Questions and Answers
What defines a function?
What is the independent variable in a function?
The input or x-value.
What is the dependent variable in a function?
The output or y-value.
What is the domain of a function?
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What is the range of a function?
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What does it mean to evaluate a function?
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Function notation is an inefficient way to write functions in mathematics.
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What is a constant function?
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What is a linear function?
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What is a quadratic function?
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What is a cubic function?
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What is an exponential function?
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The graph of an absolute value function looks like a V.
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What is discrete data?
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What is continuous data?
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What are transformations in graphing?
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What is a reflection in graphing transformations?
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What is a dilation in graphing transformations?
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What is a horizontal shift in graphing transformations?
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What is a vertical shift in graphing transformations?
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Study Notes
Functions
- A function pairs each element of the domain (input/x) with one element of the range (output/y).
- Functions can be represented using verbal descriptions, tables, graphs, or equations.
Variables
- Independent Variable: The input or x-value of a function.
- Dependent Variable: The output or y-value that relies on the input.
Function Characteristics
- Domain: All possible input values (x) for a function.
- Range: All possible output values (y) for a function.
Evaluation and Notation
- Evaluate: Process to find the output value for a specified input.
- Function Notation: Written as f(x) instead of y, representing functions in a concise manner.
Parts of an Equation
- Variable: A symbol (typically a letter) representing a number.
- Coefficient: A number multiplied by a variable.
- Constant: A number that remains unchanged.
- Operator: Symbol indicating mathematical operations (addition, subtraction, etc.).
- Exponent (Power): Indicates how many times a base is multiplied by itself.
Types of Functions
- Parent Function: The simplest form of a function from which other functions are derived.
- Constant Function: Maintains the same output regardless of input; represented as a horizontal line.
- Linear Function: Relationship where input equals output; represented as y = x.
- Quadratic Function: Involves the square of the input; produces a U-shaped graph known as a parabola (y = x²).
- Cubic Function: Involves the cube of the input.
- Square Root Function: Derives output from the square root of the input, undefined for negative values.
- Absolute Value Function: Measures distance from zero; graph appears as a V shape (y = |x|).
- Exponential Function: Base raised to a variable power, with the input as the exponent.
- Rational Function: Involves fractions; undefined for an input of zero (cannot divide by zero).
Graphing Concepts
- Graph: A representation of ordered pairs (input/output values) forming a line or curve on the coordinate plane.
- Ordered Pair: A duo of numbers that identifies a point on the coordinate plane.
- Coordinate Plane: Divided into four quadrants by the x-axis (horizontal) and y-axis (vertical).
Data Types
- Discrete Data: Can only take specific values; for example, counting items like the number of students.
- Continuous Data: Can take any value within a range; for example, heights of individuals.
Transformations
- Alterations to a parent graph include:
- Reflection: Flips the graph across a line.
- Dilation: Changes the size of the graph, either enlarging or shrinking.
- Horizontal Shift: Moves the graph left or right.
- Vertical Shift: Moves the graph up or down.
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