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| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Cartesian Coordinate System | Obsidian::STEM | geometry::coordinates |
|
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.
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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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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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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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%%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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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.
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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).
END%%
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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%%
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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).
END%%
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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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%%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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%%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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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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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%%
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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).
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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.
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%%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.
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%%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.
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%%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%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
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.
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%%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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
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.
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%%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.
END%%
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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%%
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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.
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%%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%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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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.
END%%
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Basic
Consider the graph of f(x) below. What does f(x) equal?
!
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%%
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Basic
Consider the graph of f(x) below. What does f(x) equal?
!
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?
!
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%%
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Basic
Consider the graph of f(x) below. What does f(x) equal?
!
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?
!
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?
!
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 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.
END%%
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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.
END%%
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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.
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.
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.
END%%
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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.
END%%
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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.
END%%
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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.
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%%
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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%%
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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.
END%%
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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.
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.
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%%
%%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.
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.
END%%
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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.
END%%
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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.
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.
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.
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.
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.
END%%
Reflecting
A reflection is a special case of a #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.
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.
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.
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.
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.
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.
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.
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.
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.
END%%
Bibliography
- “Cartesian Coordinate System,” in Wikipedia, October 21, 2024, https://en.wikipedia.org/w/index.php?title=Cartesian_coordinate_system.
- “James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, 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).