Course: 4.5 拦截和掩盖方法

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拦截和掩盖方法
Intercepts and the Cover-Up Method
::拦截和掩盖方法

If you know just one of the points on a line, you’ll find that isn’t enough information to plot the line on a graph. As you can see in the graph above, there are many lines—in fact, infinitely many lines—that pass through a single point. But what if you know two points that are both on the line? Then there’s only one way to graph that line; all you need to do is plot the two points and use a ruler to draw the line that passes through both of them.
::如果你只知道一条线上的一个点,你就会发现,这个信息不足以在一幅图上绘制线条。正如你在上面的图中看到的,有许多线条 — — 事实上,无限多的线条 — — 穿过一个点。但如果你知道两点都是线条上的? 那么只有一种方法可以绘制这条线;你只需要绘制两点,用一个标尺来绘制通过这两点的线条。

There are a lot of options for choosing which two points on the line you use to plot it. In this lesson, we’ll focus on two points that are rather convenient for graphing: the points where our line crosses the $\begin{aligned} x - \end{aligned}$ and $\begin{aligned} y - \end{aligned}$ axes, or intercepts . We’ll see how to find intercepts algebraically and use them to quickly plot graphs.
::选择您用于绘图的线条上的两点有很多选择。在这个教训中,我们将集中关注比较方便绘制图表的两个点:我们的线横跨 x 和 y 轴或截取的点。我们将看到如何找到代数截取器,并用它们快速绘制图表。

Look at the graph above. The $\begin{aligned} y - \end{aligned}$ intercept occurs at the point where the graph crosses the $\begin{aligned} y - \end{aligned}$ axis. The $\begin{aligned} y - \end{aligned}$ value at this point is 8, and the $\begin{aligned} x - \end{aligned}$ value is 0.
::查看上面的图表。 y- interview 是在图形横过 y- 轴的点发生的。 y- value 此时是 8, x- value 是 0 。

Similarly, the $\begin{aligned} x - \end{aligned}$ intercept occurs at the point where the graph crosses the $\begin{aligned} x - \end{aligned}$ axis. The $\begin{aligned} x - \end{aligned}$ value at this point is 6, and the $\begin{aligned} y - \end{aligned}$ value is 0.
::同样,x- interview 发生于图形横过 x- 轴的点。此点的 x- 值为 6, y- 值为 0 。

So we know the coordinates of two points on the graph: (0, 8) and (6, 0). If we’d just been given those two coordinates out of the blue, we could quickly plot those points and join them with a line to recreate the above graph.
::因此,我们知道图上两个点的坐标:0,8和6,0。如果我们刚刚得到这两个点的坐标,我们就可以快速绘制这些点,并用一条线将它们合起来,以重建上图。

Note: Not all lines will have both an $\begin{aligned} x - \end{aligned}$ and a $\begin{aligned} y - \end{aligned}$ intercept, but most do. However, horizontal lines never cross the $\begin{aligned} x - \end{aligned}$ axis and vertical lines never cross the $\begin{aligned} y - \end{aligned}$ axis.
::注:并非所有线条都有 x 和 y 和 y 界面, 但大多数线条都有。但是, 水平线从不跨越 x 轴, 垂直线从不跨越 y 轴。

For examples of these special cases, see the graph below.
::这些特殊情况的例子见下图。

Finding Intercepts by Substitution
::以替代方式查找拦截

Find the intercepts of the line $\begin{aligned} y = 13 - x \end{aligned}$ and use them to graph the function .
::查找 y=13-x 线的截取, 并用它们绘制函数图。

The first intercept is easy to find. The $\begin{aligned} y - \end{aligned}$ intercept occurs when $\begin{aligned} x = 0 \end{aligned}$ . Substituting gives us $\begin{aligned} y = 13 - 0 = 13 \end{aligned}$ , so the $\begin{aligned} y - \end{aligned}$ intercept is (0, 13).
::第一次拦截很容易找到。 y- interview 是在 x=0 时发生的。替换给我们 y= 13-0=13, y- interview is (0, 13) 。

Similarly, the $\begin{aligned} x - \end{aligned}$ intercept occurs when $\begin{aligned} y = 0 \end{aligned}$ . Plugging in 0 for $\begin{aligned} y \end{aligned}$ gives us $\begin{aligned} 0 = 13 - x \end{aligned}$ , and adding $\begin{aligned} x \end{aligned}$ to both sides gives us $\begin{aligned} x = 13 \end{aligned}$ . So (13, 0) is the $\begin{aligned} x - \end{aligned}$ intercept.
::同样,当y=0时,就会发生 x- interview 。插入 0, y 给我们 0= 13- x, 并在两侧添加 x 给我们 x= 13 。所以( 13, 0) 是 x- interview 。

To draw the graph, simply plot these points and join them with a line.
::要绘制图,只需绘制这些点,然后用一条线将它们连接起来。

Graphing Functions by Finding Intercepts
::通过查找截取来绘制函数

Graph the following functions by finding intercepts.
::通过查找拦截,绘制以下函数图。

a) $\begin{aligned} y = 2 x + 3 \end{aligned}$
::a) y=2x+3

Find the $\begin{aligned} y - \end{aligned}$ intercept by plugging in $\begin{aligned} x = 0 : \end{aligned}$
::通过插入 x=0 来查找 y- interview :

$\begin{aligned} y = 2 \cdot 0 + 3 = 3 - the y - intercept is (0, 3) \end{aligned}$

::y=20+3=3 - y - 拦截是( 0, 3)

Find the $\begin{aligned} x - \end{aligned}$ intercept by plugging in $\begin{aligned} y = 0 : \end{aligned}$
::通过插入 y=0 查找 x - 界面 :

$\begin{aligned} 0 & = 2 x + 3 - s u b t r a c t 3 f r o m b o t h s i d e s : \\ - 3 & = 2 x - d i v i d e b y 2 : \\ - \frac{3}{2} & = x - the x - intercept is (- 1.5, 0) \end{aligned}$

::0=2x+3 - 来自两侧的分数 3:- 3=2x - divide 乘以 2: - 32=x - x - 截取 (-1.5,0)

b) $\begin{aligned} y = 7 - 2 x \end{aligned}$
:b)y=7-2x

Find the $\begin{aligned} y - \end{aligned}$ intercept by plugging in $\begin{aligned} x = 0 : \end{aligned}$
::通过插入 x=0 来查找 y- interview :

$\begin{aligned} y = 7 - 2 \cdot 0 = 7 - the y - intercept is (0, 7) \end{aligned}$

::y=7 - 20=7 - y - 拦截为( 0, 7)

Find the $\begin{aligned} x - \end{aligned}$ intercept by plugging in $\begin{aligned} y = 0 : \end{aligned}$
::通过插入 y=0 查找 x - 界面 :

$\begin{aligned} 0 & = 7 - 2 x - s u b t r a c t 7 f r o m b o t h s i d e s : \\ - 7 & = - 2 x - d i v i d e b y - 2 : \\ \frac{7}{2} & = x - the x - intercept is (3.5, 0) \end{aligned}$

::0=7-2x-从两边取7:- 7=2=2x- divide 乘以-2: 72=x- x- 截取 (3.5,0)

c) $\begin{aligned} 4 x - 2 y = 8 \end{aligned}$
:c) 4x-2y=8

Find the $\begin{aligned} y - \end{aligned}$ intercept by plugging in $\begin{aligned} x = 0 : \end{aligned}$
::通过插入 x=0 来查找 y- interview :

$\begin{aligned} 4 \cdot 0 - 2 y & = 8 \\ - 2 y & = 8 - d i v i d e b y - 2 \\ y & = - 4 - the y - intercept is (0, - 4) \end{aligned}$

::40-2y=8--2y=8 - 4- y- interview is (0,- 4)

Find the $\begin{aligned} x - \end{aligned}$ intercept by plugging in $\begin{aligned} y = 0 : \end{aligned}$
::通过插入 y=0 查找 x - 界面 :

$\begin{aligned} 4 x - 2 \cdot 0 & = 8 \\ 4 x & = 8 - d i v i d e b y 4 : \\ x & = 2 - the x - intercept is (2, 0) \end{aligned}$

::4 - 20=84x=8 - divide 4:x=2 - x - 截取(2,0)

Finding Intercepts for Standard Form Equations Using the Cover-Up Method
::使用封面方法对标准形式公平进行查找的截取

Look at the last two equations in example 2. These equations are written in standard form . Standard form equations are always written “ coefficient times $\begin{aligned} x \end{aligned}$ plus (or minus) coefficient times $\begin{aligned} y \end{aligned}$ equals value ”. In other words, they look like this:
::看看例2中最后两个方程式,这些方程式以标准形式写成。标准方程式总是写成“系数乘以x+(或减以)系数乘以等值”。换句话说,它们看起来是这样的:

$\begin{aligned} a x + b y = c \end{aligned}$

::ax+by=c 轴x+by=c

where $\begin{aligned} a \end{aligned}$ has to be positive, but $\begin{aligned} b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ do not.
::a 必须是肯定的,但b和c没有。

There is a neat method for finding intercepts in standard form, often referred to as the cover-up method.
::有一种以标准形式查找拦截的整洁方法,通常称为掩盖方法。

Find the intercepts of the following equations:
::查找以下方程式的截取数据 :

To solve for each intercept, we realize that at the intercepts the value of either $\begin{aligned} x \end{aligned}$ or $\begin{aligned} y \end{aligned}$ is zero, and so any terms that contain that variable effectively drop out of the equation . To make a term disappear, simply cover it (a finger is an excellent way to cover up terms) and solve the resulting equation.
::为了解决每一次拦截,我们意识到在拦截时,正弦值为零,因此任何含有该变量的术语实际上都会从方程式中消失。要让一个术语消失,只需覆盖它(手指是掩盖条件的绝佳方法 ) , 并解决由此产生的方程式。

a) $\begin{aligned} 7 x - 3 y = 21 \end{aligned}$
::a) 7x-3y=21

To solve for the $\begin{aligned} y - \end{aligned}$ intercept we set $\begin{aligned} x = 0 \end{aligned}$ and cover up the $\begin{aligned} x - \end{aligned}$ term:
::要解决 y - 界面, 我们设置 x=0 并覆盖 x - term :

$\begin{aligned} - 3 y & = 21 \\ y & = - 7 (0, - 7) is the y - intercept. \end{aligned}$

::-3y=21y7(0,-7)是y-interview。

Now we solve for the $\begin{aligned} x - \end{aligned}$ intercept:
::现在,我们解决 x - 界面问题 :

$\begin{aligned} 7 x & = 21 \\ x & = 3 (3, 0) is the x - intercept. \end{aligned}$

::7x=21x=3(3,0)是 x- intercut 。

b) $\begin{aligned} 12 x - 10 y = - 15 \end{aligned}$
:b) 12x-10y15

To solve for the $\begin{aligned} y - \end{aligned}$ intercept $\begin{aligned} (x = 0) \end{aligned}$ , cover up the $\begin{aligned} x - \end{aligned}$ term:
::要解决y- interception (x=0) 的 y- interception (x=0) , 要覆盖 x- term :

$\begin{aligned} - 10 y & = - 15 \\ y & = 1.5 (0, 1.5) is the y - intercept. \end{aligned}$

::- 10y15y=1.5(0.1.5)是y-interfict。

Now solve for the $\begin{aligned} x - \end{aligned}$ intercept $\begin{aligned} (y = 0) \end{aligned}$ :
::现在解答 x - interception (y=0) :

$\begin{aligned} 12 x & = - 15 \\ x & = - \frac{5}{4} (- 1.25, 0) is the x - intercept. \end{aligned}$

::12x15x54(- 1. 25, 0) 是 x- interview 。

Solving Real-World Problems Using Intercepts of a Graph
::利用图的截取解决现实世界问题

Jesus has $30 to spend on food for a class barbecue. Hot dogs cost $0.75 each (including the bun) and burgers cost $1.25 (including the bun). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, without spending more than $30.
::耶稣有30美元可以花在做一个班级烧烤的食物上。热狗每人0.75美元(包括包子)和汉堡1.25美元(包括包子 ) 。绘制一张图表,显示他可以为烧烤买的热狗和汉堡的所有组合,而无需花费超过30美元。

This time we will find an equation first, and then we can think logically about finding the intercepts.
::这次我们先找到一个方程然后我们再从逻辑上考虑找到拦截物

If the number of burgers that Jesus buys is $\begin{aligned} x \end{aligned}$ , then the money he spends on burgers is $\begin{aligned} 1.25 x \end{aligned}$
::如果耶稣购买的汉堡数量是x, 那么他花在汉堡上的钱是1. 25x

If the number of hot dogs he buys is $\begin{aligned} y \end{aligned}$ , then the money he spends on hot dogs is $\begin{aligned} 0.75 y \end{aligned}$
::如果他买的热狗数量是y, 那么他花在热狗身上的钱是0.75y

So the total cost of the food is $\begin{aligned} 1.25 x + 0.75 y \end{aligned}$ .
::因此,食物的总成本是1.25x+0.755。

The total amount of money he has to spend is $30, so if he is to spend it ALL, we can use the following equation:
::他必须花的钱总额是30美元, 所以如果他要全部花掉, 我们可以使用以下方程式:

$\begin{aligned} 1.25 x + 0.75 y = 30 \end{aligned}$

::1.25x+0.75y=30

We can solve for the intercepts using the cover-up method. First the $\begin{aligned} y - \end{aligned}$ intercept $\begin{aligned} (x = 0) \end{aligned}$ :
::我们可以使用隐蔽方法解答拦截。首先, y- interview (x=0) :

$\begin{aligned} 0.75 y & = 30 \\ y & = 40 y - intercept: (0, 40) \end{aligned}$

::0.75y=30y=40y-拦截0,40)

Then the $\begin{aligned} x - \end{aligned}$ intercept $\begin{aligned} (y = 0) \end{aligned}$ :
::然后是X-拦截(y=0):

$\begin{aligned} 1.25 x & = 30 \\ x & = 24 x - intercept: (24, 0) \end{aligned}$

::1. 25x=30x=24x - 拦截: (24,0)

Now we plot those two points and join them to create our graph, shown here:
::现在,我们绘制这两个点,并加入它们来创建我们的图表,在这里显示:

We could also have created this graph without needing to come up with an equation. We know that if John were to spend ALL the money on hot dogs, he could buy $\begin{aligned} \frac{30}{.75} = 40 \end{aligned}$ hot dogs. And if he were to buy only burgers he could buy $\begin{aligned} \frac{30}{1.25} = 24 \end{aligned}$ burgers. From those numbers, we can get 2 intercepts: (0 burgers, 40 hot dogs) and (24 burgers, 0 hot dogs). We could plot these just as we did above and obtain our graph that way.
::我们还可以在不需要提出等式的情况下创建这个图表。我们知道,如果约翰把所有的钱花在热狗身上,他可以买30.75=40热狗。如果他只买汉堡,他可以买301.25=24汉堡。从这些数字中,我们可以得到两个拦截0个汉堡,40个热狗)和(24个汉堡,0个热狗 ) 。我们可以像上面那样绘制这些图,然后用这个图来获取。

As a final note, we should realize that Jesus’ problem is really an example of an inequality . He can, in fact, spend any amount up to $30. The only thing he cannot do is spend more than $30. The graph above reflects this: the line is the set of solutions that involve spending exactly $30, and the shaded region shows solutions that involve spending less than $30. We’ll work with inequalities some more in Chapter 6.
::最后一点,我们应该意识到耶稣的问题确实是一个不平等的例子。事实上,他可以花费30美元。他唯一不能做的就是花费30多美元。上面的图表反映了这一点:一线是完全花费30美元的一系列解决方案,而阴暗的区域则显示了花费不到30美元的解决办法。我们将在第6章中更多地处理不平等问题。

Examples
::实例

Example 1
::例1

Graph $\begin{aligned} 2 x + 3 y = - 6 \end{aligned}$ by finding intercepts.
::图2x+3y6 通过寻找拦截。

Find the $\begin{aligned} y - \end{aligned}$ intercept by plugging in $\begin{aligned} x = 0 : \end{aligned}$
::通过插入 x=0 来查找 y- interview :

$\begin{aligned} 2 \cdot 0 + 3 y & = - 6 \\ 3 y & = - 6 - divide by 3 : \\ y & = - 2 - the y - intercept is (0, - 2) \end{aligned}$

::2+0+3y63y6 - divide by 3:y2 - y - interview is (0,-2)

Find the $\begin{aligned} x - \end{aligned}$ intercept by plugging in $\begin{aligned} y = 0 : \end{aligned}$
::通过插入 y=0 查找 x - 界面 :

$\begin{aligned} 2 x + 3 \cdot 0 & = - 6 \\ 2 x & = - 6 - divide by 2 : \\ x & = - 3 - the x - intercept is (- 3, 0) \end{aligned}$

::2x+3062x6 - divide by 2:x3 - x- 截取 (- 3, 0)

The graph of this line is the line labeled d, the two intercepts are marked by dots.
::这条线的图是标有D的线, 两次拦截都标有点。

Example 2
::例2

Find the intercepts of $\begin{aligned} x + 3 y = 6 \end{aligned}$ using the cover-up method.
::使用掩蔽方法查找 x+3y=6 的拦截。

To solve for the $\begin{aligned} y - \end{aligned}$ intercept $\begin{aligned} (x = 0) \end{aligned}$ , cover up the $\begin{aligned} x - \end{aligned}$ term:
::要解决y- interception (x=0) 的 y- interception (x=0) , 要覆盖 x- term :

$\begin{aligned} 3 y & = 6 \\ y & = 2 (0, 2) is the y - intercept. \end{aligned}$

::3y=6y=2(2,0,2)是y-interview。

Solve for the $\begin{aligned} y - \end{aligned}$ intercept:
::解决y - interview:

$\begin{aligned} x = 6 (6, 0) is the x - intercept. \end{aligned}$

::x=6( 6, 0) 是 x- 界面。

The graph of this function and the intercepts is line c:
::此函数和拦截的图形为 c 行 :

Review
::回顾

For 1-8, find the intercepts for the following equations using substitution.
::对于 1-8, 使用替代方法找到以下方程式的拦截。

$\begin{aligned} y = 3 x - 6 \end{aligned}$
::y=3x-6 y=3x-6

$\begin{aligned} y = - 2 x + 4 \end{aligned}$
::y2x+4 y2x+4

$\begin{aligned} y = 14 x - 21 \end{aligned}$
::y=14x-21 y=14x-21

$\begin{aligned} y = 7 - 3 x \end{aligned}$
::y=7 - 3x y=7 - 3x

$\begin{aligned} y = 2.5 x - 4 \end{aligned}$
::y=2.5x-4

$\begin{aligned} y = 1.1 x + 2.2 \end{aligned}$
::y=1.1x+2.2 y=1.1x+2.2

$\begin{aligned} y = \frac{3}{8} x + 7 \end{aligned}$
::y=38x+7 y= 38x+7

$\begin{aligned} y = \frac{5}{9} - \frac{2}{7} x \end{aligned}$
::y=59-27x

For 9-16, find the intercepts of the following equations using the cover-up method.
::对于9-16,使用掩盖方法查找以下方程式的截取数据。

$\begin{aligned} 5 x - 6 y = 15 \end{aligned}$
::5x-6y=15

$\begin{aligned} 3 x - 4 y = - 5 \end{aligned}$
::3x-4y5

$\begin{aligned} 2 x + 7 y = - 11 \end{aligned}$
::2+7y11

$\begin{aligned} 5 x + 10 y = 25 \end{aligned}$
::5x+10y=25

$\begin{aligned} 5 x - 1.3 y = 12 \end{aligned}$
::5x-1.3y=12

$\begin{aligned} 1.4 x - 3.5 y = 7 \end{aligned}$
::1.4x-3.5y=7

$\begin{aligned} \frac{3}{5} x + 2 y = \frac{2}{5} \end{aligned}$
::35x+2y=25

$\begin{aligned} \frac{3}{4} x - \frac{2}{3} y = \frac{1}{5} \end{aligned}$
::34-23y=15

For 17-20, use any method to find the intercepts and then graph the following equations.
::对于 17-20, 使用任何方法查找拦截, 然后绘制下列方程。

$\begin{aligned} y = 2 x + 3 \end{aligned}$
::y=2x+3 y=2x+3

$\begin{aligned} 6 (x - 1) = 2 (y + 3) \end{aligned}$
::6(x-1-1)=2(y+3)

$\begin{aligned} x - y = 5 \end{aligned}$
::x- y=5

$\begin{aligned} x + y = 8 \end{aligned}$
::x+y=8 x+y=8

At the local grocery store strawberries cost $3.00 per pound and bananas cost $1.00 per pound.

If I have $10 to spend on strawberries and bananas, draw a graph to show what combinations of each I can buy and spend exactly $10.
::如果我有10美元花在草莓和香蕉上, 绘制一个图表来显示我每个能买到和花10美元的组合。

Plot the point representing 3 pounds of strawberries and 2 pounds of bananas. Will that cost more or less than $10?
::代表3磅草莓和2磅香蕉的分数。这要花费10美元或10美元以上吗?

Do the same for the point representing 1 pound of strawberries and 5 pounds of bananas.
::对于代表1磅草莓和5磅香蕉的点也照此办理。

::当地杂货店的草莓每磅3美元,香蕉每磅1美元。如果我有10美元花在草莓和香蕉上,请绘制一张图表,显示我每买一磅能买到和花10美元的确切组合。绘制代表3磅草莓和2磅香蕉的点数。这要花10美元或10美元以上吗?对代表1磅草莓和5磅香蕉的点数采取同样的做法。

A movie theater charges $7.50 for adult tickets and $4.50 for children. If the theater takes in $900 in ticket sales for a particular screening, draw a graph which depicts the possibilities for the number of adult tickets and the number of child tickets sold.
::电影院的成人票价为7.50美元,儿童票价为4.50美元,如果剧院为特别的筛选购买900美元的售票,请绘制一张图表,说明成人票数和售出儿童票数的可能性。

Why can't we use the intercept method to graph the following equation? $\begin{aligned} 3 (x + 2) = 2 (y + 3) \end{aligned}$
::为什么我们不能使用截截截方法来绘制以下方程式? (3(x+2)=2(y+3)

Name two more equations that we can’t use the intercept method to graph.
::列出两个无法使用截截截方法图解的方程式。

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。

4.5 拦截和掩盖方法

Section outline

General

拦截和掩盖方法

Intercepts and the Cover-Up Method
::拦截和掩盖方法

Finding Intercepts by Substitution
::以替代方式查找拦截

Graphing Functions by Finding Intercepts
::通过查找截取来绘制函数

Finding Intercepts for Standard Form Equations Using the Cover-Up Method
::使用封面方法对标准形式公平进行查找的截取

Solving Real-World Problems Using Intercepts of a Graph
::利用图的截取解决现实世界问题

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)

Section outline

General

拦截和掩盖方法

Intercepts and the Cover-Up Method ::拦截和掩盖方法

Finding Intercepts by Substitution ::以替代方式查找拦截

Graphing Functions by Finding Intercepts ::通过查找截取来绘制函数

Finding Intercepts for Standard Form Equations Using the Cover-Up Method ::使用封面方法对标准形式公平进行查找的截取

Solving Real-World Problems Using Intercepts of a Graph ::利用图的截取解决现实世界问题

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Intercepts and the Cover-Up Method
::拦截和掩盖方法

Finding Intercepts by Substitution
::以替代方式查找拦截

Graphing Functions by Finding Intercepts
::通过查找截取来绘制函数

Finding Intercepts for Standard Form Equations Using the Cover-Up Method
::使用封面方法对标准形式公平进行查找的截取

Solving Real-World Problems Using Intercepts of a Graph
::利用图的截取解决现实世界问题

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)