课程： 12.11输入输出表功能规则

章节大纲

选择节常规

折叠展开
常规

全部折叠展开全部
- 选择活动新闻通告
  
  新闻通告讨论区

输入输出表功能规则

Geri is getting ready to go through the automatic car wash with her mom. Geri’s mom gives her several dollar bills and tells her to use it to get quarters for the car wash. Geri puts a dollar in the change machine and the machine gives her 4 quarters. She puts another dollar in the machine and it gives her 4 more quarters. Geri continues until she has 26 quarters. As she walks back to the car, she begins to think about what she put into the change machine and what came out.
::Geri的妈妈给了她几张美元钞票,并告诉她用它来买洗车房。Geri在换车机里放了一美元,机器给了她4个空房。她又在机器里放了一美元,又给了她4个空房。Geri继续工作,直到她有26个空房。当她回到车上时,她开始思考她把什么放进换车机里,什么出来。

This is a table to represent the change machine’s input and output.
::这是一张代表变革机器输入和输出的表格。

Input (dollars)	Output (quarters)
6	24
5	20
4	16
3	12

What rule could Geri write to represent what happened to the input to equal the output?
::Geri能写出什么规则来代表输入发生的情况, 以等值输出 ?

In this concept, you will learn to evaluate and write function rules for an input-output table .
::在此概念中,您将学会评价和写出投入产出表的功能规则。

Writing Function Rules for Input-Output Tables
::输入输出表的写写函数规则

A function is when one variable or term depends on another according to a rule. There is a special relationship between the two variables of the function where each value in the input applies to only one value in the output. These rules are called function rules , because they explain how the function operates. The function rule is the same thing as the expression . Here are some hints for writing function rules:
::函数是当一个变量或术语根据规则取决于另一个变量或术语时。在函数的两个变量之间存在特殊关系,输入中的每个值仅适用于输出中的一个值。这些规则被称为函数规则,因为它们解释函数是如何运作的。函数规则与表达式相同。下面是写函数规则的一些提示:

Decipher the pattern of the function by asking, “What happened to the input to get the output?”
::通过询问“输入如何获得输出?” 来破解函数的图案,询问“输入是如何获得输出的?”

Write the rule as an expression.
::将规则写成表达方式。

Take a look at the following function rule and determine if it is a rule for the data in the table below.
::查看以下函数规则,并确定它是否是下表中数据的规则。

$\begin{aligned} x + 4 \end{aligned}$
::x+4x+4

Input	Output
2	5
3	6
4	7
5	8

First, substitute the input values in for $\begin{aligned} x \end{aligned}$ to see if you get the corresponding output value.
::首先, 替换 x 中的输入值, 以查看您是否得到了相应的输出值。

\begin{aligned} x + 4 \\ 2 + 4 \\ 6 \end{aligned}

This does not equal the corresponding output value of 5.
::这与相应的5个输出值不相等。

Look at the other input values. Each term in the input became the term in the output when 3 was added to it. The rule states that four was added. Therefore , this is not a viable rule.
::查看其他输入值。输入中的每个词在 3 添加到 3 时成为输出中的术语。规则指出增加了 4 个。因此, 这不是一条可行的规则。

Here is another function.
::这是另一个函数。

$\begin{aligned} 5 x \end{aligned}$
::5 x 5 x

Determine if $\begin{aligned} 5 x \end{aligned}$ is a function rule for the data in the table below.
::确定 5 x 是否是下表中数据的函数规则。

Input	Output
20	100
10	50
5	25
1	5

\begin{aligned} 5 x \\ 5 (20) \\ 100 \end{aligned}

Substitute another input value.
::替换另一个输入值。

\begin{aligned} 5 x \\ 5 (10) \\ 50 \end{aligned}

\begin{aligned} 5 x \\ 5 (5) \\ 25 \end{aligned}

\begin{aligned} 5 x \\ 5 (1) \\ 5 \end{aligned}

So, yes it is. In this case, each term in the input was multiplied by five to get the term in the output. Therefore this rule does work for this table.
::所以,是的。在这种情况下,输入中的每个术语乘以5 来获得输出中的术语。因此,这一规则对这个表格是有效的。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Geri and the change machine.
::早些时候,有人给了你一个问题关于Geri和变革机器。

Geri knows that there are 4 quarters in a dollar, and that is why she put 6 dollars in the machine and received 24 quarters. How can Geri write this as a function rule?
::Geri知道每美元有4/4美元,这就是为什么她把6美元放进机器并收到24/4美元。 Geri怎么写成这个功能规则呢?

This is a table to represent the change machine’s input and output.
::这是一张代表变革机器输入和输出的表格。

Input (dollars)	Output (quarters)
6	24
5	20
4	16
3	12

First, look at the table and ask yourself, “What happened to $\begin{aligned} x \end{aligned}$ (input) to get $\begin{aligned} y \end{aligned}$ (output)?”
::首先,看看桌子,问自己,“x(投入)为了获得(产出)发生了什么?”

What happened to 5 to get 50? What happened to 6 to get 60 and so forth? If you look carefully, you will see that the input value $\begin{aligned} (x) \end{aligned}$ is multiplied by 10 to get the output value.
::5 和 50 发生什么了? 6 和 60 等发生什么了? 如果您仔细看一看, 你会看到输入值( x) 乘以 10 以获得输出值。

Next, use a variable for the input and write the rule.
::下一步,使用变量进行输入并写入规则。

You can write it as an expression, $\begin{aligned} x (10) \end{aligned}$ or $\begin{aligned} 10 x \end{aligned}$ . This is the function rule, $\begin{aligned} 10 x \end{aligned}$ .
::您可以写成表达式, x (10) 或 10 x 。这是函数规则, 10 x 。

Then, see if the function rule $\begin{aligned} 10 x \end{aligned}$ works for each term in the table by plugging the input into the expression and seeing if it equals the listed output?
::然后, 看看函数规则 10 x 是否对表格中的每个词起作用, 将输入插入到表达式中, 并查看它是否等于列出的输出 ?

\begin{aligned} 10 x \\ 10 (5) \\ 50 \end{aligned}

\begin{aligned} 10 x \\ 10 (6) \\ 60 \end{aligned}

\begin{aligned} 10 x \\ 10 (7) \\ 70 \end{aligned}

\begin{aligned} 10 x \\ 10 (8) \\ 80 \end{aligned}

The answer is yes, this rule works for this table.
::答案是肯定的,这条规则适用于本表。

Example 2
::例2

Write a function rule to represent the data in this table.
::写入函数规则以代表本表格中的数据。

Input	Output
3	5
5	9
7	13
8	15
10	19

Here two operations were performed. The input value was multiplied by two and then one was subtracted.
::在此进行了两个操作。输入值乘以 2, 然后减去一个。

Next, use a variable for the input and write the rule.
::下一步,使用变量进行输入并写入规则。

$\begin{aligned} 2 (x) - 1 \end{aligned}$
::2 (xx) - 1

The answer or function rule is $\begin{aligned} 2 x - 1 \end{aligned}$ .
::答案或函数规则是 2 x - 1 。

Then, see if the function rule $\begin{aligned} 2 x - 1 \end{aligned}$ works for each term in the table by plugging the input into the expression and seeing if it equals the listed output.
::然后, 看看函数规则 2 x - 1 是否对表格中的每个词起作用, 将输入插入表达式, 并查看它是否等于列出的输出。

Substitute the input values in for $\begin{aligned} x \end{aligned}$ in the function $\begin{aligned} 2 x - 1 \end{aligned}$ to see if you get the results in the output column.
::将函数 2 x - 1 中的 x 的输入值替换为输入值,以查看是否在输出列中取得结果。

\begin{aligned} 2 x - 1 \\ 2 (3) - 1 \\ 6 - 1 \\ 5 (output) \end{aligned}

\begin{aligned} 2 x - 1 \\ 2 (5) - 1 \\ 10 - 1 \\ 9 (output) \end{aligned}

\begin{aligned} 2 x - 1 \\ 2 (7) - 1 \\ 14 - 1 \\ 13 (output) \end{aligned}

\begin{aligned} 2 x - 1 \\ 2 (8) - 1 \\ 16 - 1 \\ 15 (output) \end{aligned}

\begin{aligned} 2 x - 1 \\ 2 (10) - 1 \\ 20 - 1 \\ 19 (output) \end{aligned}

The answer is correct.
::答案是正确的。

Example 3
::例3

Determine whether the following rule makes sense for the input-output table.
::确定以下规则对于输入产出表是否合理。

Rule: $\begin{aligned} 4 x \end{aligned}$
::规则:第4条x

Input	Output
2	10
3	15
5	25
6	30

First, let the input value be the variable $\begin{aligned} x \end{aligned}$ .
::首先,输入值为变量 x 。

Next, substitute the input values in the expression for $\begin{aligned} x \end{aligned}$ .
::接下来,将表达式中的输入值替换为 x。

\begin{aligned} \begin{array}{rcl} 4 x & = & 10 (output) \\ 4 (2) & = & 10 \\ 8 & \neq & 10 \end{array} \end{aligned}

The answer is no, this rule does not work for this table.
::答案是否,这一规则对本表不适用。

Example 4
::例4

Determine whether the following rule makes sense for the input-output table.
::确定以下规则对于输入产出表是否合理。

Rule: $\begin{aligned} 2 x - 1 \end{aligned}$
::规则:规则2x-1

Input	Output
2	3
3	5
4	7
6	11

First, let the input value be the variable $\begin{aligned} x \end{aligned}$ .
::首先,输入值为变量 x 。

Next, substitute the input values in the expression for $\begin{aligned} x \end{aligned}$ .
::接下来,将表达式中的输入值替换为 x。

\begin{aligned} \begin{array}{rcl} 2 x - 1 & = & 3 (output) \\ 2 (2) - 1 & = & 3 \\ 4 - 1 & = & 3 \\ 3 & = & 3 \end{array} \end{aligned}

\begin{aligned} \begin{array}{rcl} 2 x - 1 & = & 5 \\ 2 (3) - 1 & = & 5 \\ 6 - 1 & = & 5 \\ 5 & = & 5 \end{array} \end{aligned}

\begin{aligned} \begin{array}{rcl} 2 x - 1 & = & 7 \\ 2 (4) - 1 & = & 7 \\ 8 - 1 & = & 7 \\ 7 & = & 7 \end{array} \end{aligned}

\begin{aligned} \begin{array}{rcl} 2 x - 1 & = & 11 \\ 2 (6) - 1 & = & 11 \\ 12 - 1 & = & 11 \\ 11 & = & 11 \end{array} \end{aligned}

The answer is yes, this rule does work for this table.
::答案是肯定的,这条规则对本表有效。

Example 5
::例5

Determine whether the following rule makes sense for the input-output table.
::确定以下规则对于输入产出表是否合理。

Rule: $\begin{aligned} 3 x \end{aligned}$
::规则:规则3x

Input	Output
2	6
3	9
4	12
6	18

First, let the input value be the variable $\begin{aligned} x \end{aligned}$ .
::首先,输入值为变量 x 。

Next, substitute the input values in the expression for $\begin{aligned} x \end{aligned}$ .
::接下来,将表达式中的输入值替换为 x。

\begin{aligned} \begin{array}{rcl} 3 x & = & 6 (output) \\ 3 (2) & = & 6 \\ 6 & = & 6 \end{array} \end{aligned}

\begin{aligned} \begin{array}{rcl} 3 x & = & 9 \\ 3 (3) & = & 9 \\ 9 & = & 9 \end{array} \end{aligned}

\begin{aligned} \begin{array}{rcl} 3 x & = & 12 \\ 3 (4) & = & 12 \\ 12 & = & 12 \end{array} \end{aligned}

\begin{aligned} \begin{array}{rcl} 3 x & = & 18 \\ 3 (6) & = & 18 \\ 18 & = & 18 \end{array} \end{aligned}

The answer is yes, this rule does work for this table.

::答案是肯定的,这条规则对本表有效。

Review
::回顾

Evaluate each given function rule to determine if the rule works for the data in the table.
::评价每个给定功能规则,以确定该规则是否适用于表中的数据。

$\begin{aligned} 2 x + 2 \end{aligned}$
::2 x + 2

Input	Output
2	6
3	8
4	10
5	12

$\begin{aligned} 3 x \end{aligned}$
::3 x 3 x

Input	Output
1	4
2	6
3	10

$\begin{aligned} 5 x + 1 \end{aligned}$
::5 x + 1 5 x + 1

Input	Output
1	6
2	11
3	16
4	21

$\begin{aligned} 2 x \end{aligned}$
::2 x 2 x 2 x

Input	Output
1	3
2	5
3	7

$\begin{aligned} 3 x - 1 \end{aligned}$
::3 x - 1 3 x - 1

Input	Output
1	2
2	5
3	8
4	11

$\begin{aligned} 2 x + 1 \end{aligned}$
::2 x + 1

Input	Output
1	3
2	4
3	6
5	10

$\begin{aligned} 4 x \end{aligned}$
::4 x 4 x

Input	Output
0	0
1	4
2	8
3	12

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

Input	Output
1	3
2	9
3	15

$\begin{aligned} 2 x \end{aligned}$
::2 x 2 x 2 x

Input	Output
0	0
1	2
2	4
3	6

$\begin{aligned} 3 x - 3 \end{aligned}$
::3 x - 3

Input	Output
1	0
2	3
4	9
5	12

Create a table for the following expressions. Use the input values: 0, 1, 2, 3, 4, 5.
::为以下表达式创建表格。使用输入值: 0, 1, 2, 3, 3, 4, 5。

$\begin{aligned} 5 x \end{aligned}$
::5 x 5 x

$\begin{aligned} 6 x + 1 \end{aligned}$
::6x1+1

$\begin{aligned} 2 x - 3 \end{aligned}$
::2 x − 3

$\begin{aligned} 3 x + 3 \end{aligned}$
::3 x + 3 3 x + 3

$\begin{aligned} 4 x + 1 \end{aligned}$
::4 x + 1 4 x + 1

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。

章节大纲

常规

输入输出表功能规则

Writing Function Rules for Input-Output Tables ::输入输出表的写写函数规则

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Writing Function Rules for Input-Output Tables
::输入输出表的写写函数规则

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源