Shadowing-Übung: Relations and Functions | Algebra - Englisch Sprechen Lernen mit Video
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in this video we're going to focus on relations and functions
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so what is a relation a relation is a set of pairs of input
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and output values here
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we have three ordered pairs in the first relation on the
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left the x value is the input value the y value
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is the output value the x values is associated with the
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domain of the relation the y values is associated with the range of the relation
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so now let's focus on part a list the domain
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and range of each relation so let's start with the domain
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so what we're gonna do is we're gonna make a list of all of the x values
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and I'm going to write it in ascendant order.
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So first we have negative 3 and then 0 and 2.
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Now let's write the range of that relation.
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So we're going to focus on the y values and it's already listed in ascendant order.
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So 1, 4, and 5.
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Now, let's do the same thing for the other relation.
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So, let's write out the domain.
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So, the lowest x value is negative 2.
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Next is 1 and then 3.
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Now, let's write out the range of that relation.
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The lowest y value is negative 2.
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And then it's 3, 4, and 7.
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So that's how you can write out the domain and range of each relation.
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Now how can we determine if the relation is a function?
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In order for the relation to be a function, every input value must have only one output value.
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If an input value corresponds to two or more output values, that relation is not a function.
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Now, let's focus on the first relation.
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So we have the ordered pair 2 -1.
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The input value is 2, the output value is 1.
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And then negative 3, 4.
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So negative 3 corresponds to 4, and then 0 corresponds to 5.
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So for the first relation, we can see that for every input value, there's only one output value.
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Now let's focus on the second relation.
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So we have the ordered pair 1, 3.
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Next is negative 2, 4.
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And then it's 3, negative 2.
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And then finally, negative 2, 7.
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So for the second relation, notice that negative 2 corresponds to two different output values.
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Now that's a problem.
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If you put in an input value of negative 2, should the output be 4 or 7?
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So whenever you have that situation, you know that relation is not a function.
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The first one is a function.
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Every input value corresponds to an output value, just one output value.
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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.
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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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I want to take a minute to talk about my website video -tutor .net.
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It's a very simple website, not too complicated,
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but for those of you who want to be notified anytime I release specialized content in the form of a video,
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an ebook, an article.
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It could be a digital course or a podcast.
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If you want to be notified, feel free to join the email list.
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And once you confirm your email, you're going to get access to a page that has all of my playlists listed on it.
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And this includes my final exam videos and also my test prep videos.
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So feel free to join the email list when you get a chance.
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And let's get back to the video.
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Now let's move on to the next example.
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Draw a mapping diagram of each relation shown below.
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So let's start with the relation on the left.
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We're going to map out the domain and the range.
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So for the domain, we have the values negative 2, 1, and 3.
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For the range, we have the y values, negative 6, 0, and 4.
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Now, negative two corresponds to zero,
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one corresponds to four, three corresponds to negative six.
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So for every input value on the domain side, there's one corresponding output value on to range aside.
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So this is a relation.
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I mean this relation is a function.
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So the answer is yes for the first relation.
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Now let's move on to the second relation.
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So let's create a map and diagram as well.
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So let's start with the domain
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The lowest x value is negative 2
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Next we have 0
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And then the last one is 3
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Now looking at the y values
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The lowest one is negative 1, and then it's going to be 1, 2, and 5.
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So negative 2 corresponds to positive 1.
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0 corresponds to 5.
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3 corresponds to 2.
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and 0 corresponds to negative 1.
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So just by seeing the repeat x values that we see here, we could tell that this is not going to be a function.
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The two x values have two different y values.
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You can see 0 points to negative 1 and 5.
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So the second relation is not a function.
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Now for this one, what we're going to do is we're going to draw a function table of the relation.
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And then we're going to determine if the relation is a function.
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So in this table, we're going to list the input values next to the output values.
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The input values represent the x values.
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The output values represent the y values.
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so the input values it corresponds to the domain and the output values corresponds to the range
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so the lowest input value that we have is negative three next is one and then we have another one
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and then after that is it's three and five
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Now, for this function table, I'm going to write the input value twice, because that's what we have here.
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When writing out the domain and the range, for repeat values, we would write repeat values once.
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Now, negative 3 corresponds to 2.
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For the table, these numbers need to match, so I'm not going to list the output values in ascending order.
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Now, for this one, we could use either one.
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So, I'm going to use 1, 2 for the next one, and then 1, 4.
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Now, when x is 3, y is 7.
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And when x is 5, y is negative 4.
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So, that's the function table and because we have two identical x values that correspond to two different y values,
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we know that this relation is not a function.
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So that's it for this problem.
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When you have a graph, the best way to determine if the graph represents a function is to use the vertical line test.
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And that's what we're going to do in this problem.
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So any way you draw a vertical line for the first graph, notice that the line only touches the graph at one point.
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Therefore, the answer is yes, it represents a function.
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For the next one on the right, if we draw a vertical line, notice at this point, or placing a line at that location,
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we have three points of intersection between the graph and the vertical line.
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If we can get two or more points on a vertical line, then the relation is not a function.
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So we're going to say no. Now, if we put the vertical line here, notice that we have five points on that line.
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So this relation is not a function.
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For the next relation, it doesn't matter where we put the graph.
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We will only get, I mean, it doesn't matter where we put the vertical line we're only going to get one point.
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If we put it here, it's only going to touch the line once.
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So we can't draw a vertical line where it touches two points.
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Therefore, this relation represents a function.
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For the circle, if we put the line here, we can get two points of intersection.
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so we're going to say yes i mean no not yes
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this is no the circle does not represent a function
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it does not pass the vertical line test
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it touches the line at two points in order for it
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to pass the vertical line test the graph must touch the line only at one point
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as we saw in these two cases.
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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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Wortschatz und Sprechhinweise zu dieser Lektion
Diese Sprechübung basiert auf dem Video „Relations and Functions“. Diese Wörter kommen am häufigsten vor: value, relation, function, negative, correspond. Dieses Video enthält 124 Sätze und 1357 Wörter zum Nachsprechen. Der gesprochene Teil dauert 12:25. Der Sprecher spricht langsam, etwa 109 Wörter pro Minute, sodass genug Zeit bleibt, jeden Laut nachzuahmen. 83 % der Wörter gehören zu den 3.000 häufigsten im Englischen; den Rest solltest du dir vor dem Üben ansehen.
Wichtiger Wortschatz in diesem Video
15 lernenswerte Wörter aus dem Video, mit Aussprache und Bedeutung:
| Wort | Aussprache | Bedeutung |
|---|---|---|
| relation Substantiv | /ɹɪˈleɪʃ(ə)n/ | Beziehung |
| correspond Verb | /ˌkoɹəˈspɑnd/ | entsprechen, korrespondieren |
| output Substantiv | /ˈaʊtpʊt/ | Output |
| domain Substantiv | /ˌdəʊˈmeɪ̯n/ | Domäne, Gebiet |
| vertical Adjektiv | /ˈvɝ.tɪ.kəl/ | vertikal, senkrecht |
| graph Substantiv | /ɡɹæf/ | Graph, Funktionsgraph |
| associate Substantiv | /əˈsoʊʃi.ət/ | Weggefährte |
| intersection Substantiv | /ˈɪntəɹˌsɛkʃən/ | Kreuzung, Straßenkreuzung |
| diagram Substantiv | /ˈdaɪ.ə.ɡɹæm/ | Diagramm |
| notify Verb | /ˈnoʊtɪfaɪ/ | mitteilen, benachrichtigen |
| complicated Adjektiv | /ˈkɑm.plɪˌkeɪ.tɪd/ | kompliziert, umständlich |
| confirm Verb | /kənˈfɝm/ | bestätigen |
| examination Substantiv | /ɪɡˌzæm.ɪˈneɪ.ʃən/ | Prüfung, Untersuchung |
| identical Adjektiv | /ɪˈdɛntɪkl̩/ | identisch |
| anytime Adverb | jederzeit, immer |
Phrasal Verbs, die du hören wirst
| Wort | Bedeutung |
|---|---|
| write out Verb | ausschreiben |
| get back to Verb | sich melden |
Aussprache, auf die du achten solltest
Der Sprecher verwendet 23 Kurzformen und abgeschwächte Formen, zum Beispiel we're, I'm, doesn't. Sprich sie in der kurzen Form, so wie du sie hörst.
- Die Laute „sh“ und „zh“: relation /ɹɪˈleɪʃ(ə)n/, associate /əˈsoʊʃi.ət/, intersection /ˈɪntəɹˌsɛkʃən/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, preposition /ˌpɹɛpəˈzɪʃən/
- Lange Wörter – auf die Betonung achten: associate /əˈsoʊʃi.ət/, intersection /ˈɪntəɹˌsɛkʃən/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, preposition /ˌpɹɛpəˈzɪʃən/
Laute, die Deutschsprachigen schwerfallen:
- Stimmhafter Auslaut — /b/, /d/, /g/, /z/, /v/ am Wortende nicht verhärten: correspond /ˌkoɹəˈspɑnd/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, ascend /əˈsɛnd/, specialize /ˈspɛʃəˌlaɪz/
- /æ/ — offener als „ä“: graph /ɡɹæf/, diagram /ˈdaɪ.ə.ɡɹæm/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, podcast /ˈpɑdˌkæst/
So übst du mit diesem Video
- Höre dir das ganze Video einmal an, ohne zu sprechen, und notiere die Wörter, die du nicht kennst.
- Sprich Satz für Satz in normaler Geschwindigkeit nach und wiederhole jeden, bis dein Rhythmus zum Sprecher passt.
- Nimm dich auf und vergleiche mit dem Original; achte dabei auf Wörter wie relation, correspond, output.
Was ist die Shadowing-Technik?
Shadowing ist eine wissenschaftlich fundierte Sprachlerntechnik, die ursprünglich für die professionelle Dolmetscherausbildung entwickelt und durch den Polyglotten Dr. Alexander Arguelles populär gemacht wurde. Die Methode ist einfach aber wirkungsvoll: Du hörst englisches Audio von Muttersprachlern und wiederholst es sofort laut — wie ein Schatten, der dem Sprecher mit nur 1–2 Sekunden Verzögerung folgt. Anders als passives Hören oder Grammatikübungen zwingt Shadowing dein Gehirn und deine Mundmuskulatur, gleichzeitig echte Sprachmuster zu verarbeiten und zu reproduzieren. Studien zeigen, dass es Aussprachegenauigkeit, Intonation, Rhythmus, verbundene Sprache, Hörverständnis und Sprechflüssigkeit signifikant verbessert — was es zu einer der effektivsten Methoden für die IELTS Speaking-Vorbereitung und reale englische Kommunikation macht.