Course: 4.2 坐标平面图 | The Way To Learn

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坐标平面中的图表

Graphs in the Coordinate Plane
::坐标平面中的图表

Once we know how to plot points on a coordinate plane , we can think about how we’d go about plotting a relationship between $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ values. Previously, you may have plotted sets of ordered pairs. A set like that is a relation , and there isn’t necessarily a consistent relationship between the $\begin{aligned} x - \end{aligned}$ values and $\begin{aligned} y - \end{aligned}$ values. If there is a relationship between the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ values, and each $\begin{aligned} x - \end{aligned}$ value corresponds to exactly one $\begin{aligned} y - \end{aligned}$ value, then the relation is called a function . Remember that a function is a particular way to relate one quantity to another.
::一旦我们知道如何在坐标平面上绘制点, 我们就可以思考如何在 x- 和 y- values 之间绘制关系。此前, 您可能已经绘制了一组有顺序的配对。类似这样的组合是一种关系, 而在 x- values 和 y- values 之间不一定存在一致的关系。如果 x- y- value 和 y- value 之间存在某种关系, 而每一个 x- value 完全对应一个 y- value, 那么关系就被称为函数。记住一个函数是将一个数量连接到另一个数量的特殊方式。

Graphing a Function
::构造函数

If you’re reading a book and can read twenty pages an hour, there is a relationship between how many hours you read and how many pages you read. You may even know that you could write the formula as either $\begin{aligned} n = 20 h \end{aligned}$ or $\begin{aligned} h = \frac{n}{20} \end{aligned}$ , where $\begin{aligned} h \end{aligned}$ is the number of hours you spend reading and $\begin{aligned} n \end{aligned}$ is the number of pages you read. To find out, for example, how many pages you could read in $\begin{aligned} 3 \frac{1}{2} \end{aligned}$ hours, or how many hours it would take you to read 46 pages, you could use one of those formulas. Or, you could make a graph of the function:
::如果您正在阅读一本书, 并且每小时可以读20页, 您读了多少小时和读了多少页之间就存在某种关系。您甚至可以知道您可以以 n=20h 或 h=n20 来写公式, 其中 h 是您所花的读数, n 是您所读数。例如, 要了解您在 312 小时里可以读多少页, 或者您读了46页需要多少小时, 您可以使用其中一种公式。或者, 您可以绘制一个函数的图表 :

Once you know how to graph a function like this, you can simply read the relationship between the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ values off the graph. You can see in this case that you could read 70 pages in $\begin{aligned} 3 \frac{1}{2} \end{aligned}$ hours, and it would take you about $\begin{aligned} 2 \frac{1}{3} \end{aligned}$ hours to read 46 pages.
::一旦您知道如何用图解这样的函数, 您可以简单地从图中读取 x - 和 y - 值之间的关系。您可以在此看到您可以在 312 小时内读取 70 页, 您需要213 小时才能读取 46 页。

Generally, the graph of a function appears as a line or curve that goes through all points that have the relationship that the function describes. If the domain of the function (the set of $\begin{aligned} x - \end{aligned}$ values we can plug into the function) is all real numbers, then we call it a continuous function . If the domain of the function is a particular set of values (such as whole numbers only), then it is called a discrete function . The graph will be a series of dots, but they will still often fall along a line or curve.
::一般而言,函数的图形显示为直线或曲线,通过函数描述的所有点。如果函数的域( x- 值组,我们可以插入函数)全部是真实数字,那么我们称之为连续函数。如果函数的域是一个特定值组(仅包括整数),那么它就被称为离散函数。图形将是一系列点,但通常仍会沿着线或曲线进行。

In graphing equations, we assume the domain is all real numbers, unless otherwise stated. Often, though, when we look at data in a table, the domain will be whole numbers (number of presents, number of days, etc.) and the function will be discrete. But sometimes we’ll still draw the graph as a continuous line to make it easier to interpret. Be aware of the difference between discrete and continuous functions as you work through the examples.
::在图形化方程式中,我们假设域名都是真实数字,除非另有说明。但是,通常情况下,当我们在表格中查看数据时,域名将是整数(显示数量、天数等 ) , 函数是分开的。但是,有时我们仍会用连续线绘制该图,以方便解释。当您通过示例工作时,要注意离散函数和连续函数之间的区别。

Using Graphs to Solve Real-World Problems
::使用图表解决现实世界问题

Sarah is thinking of the number of presents she receives as a function of the number of friends who come to her birthday party. She knows she will get a present from her parents, one from her grandparents and one each from her uncle and aunt. She wants to invite up to ten of her friends, who will each bring one present. She makes a table of how many presents she will get if one, two, three, four or five friends come to the party. Plot the points on a coordinate plane and graph the function that links the number of presents with the number of friends. Use your graph to determine how many presents she would get if eight friends show up.
::Sarah正在考虑她收到的礼物数量,以参加生日晚会的朋友人数计算。她知道她会从父母那里得到一份礼物,从祖父母那里得到一份,从叔叔和阿姨那里各得到一份。她想邀请多达10位朋友,每人带一份礼物。她准备一张表格,列出如果一个、两个、三个、四个或五个朋友来参加晚会,她会收到多少礼物。在协调平面上标出点数,并绘制将礼物数量与朋友人数挂钩的功能。使用你的图表确定如果八个朋友出现,她会收到多少礼物。

Number of Friends	Number of Presents
0	4
1	5
2	6
3	7
4	8
5	9

The first thing we need to do is decide how our graph should appear. We need to decide what the independent variable is, and what the dependent variable is. Clearly in this case, the number of friends can vary independently, but the number of presents must depend on the number of friends who show up.
::我们需要做的第一件事就是决定我们的图表应该如何显示。我们需要决定独立变量是什么,依赖变量是什么。显然,在这种情况下,朋友的数量可以独立变化,但礼物的数量必须取决于出现朋友的数量。

So we’ll plot friends on the $\begin{aligned} x - \end{aligned}$ axis and presents on the $\begin{aligned} y - \end{aligned}$ axis. Let's add another column to our table containing the coordinates that each (friends, presents) ordered pair gives us.
::因此,我们将在x-轴线上勾画朋友,在y-轴线上展示。让我们在表格中添加另一列,包含每对(朋友、赠送品)订购的一对给予我们的坐标。

Friends $\begin{aligned} (x) \end{aligned}$	Presents "> $\begin{aligned} (y) \end{aligned}$	Coordinates $\begin{aligned} (x, y) \end{aligned}$
0	4	(0, 4)
1	5	(1, 5)
2	6	(2, 6)
3	7	(3, 7)
4	8	(4, 8)
5	9	(5, 9)

Next we need to set up our axes. It is clear that the number of friends and number of presents both must be positive, so we only need to show points in Quadrant I. Now we need to choose a suitable scale for the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ axes. We only need to consider eight friends (look again at the question to confirm this), but it always pays to allow a little extra room on your graph. We also need the $\begin{aligned} y - \end{aligned}$ scale to accommodate the presents for eight people. We can see that this is still going to be under 20!
::接下来我们需要设置轴心。显然, 朋友的数量和礼物的数量都必须是正数, 所以我们只需要在 Quadrant I 中显示分数。现在我们需要选择一个适合 x - 和 y - 轴的尺度。我们只需要考虑八个朋友( 再看一个问题来证实这一点 ) , 但它总是要花钱在您的图表上留出一点额外的空间。我们还需要 Y - 比例来容纳八个人的礼物。我们可以看到这仍然要低于 20 个 !

The scale of this graph has room for up to 12 friends and 15 presents. This will be fine, but there are many other scales that would be equally good!
::本图的缩放为最多12个朋友和15个礼物。这将很好, 但还有很多其他的缩放也不错! !

Now we proceed to plot the points. The first five points are the coordinates from our table. You can see they all lie on a straight line, so the function that describes the relationship between $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ will be linear. To graph the function, we simply draw a line that goes through all five points. This line represents the function.
::现在我们开始绘制点。前面的五个点是我们表格的坐标。您可以看到它们都位于一条直线上, 因此描述 x 和 y 之间关系的函数将是线性。要绘制函数, 我们只需绘制一条贯穿所有五个点的线。此线代表函数。

This is a discrete problem since Sarah can only invite a positive whole number of friends. For instance, it would be impossible for 2.4 or -3 friends to show up. So although the line helps us see where the other values of the function are, the only points on the line that actually are values of the function are the ones with positive whole-number coordinates.
::这是一个互不关联的问题, 因为 Sarah 只能邀请一个正数的朋友。例如, 2.4 或 - 3 的朋友不可能出现。因此, 虽然线条帮助我们看到函数的其他值在哪里, 但线条上唯一真正代表函数值的点是带有正数整数坐标的点。

The graph easily lets us find other values for the function. For example, the question asks how many presents Sarah would get if eight friends come to her party. Don't forget that $\begin{aligned} x \end{aligned}$ represents the number of friends and $\begin{aligned} y \end{aligned}$ represents the number of presents. If we look at the graph where $\begin{aligned} x = 8 \end{aligned}$ , we can see that the function has a $\begin{aligned} y - \end{aligned}$ value of 12.
::图形让我们很容易地找到函数的其他值。例如, 问题询问如果八个朋友来她的聚会, Sarah会收到多少礼物。不要忘记 x 代表朋友的数量, y 代表礼物的数量。如果我们查看 x=8 的图表, 我们可以看到该函数的 Y - 值为 12 。

If 8 friends show up, Sarah will receive a total of 12 presents.
::如果8个朋友出现 Sarah将总共收到12件礼物

Graphing a Function Given a Rule
::根据规则绘制函数图

If we are given a rule instead of a table, we can proceed to graph the function in either of two ways. We will use the following example to show each way.
::如果给了我们一条规则而不是一张表格,我们就可以用两种方式中的两种方式来绘制函数图。我们将用下面的例子来显示每一种方式。

Ali is trying to work out a trick that his friend showed him. His friend started by asking him to think of a number, then double it, then add five to the result. Ali has written down a rule to describe the first part of the trick. He is using the letter $\begin{aligned} x \end{aligned}$ to stand for the number he thought of and the letter $\begin{aligned} y \end{aligned}$ to represent the final result of applying the rule. He wrote his rule in the form of an equation : $\begin{aligned} y = 2 x + 5. \end{aligned}$
::Ali试图找出他朋友给他的诡计。他的朋友首先要求他想一个数字,然后翻一番,然后在结果中增加五个。Ali写了一条规则来描述这个诡计的第一部分。他用字母x来表示他所想的数字,用字母y来表示应用规则的最后结果。他以公式的形式写了他的规则:y=2x+5。

Help him visualize what is going on by graphing the function that this rule describes.
::帮助他通过绘制此规则所描述的函数的图形来想象正在发生的事情。

Method One - Construct a Table of Values
::方法一 - 构建数值表

If we wish to plot a few points to see what is going on with this function, then the best way is to construct a table and populate it with a few $\begin{aligned} (x, y) \end{aligned}$ pairs. We’ll use 0, 1, 2 and 3 for $\begin{aligned} x - \end{aligned}$ values.
::如果我们想勾画几个要点来观察这个函数发生什么, 那么最好的方法是构建一个表格, 并用几对( X,y) 配对来填充它。我们将使用 0, 1, 2 和 3 来计算 x- value 。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y \end{aligned}$
0	5
1	7
2	9
3	11

Next, we plot the points and join them with a line.
::接下来,我们绘制点数,并排成一条线加入它们。

This method is nice and simple—especially with linear relationships, where we don’t need to plot more than two or three points to see the shape of the graph. In this case, the function is continuous because the domain is all real numbers—that is, Ali could think of any real number , even though he may only be thinking of positive whole numbers.
::这种方法既好又简单 — — 特别是在线性关系中,我们不需要绘制超过两三点的图形来查看图形的形状。在这种情况下,函数是连续的,因为域名都是真实数字 — — 也就是说,Ali可以想到任何真实数字,尽管他可能只想到正数整数。

Method Two - Intercept and
::方法二 - 拦截和

Another way to graph this function (one that we’ll learn in more detail in a later lesson) is the slope- intercept method . To use this method, follow these steps:
::绘制此函数(我们将在以后的教训中更详细地学习这一函数)的另一种方式是斜坡拦截法。要使用这种方法,请遵循以下步骤:

1. Find the $\begin{aligned} y \end{aligned}$ value of $\begin{aligned} y = 2 x + 5 \end{aligned}$ when $\begin{aligned} x = 0. \end{aligned}$
::1. 在 x=0 时查找y=2x+5的y值。

$\begin{aligned} y (0) = 2 \cdot 0 + 5 = 5, \end{aligned}$ so the $\begin{aligned} y - \end{aligned}$ intercept is (0, 5).
::y( 0) = 20+5=5, 所以y- interview is (0, 5) 。

2. Look at the coefficient multiplying the $\begin{aligned} x \end{aligned}$ .
::2. 看看乘以x的系数。

Every time we increase $\begin{aligned} x \end{aligned}$ by one, $\begin{aligned} y \end{aligned}$ increases by two, so our slope is +2.
::每次我们增加x乘以1,y 增加2,所以我们的斜坡是+2。

3. Plot the line with the given slope that goes through the intercept. We start at the point (0, 5) and move over one in the $\begin{aligned} x - \end{aligned}$ direction, then up two in the $\begin{aligned} y - \end{aligned}$ direction. This gives the slope for our line, which we extend in both directions.
::3. 将线与通过拦截的给定斜坡绘制成平面,从点(0,5)开始,在X-方向上移动一个,然后在Y-方向上移两个,这给我们的线提供了斜坡,我们向两个方向延伸。

We will properly examine this last method later in this chapter!
::我们将适当审查本章稍后部分的最后一种方法。

Examples
::实例

Example 1
::例1

The point (0, -2) is the boundary of which two quadrants ?
::点 (0, 2) 是两个四分位数的边界 ?

Since the x-value is 0, the point is on the y-axis . Since the y-value is negative, the point is on the lower half of the y-axis. This is the boundary between the 3rd and 4th quadrants.
::由于 x 值为 0, 点在 y 轴上。由于 y 值为负, 点在 y 轴下半。这是 3 和 4 之四之间的边界。

Example 2
::例2

If you move the point (-3,4) down 5, what quadrant would it be in?
::如果您将点( - 3, 4) 向下移 5 , 它会在哪个象限内 ?

Moving the point down 5 is equivalent to subtracting 5 from the y-value. $\begin{aligned} (- 3, 4 - 5) = (- 3, - 1) \end{aligned}$ . Since both coordinates are now negative, this is in the 3rd quadrant.
::向下移动点 5 等于从 Y 值中减去 5 。 (- 3-4-5) = (-3) = (- 3)-1 。由于两个坐标目前均为负,所以在第三个象限中。

Review
::回顾

Consider the graph of the equation $\begin{aligned} y = 3 \end{aligned}$ . Which quadrants does it pass through?
::考虑一下 y=3. 方程的图。它通过哪个四分位数 ?

Consider the graph of the equation $\begin{aligned} y = x \end{aligned}$ . Which quadrants does it pass through?
::考虑一下 y=x 的方程图。它通过哪个四分位数 ?

Consider the graph of the equation $\begin{aligned} y = x + 3 \end{aligned}$ . Which quadrants does it pass through?
::考虑一下 y=x+3 的方程图。它通过哪个方位 ?

The point (4, 0) is on the boundary between which two quadrants?
::点(4,0)在两个四分位数之间的边界上?

The point (0, -5) is on the boundary between which two quadrants?
::点(0, 5)在两个四分位数之间的边界上?

If you moved the point (3, 2) five units to the left, what quadrant would it be in?
::如果你把点(3,2)移到左边5个单位, 它会进入什么象限?

The following three points are three vertices of square $\begin{aligned} A B C D \end{aligned}$ . Plot them on a graph, then determine what the coordinates of the fourth point, $\begin{aligned} D \end{aligned}$ , would be. Plot that point and label it.
$\begin{aligned} A (- 4, - 4) B (3, - 4) C (3, 3) \end{aligned}$

::以下三点是 ABCD 方形的三个顶点。将其绘制在图表上, 然后确定第四点的坐标, D。绘制该点并贴上标签。 A( 4) B( 3) - 4) C( 3) C( 3) 3 ) 。

In what quadrant is the center of the square from problem 10? (You can find the center by drawing the square’s diagonals.)
::问题 10 中方方块的中心是什么? (您可以通过绘制方块的对角体找到中心 ) 。

What point is halfway between (1, 3) and (1, 5)?
::在(1,3)和(1,5)之间的中间点是什么?

What point is halfway between (2, 8) and (6, 8)?
::在(2、8)和(6、8)之间的中间点是多少?

What point is halfway between the origin and (10, 4)?
::从起源到(10,4)的中间点是多少?

What point is halfway between (3, -2) and (-3, 2)?
::在(3,2)和(3,3,2)之间的中间点是多少?

Becky has a large bag of M&Ms that she knows she should share with Jaeyun. Jaeyun has a packet of Starburst. Becky tells Jaeyun that for every Starburst he gives her, she will give him three M&Ms in return. If $\begin{aligned} x \end{aligned}$ is the number of Starburst that Jaeyun gives Becky, and $\begin{aligned} y \end{aligned}$ is the number of M&Ms he gets in return, then complete each of the following.
1. Write an algebraic rule for $\begin{aligned} y \end{aligned}$ in terms of $\begin{aligned} x \end{aligned}$ .
  ::以 x 为 y 写代数规则。
2. Make a table of values for $\begin{aligned} y \end{aligned}$ with $\begin{aligned} x \end{aligned}$ -values of 0, 1, 2, 3, 4, 5.
  ::以 X 值为 0, 1, 2, 3, 3, 4, 5 的 y 值列表
3. Plot the function linking $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ on the following scale: $\begin{aligned} 0 \leq x \leq 10, 0 \leq y \leq 10 \end{aligned}$ .
  ::在以下比例范围内绘制连接 x 和 y 的函数 : 0x10, 0y10 。
::贝基有一大袋M&Ms, 她知道她应该与Jaeyun分享。 Jaeyun 有一包Starburst 。 Becky告诉Jaeyun, 他每送一颗Starburst, 她就会给他三张M&Ms作为回报。如果x是Jaeyun给Becky的Starburst数量, 并且是他得到的M&M数量, 然后填写以下每张。写一个以 x为单位的代数规则。以 x 为x值, 以 0, 1, 2, 3, 4, 5 绘制一个数值表。将x和y 的函数依以下比例排列 : 0xxxx10, 0 y10 。

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。

Section outline

General

坐标平面中的图表

Graphs in the Coordinate Plane ::坐标平面中的图表

Graphing a Function ::构造函数

Using Graphs to Solve Real-World Problems ::使用图表解决现实世界问题

Graphing a Function Given a Rule ::根据规则绘制函数图

Method One - Construct a Table of Values ::方法一 - 构建数值表

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)