Key Concepts in Honors Precalculus

Definition: A relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.
Types of Functions:
- Linear: f(x) = mx + b
- Quadratic: f(x) = ax² + bx + c
- Polynomial: f(x) = a_nx^n + ... + a_1x + a_0
- Rational: f(x) = P(x)/Q(x), where P and Q are polynomials
- Exponential: f(x) = a * b^x
- Logarithmic: f(x) = log_b(x)

Transformations:
- Vertical/horizontal shifts
- Reflections across axes
- Stretching and compressing
Key Features:
- Intercepts (x-intercepts and y-intercepts)
- Asymptotes (vertical, horizontal)
- End behavior

Trigonometric Functions:
- Sine (sin), Cosine (cos), Tangent (tan)
- Cosecant (csc), Secant (sec), Cotangent (cot)
Unit Circle:
- Key angles: 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°
- Coordinates corresponding to angles
Identities:
- Pythagorean identities
- Angle sum and difference identities
- Double angle formulas

Form: z = a + bi (where i = √(-1))
Operations:
- Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
- Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Polar Form: z = r(cos θ + i sin θ)

Types:
- Circles: (x - h)² + (y - k)² = r²
- Ellipses: (x-h)²/a² + (y-k)²/b² = 1
- Parabolas: y = a(x - h)² + k or x = a(y - k)² + h
- Hyperbolas: (x-h)²/a² - (y-k)²/b² = 1 or vice versa
Foci and Directrices: Important for defining the shape and properties of conics.

Arithmetic Sequence: a_n = a_1 + (n - 1)d
Geometric Sequence: a_n = a_1 * r^(n - 1)
Summation Formulas:
- Arithmetic series: S_n = n/2 * (a_1 + a_n)
- Geometric series: S_n = a_1(1 - r^n) / (1 - r) for r ≠ 1

Limits: Understanding the behavior of functions as inputs approach a certain value.
Continuity: A function is continuous if:
- f(a) is defined
- lim (x→a) f(x) exists
- lim (x→a) f(x) = f(a)

Vectors: Representation, addition, scalar multiplication, dot product.
Matrices: Operations, determinants, inverses.
Probability and Statistics: Fundamental concepts, permutations, combinations, basic probability rules.

Concept: Each input corresponds to one specific output in a function.
Linear Function: Expressed as f(x) = mx + b, representing a straight line.
Quadratic Function: Formulated as f(x) = ax² + bx + c, characterized by a parabolic graph.
Polynomial Function: Written as f(x) = a_nx^n + ... + a_1x + a_0, consisting of terms with non-negative integer exponents.
Rational Function: Defined as f(x) = P(x)/Q(x), involves ratios of polynomials.
Exponential Function: Described as f(x) = a * b^x, exhibiting rapid growth or decay.
Logarithmic Function: Given by f(x) = log_b(x), representing the inverse of exponential functions.

Transformations: Adjustments made to the function's graph, including:
- Vertical and horizontal shifts to change the position.
- Reflections across axes altering orientation.
- Stretching and compressing changing the graph's shape.
Key Features:
- Intercepts: Points where the graph crosses the axes, including x-intercepts (where y=0) and y-intercepts (where x=0).
- Asymptotes: Lines that the graph approaches but never touches, both vertical and horizontal.
- End Behavior: Description of how the graph behaves as x approaches positive or negative infinity.

Trigonometric Functions:
- Sine (sin), Cosine (cos), Tangent (tan), along with their reciprocals: Cosecant (csc), Secant (sec), Cotangent (cot).
Unit Circle: Key reference for trigonometric values, with important angles being 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360° with corresponding coordinates.
Identities:
- Pythagorean Identities: Relate sine and cosine functions.
- Angle Sum and Difference Identities: Functions for combining angles.
- Double Angle Formulas: Relationships of the angles doubled.

Standard Form: A complex number represented as z = a + bi, where 'i' is the imaginary unit (√(-1)).
Operations:
- Addition: Combine real and imaginary parts separately.
- Multiplication: Apply distributive property and use i² = -1.
Polar Form: Expressed as z = r(cos θ + i sin θ), relating to magnitudes and angles.

Types of Conic Sections:
- Circles: Defined by the equation (x - h)² + (y - k)² = r² with center (h, k) and radius r.
- Ellipses: Described by (x - h)²/a² + (y - k)²/b² = 1 with semi-major and semi-minor axes.
- Parabolas: Given by y = a(x - h)² + k or x = a(y - k)² + h, featuring a vertex and focus.
- Hyperbolas: Represented as (x - h)²/a² - (y - k)²/b² = 1, involving two separate curves.
Foci and Directrices: Key points that define the attributes and orientation of each conic section.

Arithmetic Sequence: A sequence defined as a_n = a_1 + (n - 1)d where 'd' is the common difference.
Geometric Sequence: Expressed as a_n = a_1 * r^(n - 1) where 'r' is the common ratio.
Summation Formulas:
- Arithmetic Series: S_n = n/2 * (a_1 + a_n), summing terms in an arithmetic sequence.
- Geometric Series: S_n = a_1(1 - r^n) / (1 - r) for r ≠ 1, summing a geometric sequence.

Limits: Assess the behavior of functions as inputs get close to a particular value, providing insights into function behavior.
Continuity Criteria:
- A function is continuous if f(a) is defined, the limit exists as x approaches a, and the limit equals the value of the function at that point.

Vectors: Include representations, vector addition, scalar multiplication, and calculating the dot product for geometric interpretation.
Matrices: Involves operations such as addition, multiplication, determining determinants, and finding inverses.
Probability and Statistics: Fundamental principles including permutations, combinations, and basic probability rules used in analysis and predictions.