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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?

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

    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 x y 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。

    To plot a point, order matters. First, every point is written ( x , y ) , where x is the movement in the x direction and y is the movement in the y direction. If x is negative, the point will be in the 2 n d or 3 r d quadrants . If y is negative, the point will be in the 3 r d or 4 t h 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 。

    The point in the 4 t h quadrant would be (9, -5).
    ::第四象限的点将是( 9, 5 ) 。

    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 x y 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 :

    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 。

    Now, count the vertical and horizontal units along the horizontal and vertical sides ( r e d 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。

    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 2 6 or 1 3 .
    ::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。

    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。

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

    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。

    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 。

    From the slope triangle above, we see that the slope is 4 4 = 1 .
    ::从上面的斜坡三角形,我们看到斜坡是 - 441。

    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 2 n d to 4 t h quadrants. A line with positive slope will point in the opposite direction and point between the 1 s t and 3 r d 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。

    1. 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 y values and the horizontal distance is the difference between the x values. Therefore , the slope is 6 4 9 3 . We can extend this idea to any two points, ( x 1 , y 1 ) and ( x 2 , y 2 ) .
    ::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) 。

    Slope Formula : For two points ( x 1 , y 1 ) and ( x 2 , y 2 ) , the slope between them is y 2 y 1 x 2 x 1 . The symbol for slope is m .
    ::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。

    It does not matter which point you choose as ( x 1 , y 1 ) or ( x 2 , y 2 ) .
    ::您选择哪个点为 (x1,y1) 或 (x2,y2) 并不重要 。

    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。

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

    Set ( x 1 , y 1 ) = ( 4 , 1 ) and ( x 2 , y 2 ) = ( 6 , 5 ) .
    ::设置 (x1,y1) = (- 4, 1) 和 (x2,y2) = (6, 5) 。

    m = y 2 y 1 x 2 x 1 = 6 ( 4 ) 5 1 = 10 6 = 5 3

    ::my2-y1x2-x1=6-(-4)-5-1=10-653

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

    Set ( x 1 , y 1 ) = ( 9 , 1 ) and ( x 2 , y 2 ) = ( 2 , 1 ) .
    ::设置 (x1,y1) =(9,-1) 和(x2,y2) =(2,-1)。

    m = 1 ( 1 ) 2 9 = 0 7 = 0

    ::m1-(- 1)2-9=0-7=0

    Here, we have zero slope . Plotting these two points we have a horizontal line. This is because the y values are the same. Anytime the y 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

    Example 1
    ::例1

    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。

    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 3 5280 = 15 , 840 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

    12 100 = x 15 , 840 15840 12 100 = x x = 1900.8

    ::12100=x15,8401584012100=xx=1900.8

    The road drops 1900.8 feet over the 3 miles. The slope of the road is 12 100 or 3 25 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。

    Example 2
    ::例2

    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 6 8 or 2 3 .
    ::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。

    Example 3
    ::例3

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

    Use the Slope Formula. Set ( x 1 , y 1 ) = ( 2 , 7 ) and ( x 2 , y 2 ) = ( 3 , 3 ) .
    ::使用斜坡公式。 设置 (x1,y1) = (2, 7) 和 (x2,y2) = (- 3, 3) 。

    m = y 2 y 1 x 2 x 1 = 3 7 3 2 = 10 5 = 2

    ::my2 - y1x2 - x1 - x1 - 3 - 7 - 3 - 2 - 2 - 10 - 5=2

    Example 4
    ::例4

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

     Again, use the Slope Formula. Set ( x 1 , y 1 ) = ( 4 , 5 ) and ( x 2 , y 2 ) = ( 4 , 1 ) .
    ::再次使用斜坡公式。 设定 (x1, y1) = (- 4, 5) 和 (x2, y2) = (- 4, -1) 。

    m = y 2 y 1 x 2 x 1 = 1 5 4 ( 4 ) = 6 0

    ::my2-y1x2-x1_x1___1-5-4-(-4)_(-4)_________________________________________________________________________________________________________________

    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 。

    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 。

    Review
    ::回顾

    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。

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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。

    1. (-5, 6) and (-3, 0)
      :sad5-5,6)和(3,0)
    2. (1, -1) and (6, -1)
      :sad1,-1)和(6,-1)
    3. (3, 2) and (-9, -2)
      :sad3,2和3,3,2)和(9,9,2)
    4. (8, -4) and (8, 1)
      :sad8,4和8,1)
    5. (10, 2) and (4, 3)
      :sad10、2和4、3)
    6. (-3, -7) and (-6, -3)
      :sad3-3,7)和(6,6,3)
    7. (4, -5) and (0, -13)
      :sad4、5和5)和(0、13)
    8. (4, -15) and (-6, -11)
      :sad4,15,4,4,15)和(6,11)
    9. (12, 7) and (10, -1)
      :sad12,7)和(10,-1)
    10. Challenge The slope between two points ( a , b ) and (1, -2) is 1 2 . Find a and b .
      ::两点(a、b)和1点(1、2点)之间的斜坡是12点。

    Review (Answers)
    ::回顾(答复)

    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 。