Math 107
Outline of material for Test 1
Chapter 1. Fundamentals
§1.1 Coordinate Geometry
The Coordinate Plane
Coordinates of a point
Quadrants
Distance Formula
Midpoint Formula
Graphs of Equations in Two Variables
Graph of an equation in two variables
Sketching a graph by plotting points
Intercepts
x-intercept, y-intercept
Circles
Equation of a circle
Graphing a circle
Finding an equation of a circle
Identifying an equation of a circle
► Partial §; See Examples 1–5, 7–10 and related Exercises (i.e., at end of section)
§1.2 Lines
Slope of a Line
Point-Slope Form of the Equation of a Line
Slope-Intercept Form
Finding the equation of a line through two given points
Vertical & Horizontal Lines
General Equation of a Line
Graphing a linear equation
Parallel & Perpendicular Lines
► Partial §; See Examples 1–9 and related Exercises
§1.3 What Is a Function?
Functions All Around Us
Definition of Function
Function is a rule
Value/Image, Domain, Range, In/dependent variable
Ways to view a function: machine, arrow diagram
Evaluating a Function
Piecewise-defined function
Domain of a Function
Finding domains
Four Ways to Represent a Function:
Verbally (description)
Algebraically (formula)
Visually (graph) (more next §)
Numerically (table)
► Entire §; See Examples 1–7 and related Exercises
Math 107
Outline of material for Test 1
Chapter 1. Fundamentals (continued)
§1.4 Graphs of Functions
Definition of the Graph of a Function
Graphing Functions by Plotting Points
Graphing Piecewise-Defined Functions
Vertical Line Test [for a curve to be the graph of a function]
Equations that Define Functions
Frequently used functions (Table at end of §)
► Partial §; See Examples 1,4,8,9 and related Exercises
§1.5 Getting Information from the Graph of a Function
Finding Values, Domain & Range
► Partial §; See Examples 1&2 and related Exercises
§1.6 Transformations of Functions
Vertical Shifting
Horizontal Shifting
Reflecting Graphs
Vertical Stretching & Shrinking
Horizontal Stretching & Shrinking
Combining Transformations
Even & Odd functions
► Entire §; See Examples 1–8 and related Exercises
§1.7 Combining Functions
Composition of functions
Concept & Definition
Notation
In general, f○g ≠ g○f
Recognizing a Composition
► Partial §; See Examples 3,4,6 and related Exercises
§1.8 One-to-One Functions and Their Inverses
One-to-One function
Concept & Definition
Horizontal Line Test
Inverse of a [one-to-one] function
Concept & Definition, including notation, domain & range
Finding values of inverse function
Inverse Function Property
Verifying that two functions are inverses
Graphing the Inverse of a Function
–1
Obtaining graph of f from that of f
► Partial §; See Examples 1–5,9 and related Exercises
Math 107
Outline of material for Test 1
Chapter 2. Trigonometric Functions: Unit Circle Approach
§2.1 The Unit Circle
Point on Unit Circle
Terminal Point on Unit Circle determined by a real number t
Reference Number
Using reference number to find terminal point
► Entire §; See Examples 1–7 and related Exercises
§2.2 Trigonometric Functions of Real Numbers
The Trigonometric Functions
Definitions in terms of coordinates of Terminal Point
Focus on sin,cos,tan; other three are respective reciprocals
Table 1 of special values of trig. functions
[Skip for now connection to functions of angles (p.113); will come back to it.]
Values of the Trigonometric Functions
Signs of trig. functions per quadrant
Evaluating trig. functions; can use reference number
Even-oddness of trig. functions
Fundamental Identities
2
2
most important one: sin t + cos t = 1
Finding values of all trig. functions from value of one
► Partial §; See Examples 1,2,4,5 and related Exercises
§2.3 Trigonometric Graphs
Graphs of Sine & Cosine
Periodic properties of Sine & Cosine – period 2π
Graphs of Sine & Cosine on basic interval [0,2π] ; continue indefinitely according to periodicity
Think of Ferris wheel – coords. of Terminal Pt. vary as t runs from 0 to 2π quadrant by quadrant
Graphs of Transformations of Sine & Cosine
Vertical shifting & reflection
Vertical stretching: relation to Amplitude
Horizontal stretching: relation to Period
Horizontal shifting: Phase Shift
Combining Transformations
► Partial §; See Examples 1–5 and related Exercises
§2.4 More Trigonometric Graphs
Graph of Tangent
Periodic property of Tangent – period π
Graph of Tangent on basic interval (–π/2, π/2) ; continue indefinitely according to periodicity
Vertical Asymptotes at x
= ±π/2
Graph of Transformations of Tangent
Vertical shifting, reflection & stretching
Horizontal stretching: relation to Period
Horizontal shifting: Phase Shift
Combining Transformations
► Partial §; See Examples 1&2 and related Exercises
* end *