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Graphs, Functions, and Models > Transformations
Transformations
• Symmetry
• Even and Odd Functions
• Transformations of Functions
• Translations
• Reflections
• Stretching and Shrinking
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Graphs, Functions, and Models > Transformations
Symmetry
• If one object is obtained from the other by an invariant transformations, the two
objects are symmetric to each other.
• The value of the output is invariant under permutations of variables in the case of
symmetric functions.
• A binary operation is commutative if the operator, as a function of two variables, is
a symmetric function.
Sphere
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Graphs, Functions, and Models > Transformations
Even and Odd Functions
• The parity of a function does not necessarily reveal whether the function is odd or
even.
• An even function is symmetric about the y-axis.
• An odd function is symmetric with respect to 180-degree rotation about the origin.
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Graphs, Functions, and Models > Transformations
Transformations of Functions
• Transformations are ways that a function can be adjusted to create new functions.
• Transformations often preserve the original shape of the function.
• Common types of transformations include rotations, translations, reflections, and
scaling (also known as stretching/shrinking).
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Graphs, Functions, and Models > Transformations
Translations
• A translation is a function that moves every point a constant distance in a
specified direction.
• A vertical translation is generally given by the equation y = f(x) + b. These
translations shift the whole function up or down the y-axis.
• A horizontal translation is generally given by the equation y = f(x−a). These
translations shift the whole function side to side on the x-axis.
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Graphs, Functions, and Models > Transformations
Reflections
• A reflection swaps all of the x or y values across the y- or x-axis, respectively. It
can be visualized by imagining a mirror lies across that axis.
• A vertical reflection is given by the equation y = −f(x) and results in the curve
being "reflected" across the x-axis.
• A horizontal reflection is given by the equation y = f(−x) and results in the curve
being "reflected" across the y-axis.
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Graphs, Functions, and Models > Transformations
Stretching and Shrinking
• When by either f(x) or x is multiplied by a number, functions can "stretch" or
"shrink" vertically or horizontally, respectively, when graphed.
• In general, a vertical stretch is given by the equation y = pf(x). If p>1, the graph
stretches upward and downward. If p<1, the graph shrinks.
• In general, a horizontal stretch is given by the equation y = f(x/q). If q>1, the
graph shrinks horizontally, becoming more compact. If q<1, the graph stretches
horizontally.
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Appendix
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Graphs, Functions, and Models
Key terms
• additive inverse The inverse with respect to addition; the opposite.
• boolean-valued function A function of the type f : X → B, where X is an arbitrary set and where B is a boolean domain.
• commutative A binary operation is commutative if changing the order of the operands does not change the result, for example
addition and multiplication.
• euclidean space The three-dimensional space of Euclidean geometry. The term "Euclidean" distinguishes these spaces from
the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity.
• isometry A function between metric spaces (or on a single metric space) having the property that the distance between two
images is equal to the distance between their pre-images.
• parity A set with the property of having all of its elements belonging to one of two disjoint subsets, especially a set of integers
split in subsets of even and odd elements.
• quadratic A quadratic polynomial, function or equation.
• reflection a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points
• transformation The replacement of the variables in an algebraic expression by their values in terms of another set of variables;
a mapping of one space onto another or onto itself; a function that changes the position or direction of the axes of a coordinate
system.
• vector A directed quantity with both magnitude and direction.
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Graphs, Functions, and Models
Interactive Graph: Sine Function Stretching Vertically
Graph of sine function, , stretching vertically. Here, the basic sine function is shown in red while the blue function is stretched vertically by a factor of 3.
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Graphs, Functions, and Models
Sphere
In this sphere, φ is the longitude and θ the colatitude. The radius is the green dotted line. The notation of |0⟩, |1⟩, and |ψ may be unfamiliar (this is BraKet notation), but none of the lengths depicted by 0, 1 and ψ is essential to the discussion of symmetry.
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Graphs, Functions, and Models
Interactive Graph: Graph of an Even Function
Graph of is an even function. The order of x is even (2), and thus is symmetrical with respect to the vertical axis.
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Graphs, Functions, and Models
Interactive Graph: Basic Quadratic Equation Vertical Translations
The graph shows the function with the original basic function in blue. The green and red functions are representations of the function translated up or
down by two on the y-axis. Try shifting the graph vertically.
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Graphs, Functions, and Models
Interactive Graph: Basic Quadratic Equation Horizontal Translations
The graph shows the function with the original basic function in blue. The green and red functions are representations of the function translated up or
down by two on the x-axis. Try shifting the graph horizontally.
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Graphs, Functions, and Models
Interactive Graph: Graph of an Odd Function
Graph of is an odd function. The order of x is odd (3) and thus is symmetrical when rotated 180 degrees about the origin.
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Graphs, Functions, and Models
Interactive Graph: Reflected Functions
Graph of reflected function for . The green line represents the original function, while the purple and orange lines are vertical and horizontal reflections,
respectively.
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Graphs, Functions, and Models
Interactive Graph: Basic Quadratic Equation
Graph of three different y=x2 equations. The blue curve indicates the original function while the red and green curves represent two vertical translations
of that function. Try to move the graph vertically. For more of a challenge, try to move the graph horizontally.
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Graphs, Functions, and Models
Which of the following capital letters has line symmetry?
A) J
B) L
C) X
D) P
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Graphs, Functions, and Models
Which of the following capital letters has line symmetry?
A) J
B) L
C) X
D) P
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Graphs, Functions, and Models
Which of the following capital letters _______as rotational
symmetry?
A) G
B) H
C) V
D) A
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Graphs, Functions, and Models
Which of the following capital letters _______as rotational
symmetry?
A) G
B) H
C) V
D) A
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Graphs, Functions, and Models
Which of the following functions is odd?
A) f(x)=6x5-x3
B) f(x)=3x4-4x3
C) f(x)=5x4+3x2
D) f(x)=x6+8x2
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Graphs, Functions, and Models
Which of the following functions is odd?
A) f(x)=6x5-x3
B) f(x)=3x4-4x3
C) f(x)=5x4+3x2
D) f(x)=x6+8x2
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Graphs, Functions, and Models
Which of the following functions is even?
A) f(x)=x4+x2
B) f(x)=x5+x3
C) f(x)=x4+x3
D) f(x)=x3+x
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Graphs, Functions, and Models
Which of the following functions is even?
A) f(x)=x4+x2
B) f(x)=x5+x3
C) f(x)=x4+x3
D) f(x)=x3+x
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Graphs, Functions, and Models
Which of the following types of transformations results in an
image that is not that same size as the original image?
A) Rotation
B) Reflection
C) Translation
D) Scaling
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Graphs, Functions, and Models
Which of the following types of transformations results in an
image that is not that same size as the original image?
A) Rotation
B) Reflection
C) Translation
D) Scaling
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Graphs, Functions, and Models
Which of the following does NOT always stay the same during a
transformation (but it does in some cases)?
A) Orientation of the vertices
B) Angle measures
C) Parallelism
D) Length of a side
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Graphs, Functions, and Models
Which of the following does NOT always stay the same during a
transformation (but it does in some cases)?
A) Orientation of the vertices
B) Angle measures
C) Parallelism
D) Length of a side
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Graphs, Functions, and Models
Starting with the function f(x)=x4+2, which of the following
functions shifts the function UP 3 units?
A) f(x)=(x+3)4+2
B) f(x)=x4+3
C) f(x)=x4+5
D) f(x)=(x-3)4+2
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Graphs, Functions, and Models
Starting with the function f(x)=x4+2, which of the following
functions shifts the function UP 3 units?
A) f(x)=(x+3)4+2
B) f(x)=x4+3
C) f(x)=x4+5
D) f(x)=(x-3)4+2
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Graphs, Functions, and Models
Starting with the function f(x)=x3-6, which of the following
functions shifts the function to the LEFT by 4 units?
A) f(x)=(x-3)3-6
B) f(x)=x3-2
C) f(x)=(x+3)3-6
D) f(x)=x3+4
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Graphs, Functions, and Models
Starting with the function f(x)=x3-6, which of the following
functions shifts the function to the LEFT by 4 units?
A) f(x)=(x-3)3-6
B) f(x)=x3-2
C) f(x)=(x+3)3-6
D) f(x)=x3+4
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Graphs, Functions, and Models
Which equation shows what _______ would look like reflected in
the y-axis?
A) y=x2-3
B) y=-x2-3
C) y=-x2+3
D) y=x2+3
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Graphs, Functions, and Models
Which equation shows what _______ would look like reflected in
the y-axis?
A) y=x2-3
B) y=-x2-3
C) y=-x2+3
D) y=x2+3
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Graphs, Functions, and Models
Which equation shows what y=x2+3 would look like reflected in
the x-axis?
A) y=x2+3
B) y=-x2-3
C) y=x2-3
D) y=-x2+3
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Graphs, Functions, and Models
Which equation shows what y=x2+3 would look like reflected in
the x-axis?
A) y=x2+3
B) y=-x2-3
C) y=x2-3
D) y=-x2+3
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Graphs, Functions, and Models
The point (-4,5) is reflected over the line y=x. What is the result?
A) (4,-5)
B) (-5,4)
C) (5,-4)
D) (-4,-5)
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Graphs, Functions, and Models
The point (-4,5) is reflected over the line y=x. What is the result?
A) (4,-5)
B) (-5,4)
C) (5,-4)
D) (-4,-5)
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Graphs, Functions, and Models
Which of the following functions would shrink y=cosx vertically by
a factor of 2?
A) y=cos(x/2)
B) y=2cosx
C) y=cos(2x)
D) y=(cosx)/2
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Graphs, Functions, and Models
Which of the following functions would shrink y=cosx vertically by
a factor of 2?
A) y=cos(x/2)
B) y=2cosx
C) y=cos(2x)
D) y=(cosx)/2
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Graphs, Functions, and Models
Which of the following functions would stretch y=cosx horizontally
by a factor of 4?
A) y=4cosx
B) y=(cosx)/4
C) y=cos(x/4)
D) y=cos(4x)
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Graphs, Functions, and Models
Which of the following functions would stretch y=cosx horizontally
by a factor of 4?
A) y=4cosx
B) y=(cosx)/4
C) y=cos(x/4)
D) y=cos(4x)
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Graphs, Functions, and Models
Attribution
• Wiktionary. "quadratic." CC BY-SA 3.0 http://en.wiktionary.org/wiki/quadratic
• Wikidot. "Transformations - IB Math Stuff." CC BY-SA http://ibmathstuff.wikidot.com/transformations
• Wikipedia. "Symmetry in mathematics." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Symmetry_in_mathematics
• Wikipedia. "Symmetry in mathematics." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Symmetry_in_mathematics
• Wikipedia. "boolean-valued function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/boolean-valued%20function
• Wiktionary. "commutative." CC BY-SA 3.0 http://en.wiktionary.org/wiki/commutative
• Wikidot. "Transformations - IB Math Stuff." CC BY-SA http://ibmathstuff.wikidot.com/transformations
• Wikibooks. "Algebra/Absolute Value." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Absolute_Value#Translations
• Wikipedia. "Translation (geometry)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Translation_(geometry)
• Wiktionary. "vector." CC BY-SA 3.0 http://en.wiktionary.org/wiki/vector
• Wikipedia. "Even and odd functions." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Even_and_odd_functions
• Wiktionary. "additive inverse." CC BY-SA 3.0 http://en.wiktionary.org/wiki/additive+inverse
• Wiktionary. "parity." CC BY-SA 3.0 http://en.wiktionary.org/wiki/parity
• Wikipedia. "Transformation (function)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Transformation_(function)
• Wikidot. "Transformations - IB Math Stuff." CC BY-SA http://ibmathstuff.wikidot.com/transformations
• Wikipedia. "euclidean space." CC BY-SA 3.0 http://en.wikipedia.org/wiki/euclidean%20space
• Wikidot. "Transformations - IB Math Stuff." CC BY-SA http://ibmathstuff.wikidot.com/transformations
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Graphs, Functions, and Models
• Wikipedia. "Reflection (mathematics)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Reflection_(mathematics)
• Wiktionary. "reflection." CC BY-SA 3.0 http://en.wiktionary.org/wiki/reflection
• Wiktionary. "isometry." CC BY-SA 3.0 http://en.wiktionary.org/wiki/isometry
• Wiktionary. "transformation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/transformation
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