PRECALCULUS

Semester I Exam Review Sheet

Chapter Topic

P.1 Real Numbers

{1, 2, 3, 4, …} Natural (aka Counting) Numbers

{0, 1, 2, 3, 4, …} Whole Numbers

{…, -3, -2, -2, 0, 1, 2, 3, …} Integers

Can be expressed as p/q where p and q are integers (q 0)

Rational Numbers

Can not be expressed as p/q such as above Irrational Numbers

P.2 Exponents and Radicals

Properties of Exponents

Scientific Notation

Addition, Subtraction – break from std form to make exponents the same

Then add or subtract coefficients, adj to std form.

Multiplication, Division – multiply coefficients then add/subt exponents

Radicals and their Properties

Simplifying Radicals

Rationalizing

Rational Exponents

P.3 Polynomials and Factoring

Polynomials

Definitions

Coefficient

Degree

Leading coefficient

Constant term

Monomial, binomial, trinomial, polynomial

Standard form

Polynomial Operations

Addition/Subtraction

Multiplication

General (mult every term in 2nd

poly by every term in 1st poly)

FOIL

Special Products

Sum and Difference of Same Terms (Conjugates Pairs)

(ax + b)(ax – b) = a2x

2 – b

2

Square of a Binomial

(ax + b)2 = a

2x

2 + 2ab x + b

2

(ax - b)2 = a

2x

2 - 2ab x + b

2

Cube of a Binomial

u3 + v

3 = (u + v)(u

2 – uv – v

2)

u3 - v

3 = (u - v)(u

2 + uv – v

2)

Factoring

Definitions

Prime

Irreducible (over reals, over rationals, over integers)

Removing Common Factors

Factoring Special Polynomial Forms

Difference of Two Squares

Perfect Square Trinomial

Sum or Difference of Two Cubes

General Factoring of Trinomials with Binomial Factors

Leading Coefficient is 1

Leading Coefficient is not 1

Factoring by Grouping

P.4 Rational Expressions

Domain of an Algebraic Expression

Simplifying Rational Expressions

Operations with Rational Expressions

Multiplying Rational Expressions

Dividing Rational Expressions

Adding/Subtracting Rational Expressions

Find Common Denominator (LCD?)

Examples 6 and 7, Pg. 40,41

Complex Fractions (Pg. 42: Examples 8, 9, 10, 11)

Simplifying

Factoring Common Quantities with Different Rational Exponents

Difference Quotients

P.5 Solving Equations

Identity – an equation that is true for all x

Conditional equations – an equation that is true for some (or no) x

Solving Conditional Equations

Linear Equations (Pg. 50: Ex. 1-3) Also, see ex 3 for extraneous solutions

Extraneous Solutions may occur when multiplying or dividing by

a variable quantity

Quadratic Equations

Factoring

Square Root Principle

Completing the Square

Quadratic Formula

Polynomial Equations of Higher Degree (Some Cases)

Factoring

Factoring by Grouping

Radical Equations (Check for Extraneous Solutions)

Involving Rational Exponent

Isolate Rational Exponent Term

Raise both sides to reciprocal of exponent

Involving One Radical

Isolate the radical

Square both sides

Solve for x

Verify solutions

Involving Two Radicals

Isolate one radical

Square both sides

Isolate 2nd

radical

Square both sides again

Solve for x

Absolute Value

Isolate Absolute Value (on left side)

Create two non-absolute value equations

Set expression inside absolute value equal to the

other side of the equation

Set expression inside absolute value equal to the

opposite of the other side of the equation

Solve both equations for x

P.6 Solving Inequalities

Inequality solutions are sets or intervals and are called the solution set.

Solution sets may be Bounded or Unbounded Intervals

Properties of Inequalities

Two inequalities with the same solution set are equivalent.

Properties of Inequalities

Transitive a < b and b < c then a < c

Addition of Inequalities a < b and c < d then a + c < b + d

Addition of a Constant a < b then a + c < b + c

Multiplication by a Constant For c > 0, a < b then a·c < b·c

For c < 0, a < b then a·c > b·c

Linear Inequalities

Single Inequality Form 3x + 1 > 5x - 4

Double Inequality Form 5 < 4x – 3 < 7

Absolute Value Inequalities

Isolate Absolute Value quantity on left side

If Inequality is Less Than type:

Determine if inequality has No Solution or is True For All x

If not, then:

Set expression inside absolute value < right side

Set expression inside absolute value > opposite of left side

Solve both of these inequalities to find solution set

Other Types of Inequalities

Polynomial Inequalities

Move all terms to one side of inequality so other side is zero

Find the zeros of the polynomial which are called Critical Points

“n” Critical Points divide number line into “n+1” intervals

Select one convenient Test Point from each Critical Point interval

Evaluate polynomial at each Test Point

Solution is set of all Critical Point intervals that satisfy inequality

Rational Inequalities

Find the Domain of a Square Root Function (Square Root Inequality)

P.7 Errors and the Algebra of Calculus

Algebraic Errors to Avoid (See Examples in Blue Boxes in Text)

Errors Involving Parentheses

Errors Involving Fractions

Errors Involving Exponents

Errors Involving Radicals

Errors Involving Dividing Out

Some Algebra of Calculus (See Examples in Blue Boxes in Text)

Unusual Factoring

Writing with Negative Exponents

Writing a Fraction as a Sum

Inserting Factors and Terms

P.8 Graphical Representation of Data

The Cartesian (x-y) Plane

Plotting Points

Sketching a Scatter Plot

The Distance Formula

Finding a Distance

Verifying a Right Triangle

The Midpoint Formula

1.1 Graphs of Equations

Graph Sketching

Plug in points

x- and y- intercepts

symmetry (and symmetry testing)

x- axis f(x,-y) = f(x,y)

y- axis f(-x,y) = f(x,y)

origin f(-x,-y) = f(x,y)

Equation of a circle

(x – h)2 + (y – k)

2 = r

2 where center = (h,k) Standard Form

1.2 Linear Equationss in Two Variables

Using Slope

Slope-Intercept Form of linear equation y = mx + b

Point-Slope Form of linear equation y = m(x – x1) +y1

Slope:

Slope of Parallel Lines mparallel = m

Slope of Perpendicular Lines mperpendicular = -1/m

1.3 Functions

Function vs Relation

Functional Relation of x into y:

Graphical Form: Vertical Line Test

Numeric, Algebraic, Verbal Forms: For every x there is only one y

Functional Relation of y into x: For every y there is only one x

Graphical Form: Horizontal Line Test

Numeric, Algebraic, Verbal Forms: For every y there is only one x

Function Notation

Dependent variable vs independent variable

Given f(x) = 3x + 3, then f(x+x) = 3(x+x) + 3 = 3x + 3x + 3

Piecewise-Defined Functions

Step Functions

Evaluating Difference Quotients (Pg. 130: Ex. 9)

Domains of Functions

Domain of a function is all possible x- values

Range of a function is all possible y- values

Implied Domain is a restricted domain resulting from a practical

interpretation of a model (ex. volume would not be negative.)

Applications

See Pg. 130: Ex.6, 7, 8

1.4 Analyzing Graphs of Functions

Graphs

Finding Domain and Range

Vertical Line Test

Zeros of a Function

Increasing, Decreasing, and Constant Regions

Definition of Relative Minimum and Relative Maximum

Step Functions

Greatest Integer Function:

Evaluate

Graph using calculator

Piecewise-Defined Functions

Evaluate

Graph using calculator

Even and Odd Functions

Even: f(-x) = f(x)

Odd: f(-x) = -f(x)

1.5 Shifting, Reflecting, and Stretching Graphs

Shifting Left and Right

Shifts Left “c” units: f(x) = x2 g(x) = (x+c)

2

Shifts Right “c” units: f(x) = x2 g(x) = (x-c)

2

Shifting Up and Down

Shifts Up “c” units: f(x) = x2 g(x) = x

2 + c

Shifts Down “c” units: f(x) = x2 g(x) = x

2 - c

Reflecting

Reflection about the x- axis: g(x) = -f(x)

Reflection about the y- axis g(x) = f(-x)

Vertical Stretch/Shrink

1.6 Combinations of Functions

Arithmetic Combinations of Functions

(f + g)(x) = f(x) + g(x)

(f – g)(x) = f(x) – g(x)

(f·g)(x) = f(x) · g(x)

Composition of Functions

(f g) = f(g(x))

Domain of a Composite Function

Domain of (f g) = f(g(x)) is all values of x that can be plugged

into g(x) such that g(x) can be plugged into f(x)

Identifying Composite Functions

See Example 6 on Page 166

2.1 Quadratic Functions

2.2 Polynomial Functions of Higher Degree

2.3 Long and Synthetic Division of Polynomials

2.4 Complex Numbers

2.5 Zeros of Polynomial Functions

2.6 Rational Functions

2.7 Partial Fraction Decomposition

3.1 Exponential and Logarithmic Functions

3.2 Logarithmic Functions and Their Graphs

3.3 Properties of Logarithms

3.4 Exponential and Logarithmic Equations

3.5 Partial Fraction Decomposition

4.1 Radian and Degree Measure

4.2 Trigonometric Functions: The Unit Circle