BackPrecalculus Study Guide: Lines, Functions, Symmetry, Transformations, and Equations
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Lines and Their Equations
Finding the Equation of a Line
To determine the equation of a line, you need either two points on the line or one point and the slope.
Point-Slope Form: The equation of a line with slope passing through is .
Slope-Intercept Form: , where is the slope and is the y-intercept.
Standard Form: , where , , and are constants.
Example: Find the equation of the line through and .
First, calculate the slope: .
Equation: .
Parallel and Perpendicular Lines
Parallel Lines: Two lines are parallel if they have the same slope.
Perpendicular Lines: Two lines are perpendicular if the product of their slopes is .
Example: Find the equation of the line through parallel to .
Slope is (same as given line).
Use point-slope form: .
Simplify: .
Example: Find the equation of the line through perpendicular to .
Perpendicular slope: .
Equation: .
Simplify: .
Intercepts
x-intercept: Set and solve for .
y-intercept: Set and solve for .
Example: For :
x-intercept:
y-intercept:
Functions and Their Properties
Average Rate of Change
The average rate of change of a function from to is:
Example: For from to :
Average rate:
Solving for Variables
To solve for a variable, isolate it using algebraic operations.
Example: ; solve for :
Graphical Analysis
Continuity
A function is continuous at if:
is defined
exists
Example: Given a graph, check for holes, jumps, or asymptotes at the specified -values.
Symmetry
Even Function: for all in the domain. Graph is symmetric about the y-axis.
Odd Function: for all in the domain. Graph is symmetric about the origin.
Example: is even; is odd.
Transformations of Functions
Types of Transformations
Vertical Shifts: shifts up by units; shifts down by units.
Horizontal Shifts: shifts left by units; shifts right by units.
Reflections: reflects 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: is a shift left by 3 units.
Solving Absolute Value Equations and Inequalities
Absolute Value Equations
To solve , set and .
To solve , .
To solve , or .
Example: Solve :
(no solution, since absolute value cannot be negative).
Summary Table: Types of Symmetry
Type | Algebraic Test | Graphical Symmetry |
|---|---|---|
Even | y-axis | |
Odd | Origin | |
Neither | Fails both tests | No symmetry |
Summary Table: Function Transformations
Transformation | Equation | Description |
|---|---|---|
Vertical Shift Up | Up by units | |
Vertical Shift Down | Down by units | |
Horizontal Shift Left | Left by units | |
Horizontal Shift Right | Right by units | |
Reflection x-axis | Over x-axis | |
Reflection y-axis | Over y-axis | |
Vertical Stretch | , | Stretched vertically |
Vertical Compression | , | Compressed vertically |
Key Definitions
Function: A relation where each input has exactly one output.
Domain: The set of all possible input values (x-values).
Range: The set of all possible output values (y-values).
Intercept: The point where a graph crosses an axis.
Continuous Function: A function with no breaks, holes, or jumps in its graph.
Sample Problems and Solutions
Find the equation of the line through and : Slope: Equation:
Is even, odd, or neither? Not equal to or , so neither.
Solve :
Additional info: Some context and explanations have been expanded for clarity and completeness, as the original material was in exercise format.
