PreCalculus

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Functions

A Package

information packages

What is a Function?

one additional rule

Reading and Writing

language

Function Properties

characteristics, features

Number Functions

measurements

Real-Valued

Set Symbols

vocabulary

Subsets of Real Numbers

intervals

Open Sets

always room

∞

not a number

Fractions

packages

Function Graphs

Encoding Visually

pairs to dots

Decoding Visually

dots to pairs

Function Features

extreme values

Function Behavior

relative change

Formulas

The Formula Tool

part 1

The Formula Tool

part 2

Formulas and Graphs

bridge

Parentheses

grouping

Piecewise Functions

Stitching Functions

pieces and parts

Famous Piecewise

examples

Periodic Functions

repeating

Recursive Functions

formula of formulas

Graphical Language

Graph Symbols

communication

Inaccurate

suspicious

Counter Examples

dealing with all

Visual Features

Domain

inputs

Zeros

intercepts

Discontinuity

sudden breaks

End-Behavior

settling down

Visual Behavior

Change

increasing and decreasing

Optimization

minimum and maximum

Analysis

rigor

Beginning of Behavior

Proportional Change

linear

Linear Analysis

8 characteristics

Quadratic Forms

standard vertex factored

Quadratic Analysis

quadratic graphs

Library

Expected Rate

Tangent Lines

zooming in

Quadratic Analysis

behavior

Derivative

approaching

Accumulation

adding rates

Polynomial and Rational

Polynomials

characteristics

Polynomial Analysis

the polynomial story

Rational Functions

fractions of polynomials

Analysis

the rational story

Rational Analysis

with a derivative

Radical and Absolute Value

Roots and Radicals

aspects

Radical Analysis

examples

Absolute Value

traits

Abs. Value Analysis

examples

Exponential Analysis

Theory

the exponential story

Shifted Theory

the shifted story

Basic Forms

base e

Algebra

exponential form

Logarithmic Analysis

Theory

the logarithmic story

Basic Forms

basic

Trig Functions

Sine

analysis

Cosine

analysis

Tangent

analysis

Transforming

Shifting Domain

same characteristics

Shifting Range

same characteristics

Stretching

both directions

Transform Shape

graphs

Inside

domain

Outside

range

All Together

everything

Piecewise Composition

Output Is Input

evaluating

Pointwise Functions

output \(\rightarrow \) input

Linear Composition

linear linear linear

Piecewise Functions

piecewise linear

More Pieces

piecewise linear

Expected Values

Broken Values

“should be” values

Missing Values

blink

Limits

language of behavior

Space

surrounding values

Open Intervals

always space

Closeness

relative distance

Endpoints

intersection

Continuity

Discontinuity

breaks

Singularity

holes

Continuity

close as you want

Elementary Functions

nice

Function Behavior

End Behavior

approaching infinity

Limiting Behavior

limit notation

Comparing Functions

Dominance

relatively bigger

Indeterminate

no dominance

Function Zeros

Zero Property

factoring

Solving Equations

break it up

Solving Inequalities

still break up

Rate of Change

critical numbers

Equivalent Forms

Alternate Formulas

many representations

Equivalent Enough

domain watching

the Derivative

Tangent Lines

zooming in

Quadratic Analysis

behavior

From the Sides

curve pretending

Function of Change

rate as a relationship

Behavior

to algebra

Approximating Functions

tangent lines

Analyzing Functions

Analysis

8 characteristics

Analyzing

describe everything

Analyzing

describe everything

Analyzing

describe everything

A New Operation

composition

Assemble

putting functions together

Behavior

chain rule

Inverse Functions

one-to-one

Analysis

8 characteristics

More Analysis

8 characteristics

One More

analysis

Function Type

categorize

2D Numbers

Real Numbers

dots and arrows

Vectors

algebraic geometry

Numbers

Complex Numbers

Notation

\(a + b \, i\)

Operations

addition and subtraction

Complex Multiplication

products

Fractions

complex quotients

Algebraic Geometry

Polar Form

locating numbers

Graphing

polar coordinates

Direction Vectors

unit vectors

The Key

circles and triangles

Right Triangles

Similarity

unit circle again

Side Lengths

Pythagorean Theorem

Angles

inverses

3 Views

information

Trig Functions

Sine & Cosine

around the unit circle

Sine

analysis

Cosine

analysis

Tangent

quotient

As Input

composition

General Triangles

Law of Sines

proportionalities

Law of Cosine

extending Pythagorus

More Trig Functions

Reciprocals

upsidedown and backwards

Cotangent

characteristics

Secant

features

Cosecant

attributes

Algebra

relationships

Inverse Trig Functions

Inverse Functions

one-to-one

Exponential

exponential and logarithmic

ArcSine & ArcCosine

restricting the domain

Arctangent

useful

Complex Bridge

Euler’s Formula

complex exponentials

Bridge

exp, trig, hyp

Trig Identities

double and half

Function Analysis

Analyzing

describe everything

Analyzing

describe everything