Section: 查找线条的曲线 | 3.4 查找线的曲线

Section outline

The grade, or , of a road is measured as the percent of the vertical change compared to the horizontal change. For example, if a road has a grade of 7%, this means that over every 100 feet horizontally , the road will rise 7 feet vertically .

If a highway has a downgrade of 12% for 3 miles (5280 feet in a mile), how much will the road drop? We find a way to determine the grade or slope in this section.
::如果高速公路在3英里(每英里5280英尺)上降12%,公路会跌多少?我们找到办法确定这一段的等级或坡度。

Slope of a Line: Rise Over Run
::线条的斜体: 上升运行

As we saw in the previous section, the rate of change determines whether a set of data is linear or non-linear. In this section, we formalize how to find this rate of change.

::正如我们在前一节中看到的那样,变化率决定了一组数据是线性数据还是非线性数据。在本节中,我们正式确定如何找到这一变化率。

The rate of change of or the distance between the y- values compared to the x- values is called the slope . The symbol for slope is $\begin{aligned} m \end{aligned}$ . (Why it is represented by $\begin{aligned} m \end{aligned}$ is still an area of historical research). As we will see, graphically, the slope measures how steep the line is.
::Y值相对于X值的变化速度或距离称为斜坡。斜坡的符号是 m.(为什么以m表示仍然是历史研究的领域)。我们可以看到,从图形上看,斜坡测量线的陡度。

One way to determine the slope is to plot the points on a graph, draw a line through them, and then count the change in the y- values, or rise , and the change in the x- values, or run . As we saw on one number line, to find the distance between two points we can count the units in between the two numbers on the number line. We can use a similar approach with two number lines. For example, how much the graph changes vertically, or the rise, is aligned with the y- axis, so we can just count the units on that number line by counting the boxes. The approach for how the graph changes horizontally, or the run, is similar.

::确定斜坡的方法之一是在图形上绘制点,绘制一条线,然后计算Y值的变化,或上升,以及X值的变化,或运行。正如我们在一数字线上看到的那样,找到两个点之间的距离,我们可以在数字线上两个数字之间计算单位。我们可以用两个数字线来使用类似的方法。例如,图形垂直变化或上升与y轴对齐的程度,这样我们就可以通过计算框来计算该数字线上的单位。图形水平变化或运行变化的方法相似。

Let's see some examples of how this is done below.

::下面我们来看看一些例子,说明如何做到这一点。

Example 1
::例1

Find the slope of the line segment connecting the points (9, 6) and (3, 4).
::查找连接点(9,6)和点(3,4)的线段斜坡。

Solution: From the graph, we see that the rise or vertical distance is 2 boxes. That is the same as the difference between the y- values or $\begin{aligned} 6 - 4 = 2 \end{aligned}$ . Similarly, the run or horizontal distance is 6 boxes or the difference between the x- values, $\begin{aligned} 9 - 3 = 6 \end{aligned}$ . Therefore , the slope is $\begin{aligned} m = \frac{r i s e}{r u n} = \frac{2}{6} = \frac{1}{3} \end{aligned}$ .
::解答: 从图表中,我们看到上升或垂直距离为2个框。这与y值或6-4=2之间的差值相同。同样,运行或水平距离为6个框,或X值之间的差值为9-3=6。因此,斜度为m=srisrun=26=13。

Example 2
::例2

Find the slope of the line that crosses through (-4,5) and (6,-3).
::找到横穿(4、5和6、3)的线的斜坡。

Solution: Starting at the point on the left, we see that the rise is negative since we can count 8 boxes down or $\begin{aligned} - 3 - 5 = - 8 \end{aligned}$ . To find the run, we then count to the right 10 boxes or $\begin{aligned} 6 - (- 4) = 10 \end{aligned}$ . The slope is $\begin{aligned} m = \frac{r i s e}{r u n} = \frac{- 8}{10} = - \frac{4}{5} \end{aligned}$ .
::解决方案:从左边的点开始,我们看到上升为负值,因为我们可以向下数8个方框或-3-58。为了找到跑步,我们然后数到右10个方框或6-(-4)=10。斜坡是 m=srisrunn81045。

by CK-12 demonstrates how to find the slope of a line.
::通过 CK-12 演示如何找到线的斜坡。

Example 3
::例3

If a highway has a downgrade of 12% for 3 miles (5280 feet in a mile), how much will the road drop vertically?
::如果高速公路在3英里(每英里5280英尺)上降12%,

Solution: The road in the introduction slopes downward at a rate of 12% over a distance of 3 miles. We can substitute this into our formula for slope to find the vertical change.
::解决方案: 引入斜坡中的道路在3英里的距离以12%的速率向下倾斜。我们可以用这个公式来取代斜坡, 以找到垂直变化。

$\begin{aligned} m & = 12 % = \frac{12}{100} \\ m & = \frac{r i s e}{r u n} = \frac{r i s e}{3 \cdot 5, 280} \\ \frac{12}{100} & = \frac{r i s e}{15, 840} \\ 12 \cdot 15, 840 & = 100 \cdot r i s e \\ 190, 080 & = 100 \cdot r i s e \\ 1, 900.8 & = r i s e \end{aligned}$

::=12 =12 12100米中空 = 中空 3 5 28012100 中空 5 28012100 中空 15 84012+15 840 中空 190 080 中空 1 900.8 中空

The road goes down by 1,900.8 feet over 3 miles.
::这条路在三英里外下降了1,900.8英尺

Slope of a Line: The Formula
::线条的斜面:公式

As we saw in the absolute value section, the distance between two points on a number line is $\begin{aligned} | b - a | \end{aligned}$ . Since for slope the direction (up, down, left , or right) we move matters, we drop the absolute value bars.
::正如我们在绝对值部分所看到的那样,数字线两点之间的距离是 b-a。对于斜坡方向( 向上、向下、左或右) 来说, 我们移动了重要的东西, 我们下降绝对值条。

To create a formula, the rise is the change in the y - coordinates : $\begin{aligned} y_{2} - y_{1} \end{aligned}$ . The run is the change in the x -coordinates: $\begin{aligned} x_{2} - x_{1} \end{aligned}$ . Using this, w e can express rise over run as a formula.
::要创建公式, 上升是 Y 坐标的更改 : y2 - y1 。运行是 x 坐标的更改 : x2 - x1 。使用此选项, 我们可以表示作为公式运行时的上升。

Slope Formula

::斜曲公式

The slope of the line that contains two points $\begin{aligned} (x_{1}, y_{1}) \end{aligned}$ and $\begin{aligned} (x_{2}, y_{2}) \end{aligned}$ is
::包含两个点(x1,y1)和(x2,y2)的线的斜坡是

$\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \end{aligned}$

::m=y2 - y1x2 - x1

Example 4
::例4

Find the slope of the line that contains (2, 7) and (-3, -3).
::查找包含(2,7)和(3,3)的线条的斜坡。

Solution: Set $\begin{aligned} (x_{1}, y_{1}) = (2, 7) \end{aligned}$ and $\begin{aligned} (x_{2}, y_{2}) = (- 3, - 3) \end{aligned}$ . Now, using the formula, we have
::解决方案 : 设置 (x1,y1) = (2, 7) 和 (x2, y2) = (- 3, - 3) 。现在, 使用公式, 我们有了

$\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{- 3 - 7}{- 3 - 2} = \frac{- 10}{- 5} = 2 \end{aligned}$

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

When we graph the line that contains these points, the line appears to go uphill from left to right.
::当我们绘制包含这些点的线条时, 线条似乎从左向右向上移动。

Example 5
::例5

Find the slope of the line that contains (1, -4) and (-5, 6).
::查找包含(1,4)和(5,6)的线的斜坡。

Solution: Set $\begin{aligned} (x_{1}, y_{1}) = (- 4, 1) \end{aligned}$ and $\begin{aligned} (x_{2}, y_{2}) = (6, - 5) \end{aligned}$ . With the formula, we have
::解决方案: 设置 (x1,y1) = (- 4, 1) 和 (x2,y2) = (6, 5) 。

$\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{6 - (- 4)}{- 5 - 1} = \frac{10}{- 6} = - \frac{5}{3} \end{aligned}$

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

When we graph the line that contains these points, the line appears to go downhill from left to right.
::当我们绘制包含这些点的线条时, 线条似乎从左向右下坡。

by CK-12 demonstrates how to find and use the slope of a line.
::CK-12 显示如何查找和使用线的斜坡。

Example 6
::例6

Find the slope of the line that contains (9, -1) and (2, -1).
::查找包含(9)-1和(2)-1的线条的斜坡。

Solution: Set $\begin{aligned} (x_{1}, y_{1}) = (9, - 1) \end{aligned}$ and $\begin{aligned} (x_{2}, y_{2}) = (2, - 1) \end{aligned}$ . The slope is
::溶液: 设置 (x1,y1) = (9,-1) 和 (x2,y2) = (2,-1)。斜度为 :

$\begin{aligned} m = \frac{- 1 - (- 1)}{2 - 9} = \frac{0}{- 7} = 0 \end{aligned}$

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

When we graph the line that contains these points, we have a flat or horizontal line.
::当我们绘制包含这些点的线条时,我们有一个平面线或水平线。

Example 7
::例7

Find the slope of the line that contains (-4, 5) and (-4, -1).
::查找包含(-4、5)和(-4、1)的线条的斜坡。

Solution: Set $\begin{aligned} (x_{1}, y_{1}) = (- 4, 5) \end{aligned}$ and $\begin{aligned} (x_{2}, y_{2}) = (- 4, - 1) \end{aligned}$ . The slope is
::溶液: 设置 (x1,y1) = (- 4, 5) 和 (x2,y2) = (- 4, - 1)。斜坡是 :

$\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{- 1 - 5}{- 4 - (- 4)} = \frac{- 6}{0} = undefined \end{aligned}$

::my2-y1x2- x1_x1_%1-5-4-(- 4)\_60=未定义

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.
::您不能除以零。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垂直线。

WARNING
::警告

Make sure to assign a point 1 and point 2 first. It is common to switch the x- and y- coordinates between the points. For example, consider the points in Example 5. If we switch the $\begin{aligned} x_{1} \end{aligned}$ and $\begin{aligned} x_{2} \end{aligned}$ , we have $\begin{aligned} m = \frac{6 - (- 4)}{1 - (- 5)} = \frac{10}{6} = \frac{5}{3} \end{aligned}$ . We get a positive slope instead of a negative slope .
::确保首先指定一个点 1 和点 2 。在两个点之间切换 X 和 Y 坐标是常见的。例如, 考虑例 5 中的点, 如果我们切换 x1 和 x2, 我们就会有 m= 6 - (- 4) 1 - (- 5) = 106= 53。我们得到正斜度而不是负斜度。

You cannot switch the numerator and the denominator. $\begin{aligned} m \neq \frac{change in x}{change in y} \end{aligned}$
::您无法切换分子和分母。 m\\\ change in xchange in y

Do not use the phrase "no slope" to describe either zero slope or undefined slope . It is vague and zero and undefined are the mathematically accurate ways to describe these cases.
::不使用“ 无斜坡” 来描述零斜坡或未定义的斜坡。它模糊、零和未定义是描述这些情况的数学精确方法。

Classifying the Slope
::划分斜坡的分类

In general, we can see some trends in the signs of the slope.
::总的来说,我们可以看到斜坡迹象的某些趋势。

Recognizing the Slope of a Line Graphically
::图形化地承认线的曲线

As we read the graph of lines from left to right,
::当我们读到从左到右的线条图时,

Positive slope means the graph rises or goes uphill.
::阳性斜坡意味着图形上升或上升。

Negative slope means the graph falls or goes downhill.
::负斜坡意味着图形下降或下坡。

Zero slope means the line is flat or horizontal.
::零斜度表示线是平的或水平的。

Undefined slope means the line is vertical.
::未定义的斜坡表示线是垂直的。

by CK-12 demonstrates how to classify slope.
::通过 CK-12 演示如何对斜坡进行分类。

Feature: Icy Climb
::特点:攀爬冰河

by Jen Kershaw
::由Jen Kershaw著

Imagine that you are about to climb up an ice face. You have attached your harness, your ropes, and your crampons. Your ice picks are in your hands. You are ready to go, but you are not sure which climbing technique will work best. How can you decide? Well, it all has to do with slope.
::想象一下,你将要爬上冰面。你已绑上你的绳子、绳子和抽筋。你的冰锥在你的手中。你已准备好走,但你不确定哪一种攀爬技术最有效。你如何决定?这都与斜坡有关。

"Slope" refers to steepness of a line, hill, mountain, or ramp. In the sport of ice climbing, with climbing surfaces ranging from glaciers to frozen waterfalls, it is the slope of the ice that will determine which climbing method is your best choice.
::“斜坡”是指一条线、山坡、山坡或坡道的陡峭性。在攀冰运动中,从冰川到冰冻瀑布的攀岩面,正是冰的斜坡决定了哪种攀岩方法才是你的最佳选择。

For ice faces with small slopes (that are not steep), you can use a method called "flat footing." When flat footing, you widen your stance and walk like a duck. Your crampons will maintain your traction as they dig into the ice. For larger slopes, you will have to adjust your technique. You might try "side stepping," a method that involves crossing one foot over the other as you climb. The tricky part is that you can easily catch one foot on the other if you are not careful.
::对于有小斜坡的冰面,你可以使用一种叫做“斜坡基点”的方法。在平坦基点上,你可以扩大你的姿势,像鸭子一样行走。在他们挖冰时,你的抽筋将保持你的牵引力。对于较大的斜坡,你必须调整你的技巧。你可以尝试“靠边踏脚”的方法,这个方法涉及在爬上时跨过一只脚。棘手的部分是,如果你不小心,你可以轻而易举地抓住一只脚。

For most vertical climbs with very large slopes (or vertical climbs with an undefined slope), like the one pictured above, you can use "front pointing." With this technique, you kick your crampons into the ice, toe first, and then step up. Equipped with your climbing gear and your knowledge of slope, you can make your ascent in the safest and most effective way possible.
::大部分垂直攀登的斜坡( 或斜坡未定的垂直攀登) , 就像上面所描绘的一样, 您可以使用“ 前指 ” ( 前指 ) 。使用这种技术, 您可以先把抽筋踢入冰中, 先脚脚, 然后再往上爬。用你的攀爬装备和对斜坡的了解, 您可以以最安全、最有效的方式提高升幅。

by MAMMUT shows a timed climb up a vertical cliff.
::MOMMUT显示一个时间攀登到垂直悬崖上。

Summary
::摘要

Slope can be determined either by finding $\begin{aligned} m = \frac{r i s e}{r u n} \end{aligned}$ .
::斜坡可以通过找到 m=risrun 来决定。

Or, for $\begin{aligned} (x_{1}, y_{1}) \end{aligned}$ and $\begin{aligned} (x_{2}, y_{2}) \end{aligned}$ by using the formula $\begin{aligned} m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \end{aligned}$ .
::或者(x1,y1)和(x2,y2)使用公式m=y2-y1x2-x1。

Review
::回顾

Find the slope of each line.
::找到每一条线的斜坡。

1.

2.

3.

4.

5.

Find the slope of the line that contains each pair of points.
::查找包含每一对点的线条斜坡。

6. (-5, 6) and (-3, 0)
::6.(-5,6)和(-3,0)

7. (1, -1) and (6, -1)
::7.(1,-1)和(6,-1)

8. (3, 2) and (-9, -2)
::8. (3,2)和(9,2)

9. (8, -4) and (8, 1)
::9.(8,4)和(8,1)

10. (4, -15) and (-6, -11)
::10. (4,15)和(6,11)

Explore More
::探索更多

1. The slope between two points $\begin{aligned} (a, b) \end{aligned}$ and (1, -2) is $\begin{aligned} \frac{1}{2} \end{aligned}$ . Find $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ .
::1. 两个点(a、b)和(1、2)之间的斜坡为12。

2. The entrance to a business is 4 feet above the ground. The business plans to build an accessibility ramp. If the law says the slope of an accessibility ramp cannot exceed 8.33%, can the business build the ramp in the direction of the sidewalk if the distance between the sidewalk and the door is 30 feet? ¹
::2. 企业入口处离地面4英尺,商业计划建造一个无障碍坡道,如果法律规定无障碍坡道的斜坡不能超过8.33%,如果人行道和门之间的距离是30英尺,企业能否在人行道上修建坡道? 1

3. Sledding hills can be dangerous. It is recommended that the slope of the hill not exceed $\begin{aligned} \frac{3}{10} \end{aligned}$ . ² The hill in your local park is 40 feet high over a distance of 150 feet. Is it safe for sledding?
::3. 滑雪山可能很危险,建议山坡的坡度不超过310米。 2 你们当地公园的山高40英尺,距离150英尺,滑雪安全吗?

4. Casey is trying new things on her adventure vacation and is excited about riding a zipline for the first time. The beginning point is 20 meters higher than the end point, and the length of the zipline is 220 meters. What is the slope of the line?
::4. Casey在冒险度假时正在尝试新的东西,首次对乘坐拉链感到兴奋,起点比终点高20米,拉链长度为220米。

5. Alex works at a job where he makes a base pay of $250 per week and an additional amount for each sale. If Alex makes $385 in one week for 3 sales and $475 in another week for 5 sales, how much does Alex make per sale?
::5. 亚历克斯在一份工作上工作,每周基本工资为250美元,每份销售增加一笔金额,如果亚历克斯每份销售在一周内3份销售385美元,另外5份销售在一周内475美元,那么亚历克斯每份销售能赚多少?

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

References
::参考参考资料

“ Chapter 4: Ramps and Curb Ramps - United States Access Board,” .
::“第4章:Ramps和Curb Ramps——美国接入委员会”,” 。

“Sledding Hills: Managing the Risk - A Service of the Michigan Municipal League Liability and Property Pool and the Michigan Municipal League Workers’ Compensation Fund,” revised April 24, 2009, .
::2009年4月24日修订, “雪橇山:管理风险——密歇根市联盟责任和财产联营处和密歇根市联盟工人赔偿基金服务处”,2009年4月24日修订。

Section outline

Slope of a Line: Rise Over Run ::线条的斜体: 上升运行

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Slope of a Line: The Formula ::线条的斜面:公式

Slope Formula ::斜曲公式

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Example 7 ::例7

WARNING ::警告

Classifying the Slope ::划分斜坡的分类

Recognizing the Slope of a Line Graphically ::图形化地承认线的曲线

Feature: Icy Climb ::特点:攀爬冰河

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

References ::参考参考资料

Slope of a Line: Rise Over Run
::线条的斜体: 上升运行

Example 1
::例1

Example 2
::例2

Example 3
::例3

Slope of a Line: The Formula
::线条的斜面:公式

Slope Formula

::斜曲公式

Example 4
::例4

Example 5
::例5

Example 6
::例6

Example 7
::例7

WARNING
::警告

Classifying the Slope
::划分斜坡的分类

Recognizing the Slope of a Line Graphically
::图形化地承认线的曲线

Feature: Icy Climb
::特点:攀爬冰河

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX

References
::参考参考资料