3.3线性模型

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  线性模型

  You  run  a business and are analyzing how your production costs are growing. You can only afford for the costs to grow at a certain  rate . 

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  This is a common situation in many fields. You analyze a model and want to know how it is changing. In this section, we consider models that change at a linear rate. 
  ::这是许多领域常见的情况。 您分析一个模型并想知道它是如何变化的。 在本节中, 我们考虑以线性速度变化的模型 。

  

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  Recognize Linear Models
  ::认列线性模型

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  In this chapter, we will discuss linear equations and inequalities, or data that can be modeled using a line. So what makes a set of data linear in the first place?
  ::在本章中,我们将讨论线性方程和不平等问题,或者可以使用线性模型进行模拟的数据。 那么,首先哪一组数据是线性数据呢?

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  We can recognize linear models by how the input or  independent   variable , usually  $\begin{array}{r}x\end{array}$ , and output or  dependent  variable, often  $\begin{array}{r}y\end{array}$ , values change with respect to each other. By change, we mean what is the  distance  between known values. The quantity we use to compare changes is known as the  rate of change . A  rate  is a type of  ratio  or  fraction  in which you are comparing two quantities often with different units. We see a lot of these in our daily life.
  ::我们可以通过输入或独立的变量(通常是x)和输出或依附变量(往往是y)来识别线性模型。 通过变化,我们指已知值之间的距离。我们用来比较变化的数量被称为变化率。一个比率或分数类型,其中您将两个数量和不同的单位进行比较。我们在日常生活中看到很多这样的数据。

  $$\begin{array}{rlrlr}60\text{mph}& & \mathrm{\$}3.99\text{per lb.}& & 6\text{weather updates per hour}\end{array}$$

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    Rate of Change
  ::变化率变化率

   

  $$\begin{array}{r}\text{rate of change}=\frac{\text{difference in dependent variable values}}{\text{difference in independent variable values}}\end{array}$$

   

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  In the following sections, rate of change will be called   and we will find the ratio of the change in  y- values to the change in  x- values. For linear models, the rate of change is  constant , that is, does not change, regardless of which data points you choose to consider. Linear relationships have a constant rate of change. If the rate of change is not constant, the relationship is not linear.
  
  ::在以下各节中,将调用变化率,我们会发现y值变化与x值变化的比率。对于线性模型,变化率是不变的,也就是说,不改变,不管你选择考虑哪个数据点。线性关系有一个不变的变动率。如果变化率不是不变的,则关系不是线性。

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  For the moment, we will consider data without units. 
  
  ::目前,我们将考虑没有单位的数据。

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  Example 1
  ::例1

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  Determine whether the  relation  in the table below is linear or non-linear.
  ::确定下表中的关系是线性还是非线性。

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  0	3
  1	6
  2	9
  3	12

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  Solution:  The  x- values are going up by one unit at a time, while the  y- values are going up by 3 units at a time.  Therefore , the slope is:  $\begin{array}{r}\frac{3}{1}=3\end{array}$ . Since there is a constant rate of change, the relation in the table is linear. 
  ::解析度 : x 值一次上升一个单位, Y 值一次上升三个单位。 因此, 斜度是 3 1 = 3。 由于变化率不变, 表中的关系是线性 。

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  Example 2
  ::例2

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  Determine whether the relation in the table below is linear or non-linear.
  ::确定下表中的关系是线性还是非线性。

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  -1	10
  0	7
  1	4
  2	1

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  Solution:  The  x- values go up by 1 unit and the  y- values go down by 3 units.  Therefore, the slope is:  $\begin{array}{r}\frac{-3}{1}=-3\end{array}$ . Since there is a constant rate of change, the relation in the table is linear.  
  ::溶解度:x值上升1个单位,Y值下降3个单位。因此,斜度是:-3 1=-3。由于变化率不变,表中的关系是线性的。

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   by Mathispower4u demonstrates how to determine which tables represent a  linear function  or linear relationship. 
  ::通过 Mathispower4u 演示如何确定哪些表格代表线性函数或线性关系。

   

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  Example 3
  ::例3

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  Determine whether the relation in the table below is linear or non-linear.
  ::确定下表中的关系是线性还是非线性。

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  -2	5
  0	5
  2	5
  4	5

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  Solution:  The  x- values are going up by 2 units at a time. On the other hand, the  y- values are not going up or down, that is, the change in the  y- values is 0 units.  Therefore, the slope is:  $\begin{array}{r}\frac{0}{2}=0\end{array}$ . Since there is a constant rate of change, the relation in the table is linear.
  ::解析度 : x 值一次上升2 个单位。 另一方面, Y 值没有上下,也就是说, Y 值的变化是0 个单位。因此,斜度是: 0 2 = 0。由于变化率不变,表中的关系是线性的。

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  Example 4
  ::例4

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  Determine whether the relation in the table below is linear or non-linear.
  ::确定下表中的关系是线性还是非线性。

  Number of Products,  $\begin{array}{r}x\end{array}$	Costs,  $\begin{array}{r}y\end{array}$
  0	3,200
  1	3,255
  10	3,750
  100	8,700

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  Solution:  Here we are comparing the number of products produced to the costs of a business. Since the  x- values do not go up by the same amount, we need to consider the ratios between individual pairs of points. The  difference  between 1 and 0 is 1 and the difference between their respective  y- values, 3,255 and 3,200, is 55. Therefore, the slope between these two points is:  $\begin{array}{r}\frac{55}{1}=55\end{array}$ . 
  ::解决方案: 我们在这里比较生产的产品数量和企业成本。 由于 x 值不是以相同数量上升, 我们需要考虑各个点对比。 1 和 0 之间的差为 1 , 而它们各自的 y 值( 3, 255 和 3, 200) 之间的差为 55 。 因此, 这两个点之间的斜度是 55 1 = 55 。

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  Let's consider another pair of points. The difference between 10 and 1 is 9 and the difference between their respective  y- values, 3,750 and 3,255, is 495. Therefore, the slope between these two points is:  $\begin{array}{r}\frac{55}{1}=55\end{array}$ .  
  ::让我们再考虑一对点。 10和1之间的差值是 9, 各自的 y 值( 3, 750 和 3, 255) 之间的差值是 495 。 因此, 这两个点之间的斜度是 55 1 = 55 。

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  Another pair of values gives the difference between 100 and 10 is 90 and the difference between their respective  y- values, 8,700 and 3,750 is 4,950. Therefore, the slope between these two points is:  $\begin{array}{r}\frac{55}{1}=55\end{array}$ .  
  ::另一对数值显示100和10之间的差值是90,而它们各自的Y值,8,700和3,750之间的差值是4,950。因此,这两个点之间的斜度是:551=55。

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  Note, we can choose any pair of points in the table, that is, they do not have to be  consecutive . Since there is a constant rate of change, the relation is linear.
  ::注意, 我们可以在表格中选择任何一对点, 也就是说, 它们不必是连续的 。 由于变化率是不变的, 关系是线性 。

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   by widmath demonstrates how to recognize linear relationships. 
  ::通过 Widmath 演示如何识别线性关系。

  
   

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  Example 5 
  ::例5

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  Determine whether the relation in the table below is linear or non-linear.
  ::确定下表中的关系是线性还是非线性。

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  0	2
  1	3
  2	5
  3	8

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  Solution:  As the  x- values go up by 1, the  y- values do not increase by the same amount. First, the  y- values increase by 1, then 2, then 3. Since the change in the  y- values is not  consistent , this relation is non-linear.
  ::解答: 当 x 值上升 1 时, Y 值不会增加相同数额。 首先, Y 值增加 1 , 然后 2 , 然后 3 。 由于 y 值的变化不一致, 此关系是非线性 。

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  Feature: Cruise Control
  ::特点:巡航控制

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  by Deirdre Mundy
  ::由Deirdre Mundy 编辑

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  Have you ever been on a long car trip? If you know how fast you are traveling and how far you have to go, you can predict how long it will take you to get there. That is because your  speed , the distance you travel, and the time it takes to get there can be part of a linear relationship.
  ::你去过一个漫长的汽车旅行吗?如果你知道你旅行的速度有多快,要走多远,你可以预测你到那里需要多长时间。这是因为你的速度,你旅行的距离,以及到达线性关系所需的时间。

  

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  If you drive at a constant speed, you can accurately calculate how far you have traveled. Things can get complicated if your car changes speed a lot. Changing speeds frequently can waste fuel and can lead to accidents.
  ::如果您以恒定速度驾驶, 您可以准确计算您所走过的路程。 如果您的汽车变速很多, 事情可能会变得复杂。 改变速度会浪费燃料, 并可能导致事故 。

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  To make long drives safer and more efficient, many cars have cruise control systems that keep the car at a constant speed. Once activated, a computer measures the car's speed and automatically controls the engine. It helps the car travel at a constant rate, even on hills. This works well on the open highway, but you cannot use normal cruise control in stop-and-go traffic.
  ::为了让长途驾驶更安全、更有效率,许多汽车拥有使汽车保持固定速度的巡航控制系统。一旦启动,计算机将测量汽车的速度并自动控制发动机。它帮助汽车以恒定速度行驶,甚至是在山上行驶。这在开通的高速公路上运作良好,但您无法使用正常的巡航控制进行中途停留。

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  Newer systems also use radar. These cruise controls not only measure speed but also the distance to the car in front of yours. Thus, if the car ahead slows down, the system can slow your car down as well. New advances in cruise control may help make self-driving cars possible.
  ::新系统也使用雷达。 这些巡航控制不仅测量速度,而且测量你前面汽车的距离。 因此,如果汽车向前减速,系统也可以减速。 巡航控制方面的新进展可能有助于汽车自行驾驶。

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   by Bosch explains the adaptive distance and speed control system that assists the driver in keeping a safe distance from the vehicle in front of them. 
  ::Bosch解释, 路程和速度控制系统能帮助司机在车前与车辆保持安全距离。

    

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  Summary
  ::摘要

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  • Determine if the slope, or the rate of change is constant. 
    ::确定斜坡或变化率是不变的。
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  • If the rate of change is constant, the relation is linear.
    ::如果变化率不变,则关系为线性。

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  Review
  ::回顾

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  Determine whether the relation in the table is linear or non-linear.
  ::确定表格中的关系是线性还是非线性。

  1.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  0	2
  1	3
  2	5
  4	4

  2.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  1	3
  2	5
  3	7
  4	9

  3.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  2	6
  3	9
  5	15
  6	18

  4.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  2	3
  3	4
  6	7
  8	9

  5.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  8	4
  6	12
  2	8
  0	0

  6.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  0	3
  1	4
  2	5
  6	9

  7.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  5	11
  4	9
  3	7
  2	5

  8.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  1	7
  3	4
  2	9
  5	8

  9.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  1	3
  2	6
  4	12
  6	18

  10.

  $\begin{array}{r}x\end{array}$	$\begin{array}{r}y\end{array}$
  4	2
  5	3
  6	5
  7	1

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  Explore More
  ::探索更多

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  1. Dennis, the scorekeeper for the men's basketball team, has been keeping track of the team’s scoring during the weekend tournament. During the first two games, Dennis tracked the team’s shooting performance from different distances from the basket. He found that the team made approximately 50 percent of their shots when they were less than 10 feet from the basket. He also recorded their shooting averages from within 15, 20, and 25 feet of the basket. He displayed his results in the table below.
  ::1. Dennis是男子篮球队的记分员,在周末的锦标赛中一直跟踪球队的得分,在前两场比赛中,丹尼斯从篮子的不同距离跟踪球队的射击表演,发现球队在距篮子不到10英尺时大约拍了50%的球,还记录了篮子15、20和25英尺以内的射击平均数,结果显示在下表。

  播放段落 Distance from the basket,  $\begin{array}{r}d\end{array}$ ::距离篮篮子的距离,d	播放段落 Percent of shots made,  $\begin{array}{r}f\end{array}$ ::射击次数占拍摄射击次数的百分比,f
  10	50
  15	40
  20	30
  25	20

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  Is the data linear or non-linear?
  ::数据是线性数据还是非线性数据?

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  2. You have a savings account with $237.46. You save $30.00 each month. Is the data from this model linear or non-linear? (Hint: Create a table of 3 or 4 values comparing months and money in the account. )
  ::2. 您的储蓄账户为237.46美元。 您每月节省30.00美元。 数据来自这个模型线性数据还是非线性数据? (提示: 创建一个3或4美元的表格, 比较月数和账户中的钱数。 )

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  3. Sam starts a weight loss program where he loses 2 pounds per week. Is this model linear or non-linear? (Hint: Create a table of 3 or 4 values comparing weeks and weight lost.) 
  ::3. Sam开始了一个体重下降方案,他每周损失2磅,这是线性模型还是非线性模型? (提示:建立一个3或4的表格,比较周数和重量损失。 )

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  4. Tomatoes cost $1.59 per pound. Is the model of pounds of tomatoes versus cost linear or non-linear? Does it change if there is a bulk discount of $1.29 per pound for 10 pounds or more? (Hint: Create a table of 3 or 4 values comparing pounds of tomatoes and cost.)
  ::4. 番茄成本为每磅1.59美元,西红柿磅相对于线性或非线性成本模式是?如果10磅或10磅以上,每磅1.29美元的大宗折扣是每磅1.29美元,这种模式是否有变化? (提示:将西红柿和成本的磅数与3美元或4美元比较,用表格列出3美元或4美元。 )

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  5. Mariah is on a road trip, driving at 60 miles per hour. She has activated cruise control, and it is 200 miles to her next destination. How long will it take her to reach the halfway point of her trip? 
  ::5. Mariah正进行公路旅行,每小时60英里的驾驶,她已启动巡航控制,离下一个目的地还有200英里,要多久才能到达行程的中点?