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Questions and Answers
Which representation of functions is considered the most compact way?
Which representation of functions is considered the most compact way?
What characteristic defines piecewise functions?
What characteristic defines piecewise functions?
What type of function would be represented by a linear piecewise function?
What type of function would be represented by a linear piecewise function?
Which of the following best describes power functions where n is even?
Which of the following best describes power functions where n is even?
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How do vertical shifts affect the graph of a function?
How do vertical shifts affect the graph of a function?
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In composite functions, what is crucial to determine?
In composite functions, what is crucial to determine?
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What is the effect of a horizontal scale transformation on a graph?
What is the effect of a horizontal scale transformation on a graph?
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What do odd power functions exhibit in terms of symmetry?
What do odd power functions exhibit in terms of symmetry?
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Which type of function is characterized by different rules for different parts of its domain?
Which type of function is characterized by different rules for different parts of its domain?
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What effect does a reflection across the x-axis have on a function's graph?
What effect does a reflection across the x-axis have on a function's graph?
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Which representation of functions offers the most detailed visual insight?
Which representation of functions offers the most detailed visual insight?
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In the context of transformations, what is a vertical shift?
In the context of transformations, what is a vertical shift?
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Power functions where n is odd will exhibit which of the following characteristics?
Power functions where n is odd will exhibit which of the following characteristics?
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What is typically the format of representing more complex functions compactly?
What is typically the format of representing more complex functions compactly?
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Which type of function can be defined in segments that can be linear, such as tax brackets?
Which type of function can be defined in segments that can be linear, such as tax brackets?
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Which transformation is responsible for stretching or compressing a graph vertically?
Which transformation is responsible for stretching or compressing a graph vertically?
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Study Notes
Representing Functions
- Words: Least mathematical way to define functions.
- Table: Collection of data obtained from experiments.
- Graph: Provides the most illustrative representation.
- Formula: Most compact way to represent functions.
Representing Functions using Graphs
- Linear Functions: Straight line graph.
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Piecewise Functions: Different definitions based on domain partitions.
- Examples include tax brackets with varying tax rates.
- If all pieces are linear, the function is piecewise linear.
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Power and Root Functions:
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Power Functions: y = x^n
- Even n: Graph symmetric about the y-axis.
- Odd n: Graph symmetric about the origin.
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Root Functions: y= x^(1/n)
- Even n: Graph is in the first and second quadrants.
- Odd n: Graph is in all four quadrants.
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Power Functions: y = x^n
- Rational Functions: Functions defined as the ratio of two polynomials.
Representing Functions using Formulas
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Composite Functions: A function applied to the result of another function.
- Notation: f(g(x))
- Domain: The domain of the composite function is limited by the inner function's range and the outer function's domain.
Graph Transformation
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Vertical Shift: y = f(x) + c
- Shifts the graph c units upward if c is positive.
- Shifts the graph c units downward if c is negative.
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Horizontal Shift: y = f(x - c)
- Shifts the graph c units right if c is positive.
- Shifts the graph c units left if c is negative.
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Vertical Scale: y = cf(x)
- Stretches the graph vertically if c > 1.
- Compresses the graph vertically if 0 < c < 1.
- Reflects the graph across the x-axis if c < 0.
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Horizontal Scale: y = f(cx)
- Compresses the graph horizontally if c > 1.
- Stretches the graph horizontally if 0 < c < 1.
- Reflects the graph across the y-axis if c < 0.
Representing Functions
- Functions can be represented in various ways:
- Words: A descriptive, less mathematical approach.
- Table: Shows data collected through experiments.
- Graph: Offers the most visual understanding.
- Formula: The most concise way to express many functions.
Linear Functions
- Linear functions are characterized by a constant rate of change.
- Their graph is a straight line.
Piecewise Functions
- These functions have different definitions based on specific parts of their domain.
- An example is income tax, where different tax brackets have varying rates.
- A piecewise linear function has linear segments for each part of its definition.
Power and Root Functions
- Power Functions:
- When the exponent (n) is even, the graph is symmetrical about the y-axis.
- When the exponent (n) is odd, the graph is symmetrical about the origin.
- Root Functions:
- When the root (n) is even, the function is defined only for non-negative inputs.
- When the root (n) is odd, the function is defined for all real numbers.
Formula Representation
- Formulas offer a concise way to represent functions.
- Common examples include:
- f(x) = x^2 (quadratic function)
- f(x) = sin(x) (sine function)
- f(x) = ln(x) (natural logarithm function)
Composite Functions
- A composite function is created by substituting one function into another.
- The output of the inner function becomes the input of the outer function.
- Notation: f(g(x)) or f ◦ g(x)
Domain of Composite Functions
- The domain of a composite function is limited by the inner function's output, which must be included in the outer function's domain.
Graph Transformation
- Graph transformations involve altering the shape, position, or orientation of a function's graph.
- Types of transformations:
- Vertical Shift: Moving the graph up or down by adding a constant to the function.
- Horizontal Shift: Moving the graph left or right by subtracting/adding a constant inside the function.
- Vertical Scale & Reflection: Scaling the graph vertically by multiplying by a constant; a negative constant reflects the graph across the x-axis.
- Horizontal Scale & Reflection: Scaling the graph horizontally by dividing by a constant; a negative constant reflects the graph across the y-axis.
Book & Lecture Links
- Access to lecture materials and the textbook is provided through a Google Drive link.
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Description
This quiz explores various methods for representing functions including through words, tables, graphs, and formulas. It covers linear, piecewise, power, root, and rational functions along with their unique characteristics. Test your understanding of these concepts and enhance your mathematical skills.
