跟讀練習: Relations and Functions | Algebra - 通過影片學習英語口語

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in this video we're going to focus on relations and functions

so what is a relation a relation is a set of pairs of input

and output values here

we have three ordered pairs in the first relation on the

left the x value is the input value the y value

is the output value the x values is associated with the

domain of the relation the y values is associated with the range of the relation

so now let's focus on part a list the domain

and range of each relation so let's start with the domain

so what we're gonna do is we're gonna make a list of all of the x values

and I'm going to write it in ascendant order.

So first we have negative 3 and then 0 and 2.

Now let's write the range of that relation.

So we're going to focus on the y values and it's already listed in ascendant order.

So 1, 4, and 5.

Now, let's do the same thing for the other relation.

So, let's write out the domain.

So, the lowest x value is negative 2.

Next is 1 and then 3.

Now, let's write out the range of that relation.

The lowest y value is negative 2.

And then it's 3, 4, and 7.

So that's how you can write out the domain and range of each relation.

Now how can we determine if the relation is a function?

In order for the relation to be a function, every input value must have only one output value.

If an input value corresponds to two or more output values, that relation is not a function.

Now, let's focus on the first relation.

So we have the ordered pair 2 -1.

The input value is 2, the output value is 1.

And then negative 3, 4.

So negative 3 corresponds to 4, and then 0 corresponds to 5.

So for the first relation, we can see that for every input value, there's only one output value.

Now let's focus on the second relation.

So we have the ordered pair 1, 3.

Next is negative 2, 4.

And then it's 3, negative 2.

And then finally, negative 2, 7.

So for the second relation, notice that negative 2 corresponds to two different output values.

Now that's a problem.

If you put in an input value of negative 2, should the output be 4 or 7?

So whenever you have that situation, you know that relation is not a function.

The first one is a function.

Every input value corresponds to an output value, just one output value.

So a quick way to look at a relation to see if it's not going to be a function, look for repeated x values.

If you see two x values that are the same but correspond to two different y values, then you know the relation is not a function.

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And this includes my final exam videos and also my test prep videos.

So feel free to join the email list when you get a chance.

And let's get back to the video.

Now let's move on to the next example.

Draw a mapping diagram of each relation shown below.

So let's start with the relation on the left.

We're going to map out the domain and the range.

So for the domain, we have the values negative 2, 1, and 3.

For the range, we have the y values, negative 6, 0, and 4.

Now, negative two corresponds to zero,

one corresponds to four, three corresponds to negative six.

So for every input value on the domain side, there's one corresponding output value on to range aside.

So this is a relation.

I mean this relation is a function.

So the answer is yes for the first relation.

Now let's move on to the second relation.

So let's create a map and diagram as well.

So let's start with the domain

The lowest x value is negative 2

Next we have 0

And then the last one is 3

Now looking at the y values

The lowest one is negative 1, and then it's going to be 1, 2, and 5.

So negative 2 corresponds to positive 1.

0 corresponds to 5.

3 corresponds to 2.

and 0 corresponds to negative 1.

So just by seeing the repeat x values that we see here, we could tell that this is not going to be a function.

The two x values have two different y values.

You can see 0 points to negative 1 and 5.

So the second relation is not a function.

Now for this one, what we're going to do is we're going to draw a function table of the relation.

And then we're going to determine if the relation is a function.

So in this table, we're going to list the input values next to the output values.

The input values represent the x values.

The output values represent the y values.

so the input values it corresponds to the domain and the output values corresponds to the range

so the lowest input value that we have is negative three next is one and then we have another one

and then after that is it's three and five

Now, for this function table, I'm going to write the input value twice, because that's what we have here.

When writing out the domain and the range, for repeat values, we would write repeat values once.

Now, negative 3 corresponds to 2.

For the table, these numbers need to match, so I'm not going to list the output values in ascending order.

Now, for this one, we could use either one.

So, I'm going to use 1, 2 for the next one, and then 1, 4.

Now, when x is 3, y is 7.

And when x is 5, y is negative 4.

100

So, that's the function table and because we have two identical x values that correspond to two different y values,

101

we know that this relation is not a function.

102

So that's it for this problem.

103

When you have a graph, the best way to determine if the graph represents a function is to use the vertical line test.

104

And that's what we're going to do in this problem.

105

So any way you draw a vertical line for the first graph, notice that the line only touches the graph at one point.

106

Therefore, the answer is yes, it represents a function.

107

For the next one on the right, if we draw a vertical line, notice at this point, or placing a line at that location,

108

we have three points of intersection between the graph and the vertical line.

109

If we can get two or more points on a vertical line, then the relation is not a function.

110

So we're going to say no. Now, if we put the vertical line here, notice that we have five points on that line.

111

So this relation is not a function.

112

For the next relation, it doesn't matter where we put the graph.

113

We will only get, I mean, it doesn't matter where we put the vertical line we're only going to get one point.

114

If we put it here, it's only going to touch the line once.

115

So we can't draw a vertical line where it touches two points.

116

Therefore, this relation represents a function.

117

For the circle, if we put the line here, we can get two points of intersection.

118

so we're going to say yes i mean no not yes

119

this is no the circle does not represent a function

120

it does not pass the vertical line test

121

it touches the line at two points in order for it

122

to pass the vertical line test the graph must touch the line only at one point

123

as we saw in these two cases.

124

So that's how you can use the vertical line test to determine if a relation represented by a graph is a function.

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本課的詞彙與口語要點

這節口語課以影片“Relations and Functions”為素材。影片中反覆出現的詞有：value, relation, function, negative, correspond。這段影片共有 124 個句子、1357 個單詞可供跟讀。講話部分時長為 12:25。說話人語速較慢，每分鐘約 109 個詞，你有足夠的時間模仿每一個音。 83% 的單詞屬於英語最常用的 3,000 詞，其餘的詞建議在練習前先學一下。

影片中的重點詞彙

影片中 15 個值得學習的單詞，附發音和釋義：

單詞	發音	釋義
relation 名詞	/ɹɪˈleɪʃ(ə)n/	關係 /關係
correspond 動詞	/ˌkoɹəˈspɑnd/	對應 /對應
output 名詞	/ˈaʊtpʊt/	輸出 /輸出
input 名詞	/ˈɪnpʊt/	輸入 /輸入
domain 名詞	/ˌdəʊˈmeɪ̯n/	領域 /領域
vertical 形容詞	/ˈvɝ.tɪ.kəl/	豎的 /豎的, 垂直
graph 名詞	/ɡɹæf/	圖表 /圖表
associate 名詞	/əˈsoʊʃi.ət/	同行, 同事
intersection 名詞	/ˈɪntəɹˌsɛkʃən/	路口, 交叉
diagram 名詞	/ˈdaɪ.ə.ɡɹæm/	圖 /圖, 圖表 /圖表
notify 動詞	/ˈnoʊtɪfaɪ/	通知, 通告
complicated 形容詞	/ˈkɑm.plɪˌkeɪ.tɪd/	複雜 /複雜
confirm 動詞	/kənˈfɝm/	證實 /證實, 確認 /確認
examination 名詞	/ɪɡˌzæm.ɪˈneɪ.ʃən/	檢查 /檢查
identical 形容詞	/ɪˈdɛntɪkl̩/	相同, 同一

需要注意的發音

說話人用了 23 次縮略和弱讀形式，例如 we're, I'm, doesn't。請按聽到的簡短形式來說。

“sh” 和 “zh” 音: relation /ɹɪˈleɪʃ(ə)n/, associate /əˈsoʊʃi.ət/, intersection /ˈɪntəɹˌsɛkʃən/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, preposition /ˌpɹɛpəˈzɪʃən/
長單詞——注意重音位置: associate /əˈsoʊʃi.ət/, intersection /ˈɪntəɹˌsɛkʃən/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, preposition /ˌpɹɛpəˈzɪʃən/

中文母語者容易讀錯的音：

詞尾子音 — 要讀清楚，後面不要加母音: correspond /ˌkoɹəˈspɑnd/, output /ˈaʊtpʊt/, input /ˈɪnpʊt/, graph /ɡɹæf/, associate /əˈsoʊʃi.ət/
/æ/ — 嘴張大，不要讀成 /e/: graph /ɡɹæf/, diagram /ˈdaɪ.ə.ɡɹæm/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, podcast /ˈpɑdˌkæst/

如何用這段影片練習

先完整聽一遍影片，不要開口，記下不認識的單詞。
用正常速度逐句跟讀，每句重複到你的節奏與說話人一致為止。
錄下自己的聲音並與原聲對比，特別注意 relation, correspond, output 這類單詞。

什麼是跟讀法？

跟讀法 (Shadowing) 是一種有科學依據的語言學習技巧，最初開發用於專業口譯員的培訓，並由多語言者Alexander Arguelles博士普及。這個方法簡單而強大：您在聽英語母語原聲的同時立即大聲重複——就像是一個延遲1-2秒緊跟說話者的影子。與被動聽力或語法練習不同，跟讀法強迫您的大腦和口腔肌肉同時處理並模模擬實的講話模式。研究表明它能顯著提高發音準確性，語調，節奏，連讀，聽力理解和口語流利度——使其成為雅思口語備考和真實英語交流最有效的方法之一。

影子跟讀法: 閱讀完整分步指南 →

跟讀練習: Relations and Functions | Algebra - 通過影片學習英語口語

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本課的詞彙與口語要點

影片中的重點詞彙

需要注意的發音

如何用這段影片練習

什麼是跟讀法？