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

END%%

%%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).

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).

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).

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).

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.

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).

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).

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).

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).

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).

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).

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).

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).

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).

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.

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.

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.

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.

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.

END%%

%%ANKI Basic 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

END%%

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.

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.

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.

END%%

%%ANKI 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.

END%%

%%ANKI Basic 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

END%%

Bibliography