Prerequisites: Fundamental Concepts of Algebra

Basic Algebra Review

Precalculus begins with a review of fundamental algebraic concepts to ensure students are prepared for advanced topics. This includes operations with numbers, variables, and basic equations.

• Key Concepts: Arithmetic operations, properties of real numbers, order of operations, and solving simple equations.

• Example: Solve for .

• Solution:

Power Properties of Exponents

Understanding exponents is essential for manipulating algebraic expressions and solving equations.

• Definition: An exponent indicates how many times a base is multiplied by itself.

• Properties:

  • (for )

• Example:

Rational Exponents and Radicals

Rational exponents and radicals are used to express roots and fractional powers.

• Definition:

• Operations: Simplifying, multiplying, and dividing radical expressions.

• Example:

Equations and Inequalities

Linear Equations in One Variable

Linear equations are equations of the first degree, meaning the variable has an exponent of one.

• General Form:

• Solution:

• Example:

Literal (Multivariable) Equations

Literal equations involve more than one variable and require solving for a specified variable.

• Example: Solve for .

• Solution:

Quadratic Equations

Quadratic equations are second-degree equations and can be solved by factoring or using the quadratic formula.

• General Form:

• Quadratic Formula:

• Example:

Compound Inequalities and Set/Interval Notation

Compound inequalities involve two or more inequalities joined by 'and' or 'or'. Solutions are often expressed in set or interval notation.

• Example: is written as in interval notation.

Polynomial and Rational Functions

Polynomial Functions: End and Local Behavior

Polynomial functions are expressions involving powers of . Their graphs are characterized by end behavior and local maxima/minima.

• End Behavior: Determined by the leading term.

• Example: ; as , .

Rational Functions: Simplification and Restrictions

Rational functions are ratios of polynomials. Simplification and identifying restrictions (values that make the denominator zero) are key skills.

• Example: (for )

• Restrictions: Denominator cannot be zero.

Asymptotic Behavior of Rational Functions

Asymptotes are lines that the graph approaches but never touches.

• Vertical Asymptotes: Occur where the denominator is zero.

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

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

Functions and Graphs

Relations and Functions

A function is a relation in which each input has exactly one output.

• Function Notation: denotes the output for input .

• Example:

Domain and Range

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

• Example: For , domain is .

Linear Functions: Graphs and Equations

Linear functions have constant rates of change and are graphed as straight lines.

• Slope-Intercept Form:

• Example:

Algebra of Functions: Combination and Composition

Functions can be combined (addition, subtraction, multiplication, division) or composed (one function applied to another).

• Combination:

• Composition:

• Example: If , , then

Difference Quotient

The difference quotient is used to measure the average rate of change of a function.

• Formula:

• Example: For ,

Even and Odd Functions

Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

• Even:

• Odd:

• Example: is even; is odd.

Transformations of Functions

Transformations include shifts, reflections, and stretches/shrinks of parent functions.

• Vertical Shift:

• Horizontal Shift:

• Reflection: (over x-axis), (over y-axis)

• Example: ; shifts right by 2 units.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form , where and .

• Applications: Growth and decay models, compound interest.

• Example:

Logarithmic Functions

Logarithms are the inverse of exponentials. is the exponent to which must be raised to get .

• Properties:

• Change of Base Formula:

• Example:

Exponential and Logarithmic Equations

Solving equations involving exponentials and logarithms often requires applying properties and using logarithms to isolate variables.

• Example: Solve ;

• Example: Solve ;

Trigonometric Functions

The Six Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in right triangles.

• Definitions:

Right Triangle Trigonometry

Right triangle trigonometry uses the six functions to solve for unknown sides or angles.

• Example: In a triangle with ,

Sine and Cosine Values of Special Angles

Memorizing sine and cosine values for , , and is essential.

Angles and Their Measure

Angles can be measured in degrees or radians. Conversion between units is important.

• Conversion: radians

• Example: radians

Parent Functions of Sine and Cosine

The graphs of sine and cosine functions are periodic and have characteristic shapes.

• General Form:

• Period:

• Example: has period

Summary Table: Key Topics and Concepts

Additional info: This guide is based on the Math113 syllabus, referencing "College Algebra & Trigonometry" by Blitzer. For detailed practice, refer to worksheets and online assignments as indicated in the course schedule.