant)都有无限(或未定义)的斜度。

  播放段落

  Example
  ::示例示例示例示例

  播放段落

  Example 1
  ::例1

  播放段落

  Find the slopes of the lines on the graph below.
  ::在下图中查找线条的斜坡。

  

  播放段落

  Look at the lines - they both slant down (or decrease) as we move from left to right. Both these lines have negative slope .
  ::看看这些线——在我们从左向右移动时,它们都向下倾斜(或下降)。这两条线都有负斜坡。

  播放段落

  The lines don’t pass through very many convenient lattice points, but by looking carefully you can see a few points that look to have integer coordinates. These points have been circled on the graph, and we’ll use them to determine the slope. We’ll also do our calculations twice, to show that we get the same slope whichever way we choose point 1 and point 2.
  ::线条不会通过很多方便的边线点,但仔细看一看,你可以看到一些看似有整数坐标的点。 这些点在图形上已经进行了圆圈,我们会用它们来确定斜坡。 我们还将做两次计算,以显示我们得到的斜坡与我们选择第1点和第2点的方式相同。

  播放段落

  For Line $\begin{array}{r}A\end{array}$ :
  ::A行:

  播放段落

  $$\begin{array}{rlrl}& (x_1,y_1)=(-6,3)(x_2,y_2)=(5,-1)& & (x_1,y_1)=(5,-1)(x_2,y_2)=(-6,3)\\ & m=\frac{y_2-y_1}{x_2-x_1}=\frac{(-1)-(3)}{(5)-(-6)}=\frac{-4}{11}\approx -0.364& & m=\frac{y_2-y_1}{x_2-x_1}=\frac{(3)-(-1)}{(-6)-(5)}=\frac{4}{-11}\approx -0.364\end{array}$$

  
  :x1,y1) = (-6,6,3 (x2,y2) = (5,-1,1,1,1,1) = (5,1,1,1,1,1,5) = (5,1,1,1,2,2,y2) = (6,3,3,m =y2 -y1,2,x2,-x1) = (1,3) = (1,3,3,3-(6) = y2 - y1,2,x2-x1) = (1,3,3,m y2 - 6) = (1,3,3,3,7-(6) = 411,0,364)

  播放段落

  For Line $\begin{array}{r}B\end{array}$
  ::B行

  播放段落

  $$\begin{array}{rlrl}& (x_1,y_1)=(-4,6)(x_2,y_2)=(4,-5)& & (x_1,y_1)=(4,-5)(x_2,y_2)=(-4,6)\\ & m=\frac{y_2-y_1}{x_2-x_1}=\frac{(-5)-(6)}{(4)-(-4)}=\frac{-11}{8}=-1.375& & m=\frac{y_2-y_1}{x_2-x_1}=\frac{(6)-(-5)}{(-4)-(4)}=\frac{11}{-8}=-1.375\end{array}$$

  
  :x1,y1) = (-4,4,6 (x2,y2) = (4,5,5 (x1,y1) = (4,5)(x2,y2) = (4,6) = (4,6) = y2 -y1x2 -x1 = (5,6) = (4,6) = y2 - (4) - (4,6) = (4,6) = (4,6) = y2 -y1x2 - x1) = (6) - (4,6) = (6) - (4) = y2 -y1x2 - x1= (6) = (4) (4) = 11 - 8/1375

  播放段落

  You can see that whichever way round you pick the points, the answers are the same. Either way, Line $\begin{array}{r}A\end{array}$ has slope -0.364, and Line $\begin{array}{r}B\end{array}$ has slope -1.375.
  ::你可以看到,无论你从哪个方向选择点,答案都是一样的。无论哪种方式,A线都有斜度-0.364,B线有斜度-1.375。

  播放段落

  Review 
  ::回顾

  播放段落

  Use the slope formula to find the slope of the line that passes through each pair of points.
  ::使用斜坡公式找到通过每一对点的线的斜坡。

  播放段落
  1. (-5, 7) and (0, 0)
     :5,7)和(0,0)
  播放段落
  2. (-3, -5) and (3, 11)
     :3, 3, 5)和(3, 11)
  播放段落
  3. (3, -5) and (-2, 9)
     :3,5和2,9)
  播放段落
  4. (-5, 7) and (-5, 11)
     :5-5,7)和(5,11)
  播放段落
  5. (9, 9) and (-9, -9)
     :9,9)和(9,9)
  播放段落
  6. (3, 5) and (-2, 7)
     :3,5)和(2,7)
  播放段落
  7. (2.5, 3) and (8, 3.5)
     :2.5、3)和(8、3.5)

  播放段落

  For each line in the graphs below, use the points indicated to determine the slope.
  ::对于下图中的每一行,使用所标明的点来确定坡度。

  8. 

  9. 

  10. 

  播放段落
  11. For each line in the graphs above, imagine another line with the same slope that passes through the point (1, 1), and name one more point on that line.
      ::对于以上图表中的每条线, 想象另一条线, 与穿过点(1, 1) 的同一斜坡, 并在该线上再列出一个点 。

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。