Course: 4.3 线性方形图 | The Way To Learn

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线性方形图

Graphs of Linear Equations
::线性方形图

You’re stranded downtown late at night with only $8 in your pocket, and your home is 6 miles away. Two cab companies serve this area; one charges $1.20 per mile with an additional $1 fee, and the other charges $0.90 per mile with an additional $2.50 fee. Which cab will be able to get you home?
::你深夜被困在市中心,口袋里只有8美元,而你的家离这里只有6英里。两家出租车公司为这个地区服务;一家每英里收费1.20美元,额外收费1美元,另一家每英里收费0.90美元,额外收费2.50美元。哪家出租车能送你回家?

Graph a Linear Equation
::线性方形图a

A rule is a way of writing the relationship between the two quantities we are graphing. In mathematics, we tend to use the words formula and equation to describe the rules we get when we express relationships algebraically. Interpreting and graphing these equations is an important skill that you’ll use frequently in math.
::规则是写出我们正在绘制的这两个数量之间的关系的一种方式。在数学中,我们倾向于用文字公式和方程式来描述我们表达关系代数时得到的规则。解释和图形化这些方程式是数学中经常使用的重要技能。

A taxi costs more the further you travel. Taxis usually charge a fee on top of the per-mile charge to cover hire of the vehicle. In this case, the taxi charges $3 as a set fee and $0.80 per mile traveled. Here is the equation linking the cost in dollars "> $\begin{aligned} (y) \end{aligned}$ to hire a taxi and the distance traveled in miles $\begin{aligned} (x) \end{aligned}$ .
::出租车成本越高。出租车通常在每英里收费之外收费,以支付租车费用。在这种情况下,出租车收费3美元作为固定费用,每英里0.80美元。这是将美元成本与租用出租车成本和里程(x)距离联系起来的方程式。

\begin{aligned} y = 0.8 x + 3 \end{aligned}

::y=0.8x+3 y=0.8x+3

Graph the equation and use your graph to estimate the cost of a seven-mile taxi ride.
::绘制方程图,用图表估算七英里出租车的费用。

We’ll start by making a table of values. We will take a few values for $\begin{aligned} x \end{aligned}$ (0, 1, 2, 3, and 4), find the corresponding $\begin{aligned} y - \end{aligned}$ values, and then plot them. Since the question asks us to find the cost for a seven-mile journey, we need to choose a scale that can accommodate this.
::首先,我们先绘制一个数值表。我们将对 x( 0, 1, 2, 3 和 4 ) 使用几个数值, 找到相应的 y - 值, 然后绘制它们。既然问题要求我们寻找七英里行程的成本, 我们需要选择一个可以容纳这个比例的尺度。

First, here’s our table of values:
::首先,这是我们的数值表:

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y \end{aligned}$
0	3
1	3.8
2	4.6
3	5.4
4	6.2

And here’s our graph:
::以下是我们的图表:

To find the cost of a seven-mile journey, first we find $\begin{aligned} x = 7 \end{aligned}$ on the horizontal axis and draw a line up to our graph. Next, we draw a horizontal line across to the $\begin{aligned} y - \end{aligned}$ axis and read where it hits. It appears to hit around half way between $\begin{aligned} y = 8 \end{aligned}$ and $\begin{aligned} y = 9 \end{aligned}$ . Let's call it 8.5.
::为了找到七英里行程的成本, 首先我们在水平轴上找到 x=7, 然后绘制一条直线到我们的图表。接下来, 我们画一条横线到 y - 轴, 并读读它击中的位置。它似乎在 y=8 和 y=9 之间大约击中了 y=8 和 y=9 之间的一半。我们把它称为 8. 5 。

A seven mile taxi ride would cost approximately $8.50 ($8.60 exactly).
::一辆7英里的出租车将花费约8.50美元(确切数字为8.60美元)。

Here are some things you should notice about this graph and the formula that generated it:
::以下是一些你应该注意的关于这个图表和生成它的公式:

The graph is a straight line (this means that the equation is linear ), although the function is discrete and really just consists of a series of points.
::图形是一条直线线(这意味着方程式是线性),尽管函数是离散的,实际上只是由一系列点组成。

The graph crosses the $\begin{aligned} y - \end{aligned}$ axis at $\begin{aligned} y = 3 \end{aligned}$ (notice that there’s $\begin{aligned} a + 3 \end{aligned}$ in the equation—that’s not a coincidence!). This is the base cost of the taxi.
::图表横穿y=3的y-axis y=3 (注意方程式中有一个+3 - 这不是巧合!) 这是出租车的基本成本。

Every time we move over by one square we move up by 0.8 squares (notice that that’s also the coefficient of $\begin{aligned} x \end{aligned}$ in the equation). This is the rate of charge of the taxi (cost per mile).
::每次我们移动到一个平方时,我们就会向上移动0.8平方(注意这也是方程中的x系数 ) 。这就是出租车的收费率(每英里成本 ) 。

If we move over by three squares, we move up by $\begin{aligned} 3 \times 0.8 \end{aligned}$ squares.
::如果我们移动了三个方形, 我们移动了3x0.8方形。

Real-World Example: Paying off Debt
::真实世界实例:偿还债务

A small business has a debt of $500,000 incurred from start-up costs. It predicts that it can pay off the debt at a rate of $85,000 per year according to the following equation governing years in business $\begin{aligned} (x) \end{aligned}$ and debt measured in thousands of dollars "> $\begin{aligned} (y) \end{aligned}$ .
::小企业从开办费用中产生的债务为500 000美元,它预测它能够按照以下关于营业年数(x)和以千美元计量的债务的公式,以每年85 000美元的速度偿还债务。

\begin{aligned} y = - 85 x + 500 \end{aligned}

::y 85x+500

Graph the above equation and use your graph to predict when the debt will be fully paid.
::绘制上方形图,并用您的图表预测债务何时将全额偿还。

First, we start with our table of values:
::首先,我们从我们的数值表开始:

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y \end{aligned}$
0	500
1	415
2	330
3	245
4	160

Then we plot our points and draw the line that goes through them:
::然后我们划出我们的分数并划出贯穿它们的线条

Notice the scale we’ve chosen here. There’s no need to include any points above $\begin{aligned} y = 500 \end{aligned}$ , but it’s still wise to allow a little extra.
::注意我们在这里选择的尺度。没有必要包含y=500以上的任何点,但允许额外一点还是明智的。

Next we need to determine how many years it takes the debt to reach zero, or in other words, what $\begin{aligned} x - \end{aligned}$ value will make the $\begin{aligned} y - \end{aligned}$ value equal 0. We know it’s greater than four (since at $\begin{aligned} x = 4 \end{aligned}$ the $\begin{aligned} y - \end{aligned}$ value is still positive), so we need an $\begin{aligned} x - \end{aligned}$ scale that goes well past $\begin{aligned} x = 4 \end{aligned}$ . Here we’ve chosen to show the $\begin{aligned} x - \end{aligned}$ values from 0 to 12, though there are many other places we could have chosen to stop.
::接下来,我们需要确定债务需要多少年才能达到零,或者换句话说,什么是x-价值才能使y-价值等于0。我们知道它大于4。我们知道它大于4(因为在x=4时,y-价值仍然为正值),所以我们需要一个大大超过x=4的x-尺度。我们在这里选择了显示0到12的x-价值,尽管我们可以选择停止许多其他地方。

To read the time that the debt is paid off, we simply read the point where the line hits $\begin{aligned} y = 0 \end{aligned}$ (the $\begin{aligned} x - \end{aligned}$ axis). It looks as if the line hits pretty close to $\begin{aligned} x = 6 \end{aligned}$ . So the debt will definitely be paid off in six years.
::阅读债务还清的时间,我们只需读一读行点击 y=0 ( x- 轴) 的点。看起来线点击率似乎接近 x=6 。因此债务肯定会在六年内偿还。

To see more simple examples of graphing linear equations by hand, see the Khan Academy video on graphing lines at . The narrator shows how to graph several linear equations, using a table of values to plot points and then connecting the points with a line.
::要用手绘制线性方程图的更简单的例子,请见Khan学院的图解线条视频。旁白人展示了如何用一个数值表格绘制线性方程图,然后将点与线连接。

Analyze Graphs of
::分析图

We often use graphs to represent relationships between two linked quantities. It’s useful to be able to interpret the information that graphs convey. For example, the chart below shows a fluctuating stock price over ten weeks. You can read that the index closed the first week at about $68, and at the end of the third week it was at about $62. You may also see that in the first five weeks it lost about 20% of its value, and that it made about 20% gain between weeks seven and ten. Notice that this relationship is discrete, although the dots are connected to make the graph easier to interpret.
::我们经常使用图表来代表两个关联数量之间的关系。解释图形传递的信息是有益的。例如,下图显示了十周内股票价格的波动。您可以读到,第一周的指数关闭了大约68美元,第三周结束时的指数为62美元。你也可以看到,在前五周,指数损失了约20%的价值,7周到10周之间大约增加了20%。请注意,这种关系是互不关联的,尽管将点连接起来使图表更容易解释。

Analyzing graphs is a part of life - whether you are trying to decide to buy stock, figure out if your blog readership is increasing, or predict the temperature from a weather report. Many graphs are very complicated, so for now we’ll start off with some simple linear conversion graphs. Algebra starts with basic relationships and builds to more complicated tasks, like reading the graph above.
::分析图表是生命的一部分 — — 无论是你想决定购买股票,研究博客读者人数是否在增加,还是从天气报告中预测温度。许多图表非常复杂,因此我们现在先从一些简单的线性转换图开始。代数以基本关系为起点,并形成更复杂的任务,比如阅读上面的图表。

Determining Unknown Values Given a Graph
::确定一个图表的未知值

Below is a graph for converting marked prices in a downtown store into prices that include sales tax. Use the graph to determine the cost including sales tax for a $6.00 pen in the store.
::下面是将市中心商店的标志性价格转换为包括销售税在内的价格的图表。使用这个图表来确定成本,包括商店内6美元钢笔的销售税。

To find the relevant price with tax, first find the correct pre-tax price on the $\begin{aligned} x - \end{aligned}$ axis. This is the point $\begin{aligned} x = 6 \end{aligned}$ .
::为了找到相关的税价,首先在 x - 轴上找到正确的税前价格。这是点 x= 6 。

Draw the line $\begin{aligned} x = 6 \end{aligned}$ up until it meets the function, then draw a horizontal line to the $\begin{aligned} y - \end{aligned}$ axis. This line hits at $\begin{aligned} y \approx 6.75 \end{aligned}$ (about three fourths of the way from $\begin{aligned} y = 6 \end{aligned}$ to $\begin{aligned} y = 7 \end{aligned}$ ).
::向上绘制行 x=6 ,直到它符合函数,然后向 y - 轴绘制水平线。该行在 y =6. 75 处点击(从 y= 6 到 y= 7 大约四分之三的距离)。

Conversions using a Graph
::使用图表转换

The graph for converting temperature from Fahrenheit to Celsius is shown below. Use the graph to convert the following:
::将温度从法赫里内海特转换为摄氏度的图如下。

a) $\begin{aligned} 70^{\circ} \end{aligned}$ Fahrenheit to Celsius
:a) 70-华氏至摄氏度

To find $\begin{aligned} 70^{\circ} \end{aligned}$ Fahrenheit, we look along the Fahrenheit-axis (in other words the $\begin{aligned} x - \end{aligned}$ axis) and draw the line $\begin{aligned} x = 70 \end{aligned}$ up to the function. Then we draw a horizontal line to the Celsius-axis ( $\begin{aligned} y - \end{aligned}$ axis). The horizontal line hits the axis at a little over 20 (21 or 22).
::要找到 70 { } 法赫里内海- 轴, 我们沿着法赫里海- 轴( 换句话说, x- 轴) 进行查看, 然后将线x= 70 拖到函数上。然后我们绘制一条水平线到摄氏轴( y- 轴) 。水平线击中轴的时间略高于 20 ( 21 或 22 ) 。

$\begin{aligned} 70^{\circ} \end{aligned}$ Fahrenheit is approximately equivalent to $\begin{aligned} 21^{\circ} \end{aligned}$ Celsius.
::70 Fahrenheit大约相当于摄氏21摄氏度。

b) $\begin{aligned} 0^{\circ} \end{aligned}$ Celsius to Fahrenheit
:b) 华氏零摄氏度

To find $\begin{aligned} 0^{\circ} \end{aligned}$ Celsius, we look at the Fahrenheit-axis (the line $\begin{aligned} y = 0 \end{aligned}$ ). The function hits the $\begin{aligned} x - \end{aligned}$ axis just right of 30.
::为了找到零摄氏度,我们查看Fahrenheit-axis(线y=0)。函数击中x-axis正对30。

$\begin{aligned} 0^{\circ} \end{aligned}$ Celsius is equivalent to $\begin{aligned} 32^{\circ} \end{aligned}$ Fahrenheit
::0摄氏度等于32 华氏度

Examples
::实例

The graph for converting temperature from Fahrenheit to Celsius is shown below. Use the graph to convert the following:
::将温度从法赫里内海特转换为摄氏度的图如下。

Example 1
::例1

$\begin{aligned} 0^{\circ} \end{aligned}$ Fahrenheit to Celsius
::华氏至摄氏度

To find $\begin{aligned} 0^{\circ} \end{aligned}$ Fahrenheit, we just look at the $\begin{aligned} y - \end{aligned}$ axis. (Don't forget that this axis is simply the line $\begin{aligned} x = 0 \end{aligned}$ .) The line hits the $\begin{aligned} y - \end{aligned}$ axis just below the half way point between −15 and −20.
::要找到“Fahrenheitt”,我们只需看一下y-axis。 (不要忘记,这个轴只是线x=0。 ) y-axis 直达到 ~15 和 ~20 之间的半个路点下方。

$\begin{aligned} 0^{\circ} \end{aligned}$ Fahrenheit is approximately equivalent to $\begin{aligned} - 18^{\circ} \end{aligned}$ Celsius.
::0 Fahrenheit 大约相当于 - 18°C。

Example 2
::例2

$\begin{aligned} 30^{\circ} \end{aligned}$ Celsius to Fahrenheit
::30摄于华氏30摄氏度

To find $\begin{aligned} 30^{\circ} \end{aligned}$ Celsius, we look up the Celsius-axis and draw the line $\begin{aligned} y = 30 \end{aligned}$ along to the function. When this horizontal line hits the function, we draw a line straight down to the Fahrenheit-axis. The line hits the axis at approximately 85.
::要找到 30 °C, 我们向上查看摄氏轴, 然后将 y= 30 线绘制到函数上。当这条水平线击中函数时, 我们向下绘制一条直线到 Fahrenheit- agis 。这条线直达轴大约85。

$\begin{aligned} 30^{\circ} \end{aligned}$ Celsius is approximately equivalent to $\begin{aligned} 85^{\circ} \end{aligned}$ Fahrenheit.
::30摄氏度大约相当于85华氏度。

Review
::回顾

For 1-3, make a table of values for the following equations and then graph them.
::对于 1-3, 绘制以下方程式的数值表, 然后绘制图表。

$\begin{aligned} y = 2 x + 7 \end{aligned}$
::y=2x+7 y=2x+7

$\begin{aligned} y = 0.7 x - 4 \end{aligned}$
::y=0.7x-4 y=0.7x-4

$\begin{aligned} y = 6 - 1.25 x \end{aligned}$
::y=6 - 1.25x

“Think of a number. Multiply it by 20, divide the answer by 9, and then subtract seven from the result.”
1. Make a table of values and plot the function that represents this sentence.
  ::绘制一个值表并绘制代表此句的函数。
2. If you picked 0 as your starting number, what number would you end up with?
  ::如果你把0选为开始数你会用什么数字?
3. To end up with 12, what number would you have to start out with?
  ::要到12岁,你要从哪个数字开始呢?
::“想一想一个数字。乘以20,回答除以9,然后从结果中减去7。” 绘制一个数值表并绘制代表此句的函数。如果你选择 0 作为开始数,你最终会使用什么数字?要用12,你必须用什么数字开始?

At the airport, you can change your money from dollars into euros. The service costs $5, and for every additional dollar you get 0.7 euros.
1. Make a table for this and plot the function on a graph.
  ::绘制此表格并在图表中绘制函数。
2. Use your graph to determine how many euros you would get if you give the office $50.
  ::用你的图表来确定如果你给办公室50美元你能得到多少欧元
3. To get 35 euros, how many dollars would you have to pay?
  ::要拿到35欧元,你还要付多少钱?
4. The exchange rate drops so that you can only get 0.5 euros per additional dollar. Now how many dollars do you have to pay for 35 euros?
  ::汇率下跌,这样每多一美元只能得到0.5欧元。现在,你必须支付多少美元才能支付35欧元?
::在机场,你可以把钱从美元兑换成欧元。服务费用为5美元,每增加一美元,你就能得到0.7欧元。做一张表格,在图表上绘制函数。用你的图表来确定如果你给办公室50美元,你能得到多少欧元。要拿到35欧元,你必须支付多少美元?汇率下降,这样每增加1美元只能得到0.5欧元。现在,你必须支付多少美元35欧元?

For 6-9, the graph below shows a conversion chart for converting between weight in kilograms and weight in pounds. Use it to convert the following measurements.
::关于6-9,下图显示按公斤和公斤加权换算的换算图。使用该换算图转换下列测量结果。

4 kilograms into weight in pounds
::4公斤重(磅)

9 kilograms into weight in pounds
::9公斤重(磅)

12 pounds into weight in kilograms
::12磅重(公斤)

17 pounds into weight in kilograms
::17磅重(公斤)

For 10-12, use the graph from problems 6-9 to answer the following questions.
::10-12,使用问题6-9的图表回答下列问题。

An employee at a sporting goods store is packing 3-pound weights into a box that can hold 8 kilograms. How many weights can she place in the box?
::体育用品商店的雇员将3磅重的重量包装在一个能装8公斤的箱子里。她能把多少重量放在盒子里?

After packing those weights, there is some extra space in the box that she wants to fill with one-pound weights. How many of those can she add?
::在包装这些重量后,她想用一磅重量填充盒子中的一些额外空间。她可以添加多少这些空间?

After packing those, she realizes she misread the label and the box can actually hold 9 kilograms. How many more one-pound weights can she add?
::在包装这些之后,她意识到她误读了标签,盒子实际上可以持有9公斤。她还能加多少个1磅重的重量?

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

线性方形图

Graphs of Linear Equations ::线性方形图

Graph a Linear Equation ::线性方形图a

Real-World Example: Paying off Debt ::真实世界实例:偿还债务

Determining Unknown Values Given a Graph ::确定一个图表的未知值

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)