notebook/notes/geometry/cartesian.md

---
title: Cartesian Coordinate System
TARGET DECK: Obsidian::STEM
FILE TAGS: geometry::coordinates
tags:
  - geometry
---

## Overview

In plane analytic geometry, the **Cartesian coordinate system** uniquely specifies a point by a pair of real numbers called its **coordinates**. These coordinates represent signed distances to the point from two fixed perpendicular oriented lines called the **axes**. The point where the axes meet is called the **origin** and have coordinates $\langle 0, 0 \rangle$.

%%ANKI
Cloze
The {$x$-coordinate} of a point is sometimes called its {abscissa}.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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Cloze
The {$y$-coordinate} of a point is sometimes called its {ordinate}.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Basic
What is an ordinate set?
Back: A set bounded by the $x$-axis and the graph of a nonnegative function.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Basic
An ordinate set is bounded below by what?
Back: The $x$-axis, i.e. $y = 0$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Basic
An ordinate set is bounded above by what?
Back: The graph of a nonnegative function.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Cloze
The {origin} of a Cartesian coordinate system has coordinates $\langle 0, 0 \rangle$.
Reference: “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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%%ANKI
Basic
Consider point $\langle x, y \rangle$. When does this point lie in the first quadrant?
Back: When $x > 0$ and $y > 0$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Basic
Consider point $\langle x, y \rangle$. When does this point lie in the second quadrant?
Back: When $x < 0$ and $y > 0$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1731184865796-->
END%%

%%ANKI
Basic
Consider point $\langle x, y \rangle$. When does this point lie in the fourth quadrant?
Back: When $x > 0$ and $y < 0$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1731184865799-->
END%%

%%ANKI
Basic
Consider point $\langle x, y \rangle$. When does this point lie in the third quadrant?
Back: When $x < 0$ and $y < 0$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Basic
The "vertical line test" of a Cartesian coordinate system is used to determine what?
Back: Whether the tested graph depicts a function or not.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Basic
In Cartesian coordinate systems, why does the vertical line test work?
Back: A function is single-valued. A vertical line that intersects a graph multiple times immediately contradicts this.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

## Cartesian Equations

An equation that completely characters a figure within the Cartesian coordinate system is called a **Cartesian equation**.

%%ANKI
Basic
What is a Cartesian equation?
Back: An equation that completely characterizes a figure within the Cartesian coordinate system.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Basic
What is the Cartesian equation of a circle centered around the origin with radius $r$?
Back: $x^2 + y^2 = r^2$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

%%ANKI
Basic
What figure does the following Cartesian equation characterize? $x^2 + y^2 = r^2$
Back: A circle with radius $r$ centered around the origin.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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END%%

## Translations

There are two kinds of translations that we can do to a graph: **shifting** and **scaling**. A **reflection** is a special case of scaling.

%%ANKI
Basic
What are the two kinds of translations that can be done to a graph?
Back: Shifting and scaling.
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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END%%

%%ANKI
Basic
Which of the two kinds of translations is reflection a special case of?
Back: Scaling.
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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### Shifting

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.

%%ANKI
Basic
What does it mean for a shift of a graph to be rigid?
Back: A shift does not change the size or shape of the graph.
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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%%ANKI
Basic
A {vertical} shift adds a constant to the {$y$}-coordinates of a graph.
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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END%%

%%ANKI
Cloze
A {horizontal} shift adds a constant to the {$x$}-coordinates of a graph.
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).
<!--ID: 1733520215005-->
END%%

%%ANKI
Basic
Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $f(x + k)$?
Back: A horizontal shift.
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).
<!--ID: 1733520215013-->
END%%

%%ANKI
Basic
Let $f(x)$ be a function and $k$ be a constant. What kind of translation is $f(x) + k$?
Back: A vertical shift.
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).
<!--ID: 1733520215016-->
END%%

%%ANKI
Cloze
Let $f(x)$ be a function and $k$ be a constant. $f(x + k)$ horizontally shifts {left} when {$k > 0$}.
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).
<!--ID: 1733520215019-->
END%%

%%ANKI
Cloze
Let $f(x)$ be a function and $k$ be a constant. $f(x) + k$ vertically shifts {down} when {$k < 0$}.
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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END%%

%%ANKI
Cloze
Let $f(x)$ be a function and $k$ be a constant. $f(x) - k$ vertically shifts {up} when {$k > 0$}.
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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END%%

%%ANKI
Cloze
Let $f(x)$ be a function and $k$ be a constant. $f(x + k)$ horizontally shifts {right} when {$k < 0$}.
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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%%ANKI
Basic
Consider the graph of $f(x)$ below. What does $f(x)$ equal?
![[abs-right.png]]
Back: $f(x) = \lvert x - 2 \rvert$
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).
<!--ID: 1733520215034-->
END%%

%%ANKI
Basic
Consider the graph of $f(x)$ below. What does $f(x)$ equal?
![[abs-left.png]]
Back: $f(x) = \lvert x + 2 \rvert$
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).
<!--ID: 1733520215040-->
END%%

%%ANKI
Basic
Consider the graph of $f(x)$ below. What does $f(x)$ equal?
![[abs-up.png]]
Back: $f(x) = \lvert x \rvert + 2$
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).
<!--ID: 1733520215046-->
END%%

%%ANKI
Basic
Consider the graph of $f(x)$ below. What does $f(x)$ equal?
![[abs-down.png]]
Back: $f(x) = \lvert x \rvert - 2$
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).
<!--ID: 1733520215049-->
END%%

%%ANKI
Basic
Consider the graph of $f(x)$ below. What does $f(x)$ equal?
![[abs-right-down.png]]
Back: $f(x) = \lvert x - 2 \rvert - 2$
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).
<!--ID: 1733520215053-->
END%%

%%ANKI
Basic
Consider the graph of $f(x)$ below. What does $f(x)$ equal?
![[abs-left-down.png]]
Back: $f(x) = \lvert x + 2 \rvert - 2$
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).
<!--ID: 1733520215056-->
END%%

## Bibliography

* “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).
* “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).
* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).