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 $\langle 0, 0 \rangle$.
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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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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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).
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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).
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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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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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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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## Cartesian Equations
An equation that completely characters a figure within the Cartesian coordinate system is called a **Cartesian equation**.
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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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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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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).
There are two kinds of translations that we can do to a graph: **shifting** and **scaling**. A **reflection** is a special case of scaling.
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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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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).
{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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{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).
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.
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).
Which of the two fundamental graph translations 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).
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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).
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).
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).
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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).
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).
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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).
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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).
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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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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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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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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).
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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).
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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).
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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).
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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).
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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).
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**.
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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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Basic
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).
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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).
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Basic
Let $f(x)$ be a function and $k$ be a constant. What kind of translation 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).
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Let $f(x)$ be a function and $k$ be a constant. What kind of translation 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).
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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).
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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).
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Cloze
Let $f(x)$ be a function and $k$ be a constant. $f(kx)$ is horizontally {stretched} when {$0 < \lvertk \rvert<1$}.
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).
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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).
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Cloze
Let $f(x)$ be a function and $k$ be a constant. $kf(x)$ is vertically {compressed} when {$0 < \lvertk \rvert<1$}.
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).
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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).
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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).
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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).
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%%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).
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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).
* “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).
