Precalculus Study Guide: Algebraic Expressions, Functions, and Graphs

Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

Algebraic Expressions and Polynomial Operations

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. This is useful for simplifying expressions and solving equations.

Key Point: To factor a quadratic or higher-degree polynomial, look for common factors and apply techniques such as grouping or the distributive property.
Example: Factor .
Solution: Group and factor to get .

Simplifying Radical Expressions

Radical expressions can often be simplified by combining like terms and using properties of exponents.

Key Point: Use the distributive property and exact values for simplification.
Example: Simplify .
Solution: Expand to get .

Multiplying Polynomials

Multiplying polynomials involves distributing each term in one polynomial to every term in the other.

Key Point: Apply the distributive property (FOIL for binomials).
Example: Multiply .
Solution: Expand to get .

Simplifying Complex Numbers

Complex numbers are numbers of the form , where is the imaginary unit ().

Key Point: Use properties of exponents and to simplify expressions.
Example: Simplify .
Solution: Use cycle: , , , , repeat. Find each power and simplify.

Solving Equations

Solving by Factoring

Factoring allows us to set each factor equal to zero to find solutions.

Key Point: Set each factor to zero and solve for the variable.
Example: Solve .
Solution: Rearrange and factor to find and .

Quadratic Formula

The quadratic formula solves equations of the form .

Formula:
Example: Solve .
Solution: Substitute , , into the formula.

Completing the Square

Completing the square is a method to solve quadratic equations by rewriting them in the form .

Key Point: Rearrange and add/subtract terms to form a perfect square trinomial.
Example: Solve by completing the square.

Functions and Their Properties

Domain of Functions

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Key Point: For square root functions, the radicand must be non-negative.
Example: Find the domain of .
Solution: Set and solve for .

Undefined Expressions

An expression is undefined when its denominator equals zero.

Key Point: Set the denominator equal to zero and solve for the variable.
Example: is undefined when .

Inverse Functions

The inverse of a function , denoted , reverses the effect of .

Key Point: To find the inverse, solve for in terms of , then swap and .
Example: For , the inverse is .

Function Composition

Function composition means applying first, then to the result.

Key Point: is found by substituting into .
Example: If and , then .

Graphing and Analyzing Functions

Quadratic Functions and Their Graphs

Quadratic functions are of the form and graph as parabolas.

Key Point: The vertex form is , where is the vertex.
Example: has vertex and x-intercepts found by setting .

Transformations of Functions

Transformations include shifts, stretches, compressions, and reflections.

Key Point: For , the graph of is stretched vertically by 2, compressed horizontally by 1/4, shifted right by , and up by 3.

Point-Slope and Slope-Intercept Form

Linear equations can be written in point-slope form and slope-intercept form .

Key Point: Use two points to find the slope , then substitute into the point-slope form.
Example: For points and , find and write the equation.

Special Topics

Direct and Inverse Variation

Direct variation means one variable increases as another increases; inverse variation means one decreases as the other increases.

Key Point: If varies directly as and inversely as the cube root of , then .
Example: Rearranging for gives .

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes sign over an interval, it must have a zero in that interval.

Key Point: If and have opposite signs, there is at least one real zero between and .
Example: For , if and , there is a zero between 0 and 1.

Tables

Sample Table: Function Composition and Domain

Function	Composition	Domain

Sample Table: Quadratic Equation Solutions

Equation	Method	Solution Set
	Completing the Square
	Quadratic Formula

Sample Table: Intermediate Value Theorem

Interval			Zero Exists?
[0, 1]	-2	8	Yes

Additional info:

Some problems involve graph matching and transformations, which are key Precalculus skills.
Complex numbers and their operations are included, which are part of Precalculus curriculum.
All problems are representative of Precalculus college-level exam questions.