Pre-Calculus Pre AP 2015-16
I. Functions and Graphs. Chapter 1
1.1 Modeling and Equations Solving:
1. Concepts, Terms, Definitions, and Theorems. a) Mathematical Model b) Numerical Model c) Algebraic Model d) Graphical Model e) Zero Product Theorem f) X-Intercept= Root =Solution=zero of the function
2. Assignments 1.1
pp 74-79:3, 6, 9,11, 15, 16, 22, 30, 31, 33, 37, 48,
55-60.
3. Computer aided instruction : My Math Lab; Videos; Desmos
1.2 Functions and Their Properties.
1. Concepts, Definitions, and Theorems. a) Definition of Function: A function is a rule of mapping such that for each element in the Domain there is exactly one element in the Range. If the order pair is (x, y), then this is translated that for each element, x, in the
Domain, there is exactly one, y, in the Range.
b) Continuity of a Function:
A function, f(x) is continuous at x = c if:

f ( c ) is defined.

The limit of f(x) exits at x = c ( lim 𝑥→𝑐 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠).

The limit of f(x) exists at x = c and equals to f ( c ). lim 𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐).
c) Removable discontinuity: A discontinuity at x = c is called removable if f(x) can be made continuous by defining or redefining f( c ). d) Non-removable discontinuity is when f(x) can not be made continuous at x = c or if f(c ) is undefined such that a vertical asymptote at x = c. e) Increasing Function: A function, f , is increasing on an interval if for any two numbers x
1
and x
2
in the interval where x
1
< x
2,
then f (x
1
) < f(x
2
). f) Decreasing Function: A function, f , is decreasing on an interval if for any two numbers x
1
and x
2
in the interval where x
1
> x
2,
then

f (x
1
) > f(x
2
). g) Constant Function: A function , f , is a constant function on an interval if for any 2 points in the interval the slope is 0. ( ∆𝑦 = 0) h) Bounded Function:

A function , f, is bounded below if:

A function , f, is bounded above if:

A function , f, is bounded if: i) Relative or Local extrema of a Functions:

If there is an open interval containing , c , on which f ( c ) is a maximum then f( c ) is called a relative or local maximum of f , or f has a relative maximum at (c, f( c ) ).

f there is an open interval containing , c , on which f ( c ) is a minimum then f( c ) is called a relative or local minimum of f
, or f has a relative minimum at (c, f( c ) ). j) Symmetry of a Function:

X axis: For all x in the domain of f , f( x ) = - f(x).
(x, y) → (x, y).

Y axis: For all x in the domain of f, f( x ) = f( -x).
(x, y) → ( x, y). f(x) is an even function.

Origin: For all x in the domain of f , f( x ) = f( -x).
(x, y) → ( - x, y). f(x) is an odd function. k) Asymptotes:

Vertical Asymptote:

Horizontal Asymptote:
The line y = L is a Horizontal Asymptote of the graph of f , if lim 𝑥→∞ 𝑓(𝑥) = 𝐿 𝑜𝑟 lim 𝑥↑→ −∞ 𝑓(𝑥) = 𝐿 .
2. Assignments 1.2
pp. 94-98: 3,12,14,15, 18, 20 22, 23, 27, 28, 35-40,
47- 53, 55, 57-59, 62-67, 68, 73, 79, 80, 81, 84.
3. Computer aided instruction: My Math Lab; Videos; Desmos

1.3 Twelve Basic Functions.
1. Concepts, Definitions, and Theorems. a) Graph the twelve Basic Functions:

f(x) = x The Identity Function of a Linear equation

f(x) = 𝑥 2
The Quadratic Function

f(x) = x 3 The Cubic Function

f(x) =
1 𝑥
The Reciprocal Function

f(x) = √𝑥 The Square Root Function

f(x) = 𝑒 𝑥
The Exponential Function

f(x) = ln x The Logarithm Function

f(x) = sin x f(x) = cos x Two of the six trig functions

f(x) = |𝑥| The Absolute Value Function

f(x) = int(x) The Greatest Integer Function

f(x) =
1
1+𝑒 −𝑥
The Logistic Function

**f(x) = [𝑥] The Greatest Integer Function**
In each of these functions find: Domain and Range;
Intercepts; Continuity; Vertical Asymptotes; Horizontal
Asymptotes; Symmetry; Boundedness; Local (Relative)
Maximum and Local (Relative) Minimum Points; End Behavior: lim 𝑥→∞ 𝑓(𝑥) 𝑎𝑛𝑑 lim 𝑥↑→ −∞ 𝑓(𝑥) b) Piecewise Functions
2. Assignments 1.3
pp.
106-109: 1-28, 30-32, 33, 45, 48, 51, 60.
3. Computer aided instruction: My Math Lab; Videos; Demos
1.4 Combining Functions Algebraically.
1. Concepts, Definitions, and Theorems. a) b)
Definition of Sum, Difference, Product, and Quotient of f(x) and g(x):
Definition of Composition of Functions:

2. Assignments of 1.4
pp.116-118: 3, 4, 7, 8, 12, 14, 15-20, 22, 24, 27,
28, 33, 37, 41, 45, 46, 47, 50.
3. Computer aided instruction: My Math Lab; Videos; Demos
1.5 Parametric Relations and Inverses.
1. Concepts, Definitions, and Theorems. a) Relations Defined Parametrically: Give Example. b) Definition of Inverse Relation: c) Horizontal Line Test: d) Definition of Inverse Function: e) Inverse Reflection Principle: f) Inverse Composition Rule:
2. Assignments 1.5 pp. 125-128: 3, 4, 9-11, 15-18, 20, 21, 27, 29, 34-36
42, 43, 45, 46.
3. Computer aided instruction: My Math Lab; Videos; Desmos
1.6 Graphical Transformations.
1. Concepts, Definitions, and Theorems. a) Definition of Transformations:

Rigid Transformations:

Non-rigid Transformations: b) Translations:

Vertical:

Horizontal: c) Reflections across Axis:

Across the x axis:

Across the y axis:


Across the origin: d) Horizontal Stretches or Shrinks: e) Vertical Stretches or Shrinks:
2. Assignments 1.6 pp.: 136-139: 3,5,6, 8-12, 17, 18, 20, 22-25, 27, 29, 32,
33, 39, 42, 43, 45, 47, 49, 50-52, 59-64.
3. Computer aided instruction: My Math Lab; Videos; Desmos
Possible Extensions for Mastery
Review Exercises: pp. 152-3: 1-10, 14-18, 20, 21-23, 25, 30-31, 33, 34, 37, 38, 41-44,
47, 48, 53, 54, 59-61.
Pretest Provided.