Transcript University Physics I - Cloud County Community College

Trigonometry
Cloud County Community College
Spring, 2012
Instructor: Timothy L. Warkentin
Course Overview
• The importance of study and completion of homework.
• Resources on the Cloud website: syllabus (homework
assignments), chapter outlines, homework solutions,
handouts, and class notes.
• The importance of memorization in the study of
Trigonometry.
• Using technology in Calculus I (TI-84 calculators,
Graphing Calculator, Internet Resources). Lab:
Introduction to Wolfram Alpha
Chapter 1: Functions and Graphs
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Equations and Inequalities
A Two-Dimensional Coordinate System and Graphs
Introduction to Functions
Properties of Graphs
The Algebra of Functions
Inverse Functions
Modeling Data Using Regression
Chapter 1 Overview
• Chapter 1 reviews important material needed to begin a
study of Trigonometry. A through understanding of
functions, function inverses and the notation used in their
descriptions is an essential prerequisite to any
understanding of Trigonometry.
1.1: Equations and Inequalities 1
• The Complex Number System
1.1: Equations and Inequalities 2
• Three ways of writing set solutions:
– Graphing
– Interval Notation
– Set Builder Notation
• The Absolute Value of a Number: the distance the
number is from the origin.
• The Distance Between Two Numbers: the absolute value
of the difference between the numbers.
• Any equation that can be put into the form ax + b = 0 is a
Linear Equation. Example 1
1.1: Equations and Inequalities 3
•
Solving Literal Equations: Example 2
– Clear denominators.
– Complete multiplications.
– Separate terms with target variable over the equality sign from
other terms.
– Factor out the target variable.
– Divide to isolate the target variable.
• Solving Quadratic Equations: Examples 3 & 4
–
–
–
–
Taking square roots.
Factoring (Zero Product Property).
Completing the Square.
Quadratic Formula.
1.1: Equations and Inequalities 4
• Solving Inequalities: Examples 5 & 6
– When an inequality is multiplied or divided by a negative number
the direction of the inequality changes.
– The Critical Point Method.
– The Graphical Method.
• Solving Absolute Value Inequalities: Examples 7 & 8
– Using sign switches.
– Using the distance between two points and a number line.
1.2: A Two-Dimensional Coordinate
System and Graphs 1
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•
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The Cartesian Coordinate System.
The Distance and Midpoint Formulas. Example 1
Graphing by using points. Examples 2-4
Graphing using the TI-84 calculator. Examples 2-4
Finding x (set y = 0) and y (set x = 0) intercepts.
Example 5
• Finding x intercepts with a TI-84 calculator. Example 5
• The Equation of a Circle Example 6
– Switching from Standard Form to General Form (expand the
squares)
– Switching from General Form to Standard Form (double
completion of the square) Example 7
1.3: Introduction to Functions 1
• Relations, Functions and 1-to-1 Functions
1.3: Introduction to Functions 2
• Domain: pre-images, x-values, independent values,
inputs.
• Range: images, y-values, dependent values, outputs.
• Function: Every pre-images has exactly one image.
• Functions can be described using function notation,
ordered pairs, Venn diagrams, input/output machine
diagrams, and tables.
• Function notation and dummy variables. Example 1
• Piecewise Functions (TI-84 calculators). Example 2
• Identifying Functions (The Vertical Line Test - VLT).
Examples 3 & 6
1.3: Introduction to Functions 3
• Domain Issues: Example 4
– Division by Zero
– Even Roots of Negative Numbers
– Physical Constraints
• Graphing functions using points (TI-84 tables) Example 5
• Graphing functions using the TI-84 calculator and
Graphing Calculator.
• Increasing and Decreasing Functions.
• 1-to-1 functions (The Horizontal Line Test – HLT)
• The Greatest Integer Function (The Floor Function, TI-84
int( command) Example 7
• Using functions to solve applications. Examples 8-10
1.4: Properties of Graphs 1
• Relation Symmetry: Examples 1 & 2
– y-axis symmetry: -x replacing x yields equivalent equation.
– x-axis symmetry: -y replacing y yields equivalent equation.
– Origin symmetry: -x replacing x & -y replacing y yields equivalent
equation.
• Function Parity: Example 3
– Even: f[-x] = f[x]
– Odd: f[-x] = -f[x]
• Function Translation: Example 4
– Vertical: g[x] = f[x] + k is f[x] translated vertically by k units.
– Horizontal: g[x] = f[x-h] is f[x] translated horizontally by h units.
1.4: Properties of Graphs 2
• Function Reflection: Example 5
– -f[x] is f[x] reflected over the x-axis.
– f[-x] is f[x] reflected over the y-axis.
• Function Vertical Elasticity: Example 6
– If a > 1 then g[x] = a·f[x] is f[x] stretched vertically by a factor of
a.
– If 0 < a < 1 then g[x] = a·f[x] is f[x] compressed vertically by a
factor of a.
1.4: Properties of Graphs 3
• Function Horizontal Elasticity: Example 7
– If a > 1 then g[x] = f[a·x] is f[x] compressed horizontally by a
factor of 1/a.
– If 0 < a < 1 then g[x] = f[a·x] is f[x] stretched horizontally by a
factor of 1/a.
•
1
Function Transformation: y  a  f [ ( x  h)]  k
w
– a is the vertical scaling factor.
– w is the horizontal scaling factor.
– h is the horizontal shift.
– k is the vertical shift.
1.5: The Algebra of Functions 1
• Operations on Functions Examples 2, 5, 6 & 7
( f  g )[x]  f [ x]  g[ x]
( f g )[x]  f [ x] g[ x]
f
f [ x]
 [ x] 
, g[ x ]  0
g[ x ]
g
( f  g )[x]  f [ g[ x]]
• Domains of Combined Functions Example 1
• The Difference Quotient: Examples 3 & 4
f x1  h  f x1 
DQ 
h
1.6: Inverse Functions 1
• The inverse of a Relation is that Relation that switches
the order of the ordered pair elements. Every Relation
has an Inverse.
• A Function will have an Inverse Function IFF it is a 1-to-1
Function.
• Identifying 1-to-1 Functions (The Horizontal Line Test HLT). Example 1
• Proving that a pair of functions are inverses. Example 2
• Finding an Inverse (Switch Method). Examples 3 & 4
• Restricting the domain of a function (domain surgery).
Examples 5 & 6
1.7: Modeling Data Using Regression 1
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Linear Regression Models. Example 1
The Correlation Coefficient.
The Coefficient of Determination.
Quadratic Regression Models. Example 2
Using the TI-84 to model data.