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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%%ANKI
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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END%%

%%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).
<!--ID: 1731184865794-->
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).
<!--ID: 1731184865801-->
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).
<!--ID: 1731185808236-->
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).
<!--ID: 1731185808246-->
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).
<!--ID: 1731185808249-->
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%%

## Transformations

There are two kinds of transformations 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 transformations 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 transformations 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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END%%

%%ANKI
Cloze
{1:Adding} is to {2:shifting} as {2:multiplying} is to {1: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
Cloze
{1:Dividing} is to {2:scaling} as {2:subtracting} is to {1:shifting}.
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: 1733522558480-->
END%%

%%ANKI
Basic
Given function $f$, which of $a$, $b$, $c$, and/or $d$ relate to shifting? $$y = af(b[x - c]) + d$$
Back: $c$ and $d$
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: 1734620943896-->
END%%

%%ANKI
Basic
Given function $f$, which of $a$, $b$, $c$, and/or $d$ relate to a horizontal transformation? $$y = af(b[x - c]) + d$$
Back: $b$ and $c$
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: 1734620943904-->
END%%

%%ANKI
Basic
Given function $f$, which of $a$, $b$, $c$, and/or $d$ relate to a vertical transformation? $$y = af(b[x - c]) + d$$
Back: $a$ and $d$.
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: 1734620943910-->
END%%

%%ANKI
Basic
Given function $f$, which of $a$, $b$, $c$, and/or $d$ relate to scaling? $$y = af(b[x - c]) + d$$
Back: $a$ and $b$.
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: 1734620943916-->
END%%

%%ANKI
Basic
Given function $f$, which of $a$, $b$, $c$, and/or $d$ apply a transformation that "acts normally"? $$y = af(b[x - c]) + d$$
Back: $a$ and $d$.
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: 1734620943921-->
END%%

%%ANKI
Basic
Given function $f$, which of $a$, $b$, $c$, and/or $d$ apply a transformation that "acts inversely"? $$y = af(b[x - c]) + d$$
Back: $b$ and $c$.
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: 1734620943926-->
END%%

%%ANKI
Basic
Which of horizontal and/or vertical transformations "act inversely"?
Back: Horizontal transformations.
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: 1734620943931-->
END%%

%%ANKI
Basic
Which of horizontal and/or vertical transformations "act normally"?
Back: Vertical transformations.
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: 1734620943937-->
END%%

%%ANKI
Basic
How is the graph of $y = f(x)$ transformed in the graph of $y = \frac{1}{2}f(\frac{x}{3})$?
Back: It's horizontally scaled by $3$ and vertically compressed by $2$.
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: 1734656174535-->
END%%

%%ANKI
Basic
Consider $y = f(x)$. Is $y = af(bx)$ first scaled vertically or horizontally?
Back: N/A. There is no order of operations in this case.
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: 1734656174588-->
END%%

%%ANKI
Basic
How is the graph of $y = f(x)$ transformed in the graph of $y = f(x - 3) + 2$?
Back: It's shifted right by $3$ and shifted up by $2$.
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: 1734656174596-->
END%%

%%ANKI
Basic
Consider $y = f(x)$. Is $y = f(x + a) + b$ first shifted vertically or horizontally?
Back: N/A. There is no order of operations in this case.
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: 1734656174604-->
END%%

%%ANKI
Basic
How is the graph of $y = f(x)$ transformed in the graph of $y = 2f(x) + 5$?
Back: It's vertically scaled by $2$ and then shifted up by $5$.
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: 1734656174610-->
END%%

%%ANKI
Basic
Consider $y = f(x)$. Is $y = af(x) + b$ first scaled or shifted?
Back: Scaled.
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: 1734656174615-->
END%%

%%ANKI
Basic
Consider $y = f(x)$. Is $y = f(ax) + b$ first scaled or shifted?
Back: Scaled.
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: 1734656174619-->
END%%

%%ANKI
Basic
How is the graph of $y = f(x)$ transformed in the graph of $y = f(2x - 3)$?
Back: It's horizontally compressed by $2$ and then shifted right by $3/2$.
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: 1734656174624-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $2f(x - 3) - 5$?
Back: $(1, 8)$
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: 1734657098882-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the range of $2f(x - 3) - 5$?
Back: $[3, 11]$
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: 1734657098887-->
END%%

### Shifting

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.

%%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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END%%

%%ANKI
Basic
Which of the two fundamental graph transformations is considered "rigid"?
Back: Shifts.
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: 1733522558483-->
END%%

%%ANKI
Cloze
A {vertical} shift adds/subtracts 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).
<!--ID: 1734278286937-->
END%%

%%ANKI
Cloze
A {horizontal} shift adds/subtracts 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
Cloze
A {horizontal} shift corresponds to adding/subtracting the {input} of a function.
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: 1733522558486-->
END%%

%%ANKI
Cloze
A {vertical} shift corresponds to adding/subtracting the {output} of a function.
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: 1733522558489-->
END%%

%%ANKI
Basic
Let $f(x)$ be a function and $k$ be a constant. What kind of transformations 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 transformations 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).
<!--ID: 1733520215023-->
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).
<!--ID: 1733520215027-->
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).
<!--ID: 1733520215031-->
END%%

%%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%%

%%ANKI
Basic
Horizontal shifting of a function affects which of its range and/or domain?
Back: The domain.
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: 1734656938097-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(x - 2)$?
Back: $(0, 7)$
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: 1734656938102-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(x + 2)$?
Back: $(-4, 3)$
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: 1734656938105-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(x) + 2$?
Back: $(-2, 5)$
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: 1734656938109-->
END%%

%%ANKI
Basic
Vertical shifting of a function affects which of its range and/or domain?
Back: The range.
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: 1734656938112-->
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%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(x - 2)$?
Back: $[4, 8]$
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: 1734656938116-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(x) - 2$?
Back: $[2, 6]$
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: 1734656938120-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(x) + 2$?
Back: $[6, 10]$
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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### Scaling

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.

Scaling is also known as **stretching** and **compressing**.

%%ANKI
Basic
What does it mean for a scaling of a graph to be non-rigid?
Back: A scaling changes the size and/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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END%%

%%ANKI
Cloze
A {vertical} scaling multiplies/divides 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).
<!--ID: 1734299397501-->
END%%

%%ANKI
Cloze
A {horizontal} scaling multiplies/divides 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: 1733522558499-->
END%%

%%ANKI
Basic
Let $f(x)$ be a function and $k$ be a constant. What kind of transformation is $kf(x)$?
Back: A vertical 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).
<!--ID: 1733522558503-->
END%%

%%ANKI
Basic
Let $f(x)$ be a function and $k$ be a constant. What kind of transformation is $f(kx)$?
Back: A horizontal 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).
<!--ID: 1733522558507-->
END%%

%%ANKI
Cloze
A {vertical} scaling corresponds to multiplying/dividing the {output} of a function.
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: 1733522558513-->
END%%

%%ANKI
Cloze
A {horizontal} scaling corresponds to multiplying/dividing the {input} of a function.
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: 1733522558519-->
END%%

%%ANKI
Cloze
Let $f(x)$ be a function and $k$ be a constant. $f(kx)$ is horizontally {stretched} when {$0 < \lvert k \rvert < 1$}.
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: 1733522558523-->
END%%

%%ANKI
Cloze
Let $f(x)$ be a function and $k$ be a constant. $kf(x)$ is vertically {stretched} when {$\lvert k \rvert > 1$}.
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: 1733522558528-->
END%%

%%ANKI
Cloze
Let $f(x)$ be a function and $k$ be a constant. $kf(x)$ is vertically {compressed} when {$0 < \lvert k \rvert < 1$}.
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: 1733522558532-->
END%%

%%ANKI
Cloze
Let $f(x)$ be a function and $k$ be a constant. $f(kx)$ is horizontally {compressed} when {$\lvert k \rvert > 1$}.
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: 1733522558536-->
END%%

%%ANKI
Cloze
Let $f(t)$ be a function of time. Vertically compressing $f(t)$ means it takes {more} time to reach a value.
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: 1733522558540-->
END%%

%%ANKI
Cloze
Let $f(t)$ be a function of time. Horizontally compressing $f(t)$ means it takes {less} time to reach a value.
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: 1733522558545-->
END%%

%%ANKI
Cloze
Let $f(t)$ be a function of time. Vertically stretching $f(t)$ means it takes {less} time to reach a value.
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: 1733522558550-->
END%%

%%ANKI
Cloze
Let $f(t)$ be a function of time. Horizontally stretching $f(t)$ means it takes {more} time to reach a value.
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: 1733522558555-->
END%%

%%ANKI
Basic
Horizontal scaling of a function affects which of its range and/or domain?
Back: The domain.
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: 1734656938129-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(2x)$?
Back: $(-1, 5/2)$
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: 1734656938133-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $2f(x)$?
Back: $(-2, 5)$
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: 1734656938137-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(x / 2)$?
Back: $(-4, 10)$
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: 1734656938141-->
END%%

%%ANKI
Basic
Vertical scaling of a function affects which of its range and/or domain?
Back: The range.
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: 1734656938145-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(2x)$?
Back: $[4, 8]$
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: 1734656938149-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the range of $2f(x)$?
Back: $[8, 16]$
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: 1734656938152-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the range of $f(x) / 2$?
Back: $[2, 4]$
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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### Reflecting

A **reflection** is a special case of a [[#Scaling|scaling]].

%%ANKI
Basic
A reflection is a special case of what other kind of transformation?
Back: A 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).
<!--ID: 1734620943942-->
END%%

%%ANKI
Cloze
To reflect a function's graph about the {$y$}-axis, multiply its {inputs} by $-1$.
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: 1734620943947-->
END%%

%%ANKI
Cloze
To reflect a function's graph about the {$x$}-axis, multiply its {outputs} by $-1$.
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: 1734620943953-->
END%%

%%ANKI
Basic
Consider the graph of $f(x)$. Is $f(-x)$ its horizontal or vertical reflection?
Back: Horizontal (reflection along the $y$-axis).
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: 1734656938160-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the domain of $f(-x)$?
Back: $(-5, 2)$
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: 1734656938163-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has domain $(-2, 5)$. What is the range of $f(-x)$?
Back: The same as that of $f(x)$.
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: 1734656938167-->
END%%

%%ANKI
Basic
Consider the graph of $f(x)$. Is $-f(x)$ its horizontal or vertical reflection?
Back: Vertical (reflection along the $x$-axis).
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: 1734656938171-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the domain of $-f(x)$?
Back: The same as that of $f(x)$.
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: 1734656938174-->
END%%

%%ANKI
Basic
Suppose $f(x)$ has range $[4, 8]$. What is the range of $-f(x)$?
Back: $[-8, -4]$
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: 1734656938178-->
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## 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).