Functions and Linear Functions Overview

Functions

• Functions are relationships between inputs (domain) and outputs (range). A function maps each input to one and only one output.
• Representing functions: Equations, tables, graphs, and verbal descriptions.
• Domain and Range: The set of all possible input values (domain) and the set of all possible output values (range).
• Evaluating functions: Substituting input values into the function's equation to find the corresponding output.
• Function notation: f(x) represents the output of function f for input x.
• Types of functions: Linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric. Understanding the characteristics of each type (e.g., graphs, equations, transformations).

Linear Functions

• Linear functions have a constant rate of change and a graph that's a straight line.
• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
• Point-slope form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.
• Finding the equation of a line given two points or a point and the slope.
• Parallel and perpendicular lines: Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals of each other.

Quadratic Functions

• Quadratic functions have a graph that's a parabola.
• Standard form: f(x) = ax² + bx + c.
• Vertex form: f(x) = a(x - h)² + k, where (h, k) is the vertex.
• Finding the vertex, axis of symmetry, and x-intercepts (roots/zeros) of a quadratic function.
• Solving quadratic equations using factoring, completing the square, and the quadratic formula.

Polynomials

• Polynomials are functions involving sums of powers of x (e.g., x³, x², x, and constants).
• Understanding the relationship between the degree of a polynomial and its graph's end behavior.
• Finding zeros (roots) of a polynomial.

Rational Functions

• Rational functions are quotients of polynomials.
• Understanding asymptotes (vertical, horizontal, oblique/slant).
• Finding domains of rational functions, considering where denominators are zero.

Exponential and Logarithmic Functions

• Exponential functions involve bases raised to variable powers.
• Logarithmic functions are the inverses of exponential functions.
• Understanding properties of exponents and logarithms, and how to apply them to solve equations.
• Graphing exponential and logarithmic functions.

Trigonometric Functions

• Trigonometric functions relate angles and sides of right triangles.
• Understanding sine, cosine, tangent, cosecant, secant, and cotangent functions.
• Basic trigonometric identities.
• Graphs of trigonometric functions, including period and amplitude.

Transformations

• Transformations of functions (horizontal shifts, vertical shifts, reflections, stretches, compressions) with an understanding of how they affect the graph.
• Understanding how transformations affect domain and range.

Systems of Equations and Inequalities

• Solving systems of linear equations using graphing, substitution, and elimination methods. Understanding when systems have no solutions or infinitely many solutions.
• Solving systems of inequalities by graphing.

Conic Sections

• Identifying and graphing parabolas, circles, ellipses, and hyperbolas. Understanding equations that define each.

Sequences and Series

• Arithmetic and geometric sequences.
• Finite and infinite series; sum formulas.

Other Important Topics

• Function composition
• Inverse functions
• Factoring techniques
• Solving inequalities
• Graphing techniques
• Identifying key characteristics (roots, asymptotes, intercepts, extrema, etc.) of functions
• Problem-solving strategies and applications involving functions.

This quiz covers the essential concepts of functions, including their representations, domain and range, and function notation. Dive into linear functions and learn about their characteristics, forms, and evaluating procedures. Ideal for those studying algebra or mathematics.