Graphs, Functions, and Models

Distance and Midpoint Between Points

Understanding the geometric relationship between points is fundamental in algebra and analytic geometry.

• Distance Formula: The distance between two points and is given by:

• Midpoint Formula: The midpoint between two points and is:

• Example: Find the distance and midpoint between and . Distance: Midpoint:

Slope and Equations of Lines

The slope of a line measures its steepness and direction.

• Slope Formula: For points and :

• Point-Slope Form:

• Slope-Intercept Form:

• Example: Find the equation of the line passing through and . Slope: Equation:

Functions and Their Properties

Linear, Quadratic, and Polynomial Functions

Functions describe relationships between variables. Linear and quadratic functions are foundational in algebra.

• Linear Function:

• Quadratic Function:

• Polynomial Function:

• Applications: Used to model real-world phenomena such as projectile motion (quadratic) or population growth (polynomial).

Composition of Functions

The composition of functions involves applying one function to the result of another.

• Definition:

• Example: If and , then

Inverse Functions

An inverse function reverses the effect of the original function.

• Definition:

• Finding the Inverse: Solve for in terms of , then interchange and .

• Example: So,

Quadratic Equations and Inequalities

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula:

• Example: Solve Factoring:

Solving Inequalities

Inequalities can be solved using algebraic manipulation and by analyzing sign charts.

• Example: Solve

Polynomial and Rational Functions

Domain and Range

The domain is the set of all possible input values; the range is the set of possible output values.

• Finding Domain: Exclude values that make the denominator zero or result in even roots of negative numbers.

• Example: has domain

Asymptotes

Asymptotes are lines that a graph approaches but never touches.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is nonzero.

• Horizontal Asymptotes: Determined by the degrees of the numerator and denominator.

• Example: has a vertical asymptote at and a horizontal asymptote at

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions model growth and decay processes.

• General Form:

• Example: models doubling growth.

Logarithmic Functions

Logarithms are the inverses of exponential functions.

• Definition:

• Properties:

• Example: because

Solving Exponential and Logarithmic Equations

• Example: Solve because

• Example: Solve because

Systems of Equations and Matrices

Solving Systems of Equations

Systems of equations can be solved using substitution, elimination, or matrix methods.

• Two Variables: Solve Add: , then

• Three Variables: Use elimination or matrices to solve systems with three unknowns.

Applications of Systems

• Applied Problems: Systems can model mixtures, investments, and other real-world scenarios.

Additional Topics

Growth and Decay Applications

Exponential models are used for population growth, radioactive decay, and half-life problems.

• Half-Life Formula: where is the half-life.

• Example: If grams and the half-life is 5 years, after 10 years: grams

Summary Table: Key Concepts and Methods

Additional info: This guide covers all major topics listed in the exam review outline, providing definitions, formulas, and examples for each. For more advanced applications, consult your textbook or instructor.