Equations and Inequalities

Solving Linear and Quadratic Equations

Solving equations is a foundational skill in algebra, involving finding the value(s) of the variable that make the equation true.

• Linear Equations: Equations of the form .

• Quadratic Equations: Equations of the form .

• Factoring, quadratic formula, and completing the square are common methods for solving quadratics.

• Example: Solve using the quadratic formula:

Solving Rational and Radical Equations

• Rational Equations: Equations involving fractions with polynomials in the numerator and denominator.

• Radical Equations: Equations involving roots, such as square roots.

• Example: Solve by finding a common denominator.

Functions and Graphs

Definition and Properties of Functions

A function is a relation in which each input has exactly one output. Functions can be represented by equations, tables, or graphs.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Even/Odd Functions: Even if , odd if .

• Example: is even; is odd.

Graphing Functions

• Plot points by substituting values for and finding corresponding values.

• Identify intercepts, asymptotes, and general shape.

• Example: Graph by plotting several points and drawing the parabola.

Transformations of Functions

• Vertical and horizontal shifts, reflections, and stretches/compressions.

• Example: is shifted right 2 units and up 1 unit.

Polynomial and Rational Functions

Polynomial Functions

• Functions of the form .

• Degree determines the end behavior and number of possible real roots.

• Example: is a cubic polynomial.

Rational Functions

• Functions of the form where .

• Vertical asymptotes occur where ; horizontal asymptotes depend on the degrees of and .

• Example: has vertical asymptotes at and .

Exponential and Logarithmic Functions

Exponential Functions

• Functions of the form .

• Used to model growth and decay, such as population or compound interest.

• Example: models population growth.

Logarithmic Functions

• Inverse of exponential functions: means .

• Properties: , , .

• Example: Solve for .

Systems of Equations and Inequalities

Solving Systems of Linear Equations

• Methods: Substitution, elimination, and graphing.

• Systems can have one solution, no solution, or infinitely many solutions.

• Example: Solve by elimination.

Applications of Systems

• Word problems involving mixtures, investments, and optimization.

• Example: A hotel rents rooms with and without kitchens at different rates; set up a system to find the number of each type rented.

Matrices and Determinants

Matrix Operations

• Addition, subtraction, and multiplication of matrices.

• Determinants are used to solve systems of equations and find inverses.

• Example: For , .

Conic Sections

Parabolas, Circles, Ellipses, and Hyperbolas

• Standard equations for each conic section.

• Identify and graph conic sections from their equations.

• Example: The equation represents a parabola.

Sequences, Induction, and Probability

Arithmetic and Geometric Sequences

• Arithmetic:

• Geometric:

• Example: Find the 10th term of the sequence

Probability Basics

• Probability of an event:

• Applications in counting and arrangements.

Additional Topics

Compound Interest

• Formula:

• Used to calculate future value of investments.

• Example: Compare yields for different compounding periods.

Inverse Functions

• Find by solving for and interchanging and .

• Example: If , solve for .

Sample Table: Properties of Logarithms

Additional info: This guide covers all major topics from equations and inequalities to functions, systems, matrices, conic sections, and sequences, as reflected in the provided exam study guide.