Add missing scaling flashcards.
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"_journal/2024-11/2024-11-08.md": "806bbade5f8339579287687f9433334e",
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"x86-64/directives.md": "019c1c1d04efb26c3e8758aac4543cc7",
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"geometry/cartesian.md": "662f241b94c78dfb8ac1e7fd0c1120e8",
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"geometry/cartesian.md": "67d6920732d1ab7dd72e12fba28cf8bc",
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"geometry/index.md": "cac68c1b624dbb0552e56cce47bcc21d",
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"_journal/2024-11/2024-11-09.md": "46f3a640223ef533f4523837b67b57c3",
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@ -996,7 +996,7 @@
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"_journal/2024-12/2024-12-03.md": "54480a38f1e16e48529cbb99c5349c74",
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"_journal/2024-12-05.md": "efc79d779a0b0354fc0e8658b074a693",
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"_journal/2024-12/2024-12-04.md": "965f6619edf1002d960203e3e12a413b",
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"_journal/2024-12-06.md": "8ab4dc393cc7d68012b47e28029fd16d",
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"_journal/2024-12-06.md": "d75323d0fec57f4fc1f13cb4370df18d",
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"_journal/2024-12/2024-12-05.md": "4f3b1e7a43e01cc97b0eed6fbc6c1f96",
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"calculus/integrals.md": "a7ef5031ca474cd9d37c1aea85e96237"
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},
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@ -9,4 +9,4 @@ title: "2024-12-06"
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- [ ] Korean (Read 1 Story)
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* Begin notes on [[integrals]], starting with those of step functions.
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* Notes on [[cartesian#Shifting|shifting]] graphs.
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* Notes on [[cartesian#Shifting|shifting]] and [[cartesian#Scaling|scaling]] graphs.
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@ -151,9 +151,23 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces
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<!--ID: 1733520269110-->
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END%%
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%%ANKI
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Cloze
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{1:Adding} is to {2:shifting} as {2:multiplying} is to {1:scaling}.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558471-->
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END%%
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%%ANKI
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Cloze
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{1:Dividing} is to {2:scaling} as {2:subtracting} is to {1:shifting}.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558480-->
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END%%
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### Shifting
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A **vertical shift** adds a constant to every $y$-coordinate of a graph, leaving the $x$-coordinate unchanged. A **horizontal shift** adds a constant to every $x$-coordinate of a graph, leaving the $y$-coordinate unchanged.
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A **vertical shift** adds/subtracts a constant to every $y$-coordinate of a graph, leaving the $x$-coordinate unchanged. A **horizontal shift** adds/subtracts a constant to every $x$-coordinate of a graph, leaving the $y$-coordinate unchanged.
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%%ANKI
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Basic
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@ -165,18 +179,40 @@ END%%
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%%ANKI
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Basic
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A {vertical} shift adds a constant to the {$y$}-coordinates of a graph.
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Which of the two fundamental graph translations is considered "rigid"?
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Back: Shifts.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558483-->
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END%%
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%%ANKI
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Basic
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A {vertical} shift adds/subtracts a constant to the {$y$}-coordinates of a graph.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733520214996-->
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END%%
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%%ANKI
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Cloze
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A {horizontal} shift adds a constant to the {$x$}-coordinates of a graph.
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A {horizontal} shift adds/subtracts a constant to the {$x$}-coordinates of a graph.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733520215005-->
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END%%
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%%ANKI
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Cloze
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A {horizontal} shift corresponds to adding/subtracting the {input} of a function.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558486-->
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END%%
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%%ANKI
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Cloze
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A {vertical} shift corresponds to adding/subtracting the {output} of a function.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558489-->
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END%%
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%%ANKI
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Basic
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Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $f(x + k)$?
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@ -275,6 +311,120 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces
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<!--ID: 1733520215056-->
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END%%
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### Scaling
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A **vertical scaling** will multiply/divide every $y$-coordinate of a graph, leaving the $x$-coordinate unchanged. A **horizontal scaling** will multiply/divide every $x$-coordinate of a graph, leaving the $y$-coordinate unchanged.
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Scaling is also known as **stretching** and **compressing**.
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%%ANKI
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Basic
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What does it mean for a scaling of a graph to be non-rigid?
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Back: A scaling changes the size and/or shape of the graph.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558492-->
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END%%
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%%ANKI
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Basic
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A {vertical} scaling multiplies/divides the {$y$}-coordinates of a graph.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558496-->
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END%%
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%%ANKI
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Cloze
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A {horizontal} scaling multiplies/divides the {$x$}-coordinates of a graph.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558499-->
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END%%
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%%ANKI
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Basic
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Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $kf(x)$?
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Back: A vertical scaling.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558503-->
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END%%
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%%ANKI
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Basic
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Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $f(kx)$?
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Back: A horizontal scaling.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558507-->
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END%%
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%%ANKI
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Cloze
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A {vertical} scaling corresponds to multiplying/dividing the {output} of a function.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558513-->
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END%%
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%%ANKI
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Cloze
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A {horizontal} scaling corresponds to multiplying/dividing the {input} of a function.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558519-->
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END%%
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%%ANKI
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Cloze
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Let $f(x)$ be a function and $k$ be a constant. $f(kx)$ is horizontally {stretched} when {$0 < \lvert k \rvert < 1$}.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558523-->
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END%%
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%%ANKI
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Cloze
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Let $f(x)$ be a function and $k$ be a constant. $kf(x)$ is vertically {stretched} when {$\lvert k \rvert > 1$}.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558528-->
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END%%
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%%ANKI
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Cloze
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Let $f(x)$ be a function and $k$ be a constant. $kf(x)$ is vertically {compressed} when {$0 < \lvert k \rvert < 1$}.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558532-->
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END%%
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%%ANKI
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Cloze
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Let $f(x)$ be a function and $k$ be a constant. $f(kx)$ is horizontally {compressed} when {$\lvert k \rvert > 1$}.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558536-->
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END%%
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%%ANKI
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Cloze
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Let $f(t)$ be a function of time. Vertically compressing $f(t)$ means it takes {more} time to reach a value.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558540-->
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END%%
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%%ANKI
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Cloze
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Let $f(t)$ be a function of time. Horizontally compressing $f(t)$ means it takes {less} time to reach a value.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558545-->
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END%%
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%%ANKI
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Cloze
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Let $f(t)$ be a function of time. Vertically stretching $f(t)$ means it takes {less} time to reach a value.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558550-->
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END%%
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%%ANKI
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Cloze
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Let $f(t)$ be a function of time. Horizontally stretching $f(t)$ means it takes {more} time to reach a value.
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Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
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<!--ID: 1733522558555-->
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END%%
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## Bibliography
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* “Cartesian Coordinate System,” in _Wikipedia_, October 21, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system](https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system&oldid=1252434514).
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