Ms. Liu Math Final: Unit 1

Complex Numbers

Numbers in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1).

Real Numbers

All numbers on the number line, including rational and irrational numbers.

Imaginary Numbers

Numbers that involve i, e.g., 3i, where i² = -1.

Rational Numbers

Numbers that can be expressed as p/q, where p and q are integers, and q ≠ 0.

Irrational Numbers

Numbers that cannot be expressed as p/q, e.g., π, 2√2.

Integers

Whole numbers, including negative numbers, zero, and positive numbers (…, -2, -1, 0, 1, 2, …).

Whole Numbers

Non-negative integers (0, 1, 2, 3, …).

Natural Numbers

Positive integers starting from 1 (1, 2, 3, …).

Interval Notation

Represents subsets of real numbers.

Open Interval

Not includes a and b; represented as (a, b).

Closed Interval

Includes a and b; represented as [a, b].

Half-open Intervals

Includes one endpoint but not the other; represented as [a, b) or (a, b].

Set Notation

Describes a set of elements using curly braces, e.g., {x | x > 3}.

Relation

A set of ordered pairs (x, y).

Function

A relation where each x-value has exactly one y-value.

Vertical Line Test

A graph represents a function if any vertical line intersects it at most once.

Domain

All possible x-values (inputs) of a function.

Range

All possible y-values (outputs) of a function.

Function Evaluation

Substituting the input value into the function.

Function Composition

Combining two functions: (f∘g)(x) = f(g(x)).

Real-world function application

Compositions model scenarios where one process depends on another.

Vertical Shift Upward

f(x) + c shifts the graph upward by c.

Vertical Shift Downward

f(x) - c shifts the graph downward by c.

Horizontal Shift Left

f(x + c) shifts the graph left by c.

Horizontal Shift Right

f(x - c) shifts the graph right by c.

Reflection Across the x-axis

−f(x) reflects the graph across the x-axis.

Reflection Across the y-axis

f(−x) reflects the graph across the y-axis.

Vertical Stretch

af(x) where a > 1 stretches the graph vertically.

Vertical Compression

af(x) where 0 < a < 1 compresses the graph vertically.

Horizontal Compression

f(bx) where b > 1 compresses the graph horizontally.

Horizontal Stretch

f(bx) where 0 < b < 1 stretches the graph horizontally.

Multiple Transformations

More than one transformation can be applied in sequence.