BackPrecalculus Study Notes: Functions, Graphs, and Rates of Change
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Functions and Their Representations
Definition of a Function
A function is a relation that assigns to each element in a set called the domain exactly one element in a set called the range.
Notation: If is a function, denotes the output when the input is .
Example: assigns to each the value .
Ways to Represent Functions
Algebraic: Using an equation, e.g., .
Graphical: Plotting points on a coordinate plane.
Tabular: Listing input-output pairs in a table.
Verbal: Describing the relationship in words.
Linear Functions
A linear function has the form , where is the slope and is the y-intercept.
Slope (): Measures the rate of change; calculated as .
Y-intercept (): The value of when .
Example: is a linear function with slope and y-intercept $3$.
Graphing and Analyzing Functions
Graphing Linear Functions
Plot the y-intercept .
Use the slope to find another point: from , move up/down units and right 1 unit.
Draw a straight line through the points.
Interpreting Graphs
Increasing Function: The graph rises as increases.
Decreasing Function: The graph falls as increases.
Constant Function: The graph is a horizontal line; for all .
Example: is constant; is increasing; is decreasing.
Key Features of Graphs
Intercepts: Points where the graph crosses the axes.
Domain: All possible input values (-values).
Range: All possible output values (-values).
Symmetry: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.
Rates of Change
Average Rate of Change
The average rate of change of a function from to is given by:
Represents the slope of the secant line connecting and .
Example: For , the average rate of change from to is .
Instantaneous Rate of Change
The instantaneous rate of change at a point is the slope of the tangent line at that point. In calculus, this is the derivative, but in precalculus, we estimate it by considering values very close to the point of interest.
For linear functions, the instantaneous rate of change is constant and equal to the slope .
For nonlinear functions, it varies depending on .
Transformations of Functions
Types of Transformations
Vertical Shifts: shifts the graph up by units if , down if .
Horizontal Shifts: shifts the graph right by units if , left if .
Reflections: reflects the graph over the x-axis; reflects over the y-axis.
Vertical Stretch/Compression: stretches if , compresses if .
Horizontal Stretch/Compression: compresses horizontally if , stretches if .
Example of Transformation
Given , is shifted right by 2 units and up by 3 units.
Special Types of Functions
Piecewise Functions
A piecewise function is defined by different expressions for different intervals of the domain.
Example:
if
if
Absolute Value Function
The absolute value function is defined as .
Graph is V-shaped, with vertex at the origin.
Can be written as a piecewise function:
if
if
Tables: Summary of Function Types and Properties
Function Type | General Form | Graph Shape | Rate of Change |
|---|---|---|---|
Linear | Straight line | Constant () | |
Quadratic | Parabola | Variable | |
Absolute Value | V-shape | Piecewise constant | |
Piecewise | Varies | Varies | Varies |
Additional info:
Some content was inferred from context due to unclear handwriting and fragmented notes. Standard definitions and examples were added for completeness.
Key terms such as "rate of change," "domain," and "range" were expanded for clarity.
