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Math 2200 Chapter 1 Arithmetic and Geometric Sequences and Series Review

Key Ideas Description or Example

Sequences An ordered list of numbers where a mathematical pattern can be used to determine the next terms.

Example: 1, 5, 9, 13, 17... or 1000, 100, 10, 1... n is the term position or the number of terms, n must be a natural number

Series The sum of all the terms of a finite sequence Example: 5 + 10 + 15 + 20

1 + 0.5 + 0.25 + 0.125...

Arithmetic

Sequence A sequence that has a common difference, 1n nd t t

Example: 2, 4, 6, 8, 10, 12, 14,... where d = 2

Graph of an

Arithmetic

Sequence

Always discrete since the n values or the term position must be natural numbers.

Related to a linear function y = mx + b where m = d and 1b t m .

The slope of the graph represents the common difference of the general term of the sequence.

t1 = b + m , add the y-intercept to the slope to get the value of the first term of the sequence.

Arithmetic

Series The sum of an arithmetic sequence.

Use 12

n n

nS t t when you know the first term, last term, and the number of terms.

Use 12 12

n

nS t n d when you know the first term, the common difference,

and the number of terms.

You may need to determine the number of terms by using 1 1nt t n d .

Geometric

Sequence A sequence that has a common ratio. 11

n

nt t r

Example: 3, 9, 27, 82, 243, 729, 2187... where r = 3

Graph is discrete, not continuous, and not linear.

n

tn

2

Problem

Finite Geometric

Series

A finite geometric series is the expression for the sum of the terms of a finite geometric

sequence.

The General formula for the Sum of the first n terms

Known Values are: Known Values are:

t1, r and n t1, r and tn

Problem

Infinite Geometric

Series

A geometric series that does not end or have a final term.

It may be convergent (sum approaches a value, there is a formula for this) or divergent

(sum gets infinitely larger).

The series is convergent if the absolute value of r is a fraction or decimal less than one:

|r|1, -1 > r > 1

Sum of an Infinite

Geometric Series,

only if it

converges

You must know the value of the first term and the common ratio.

General or explicit

formula The unique parameters are substituted into the formula. The parameters are 1t and r for a geometric sequence or d for arithmetic sequence.

Example: 13 2nnt or 2 3nt n

1( 1) , 11

n

n

t rS r

r

1 , 11

nn

rt tS r

r

1 , 11

tS r

r

3

Math 2200 Chapter 2 Trigonometry Review

Key Ideas Description or Example

Sketching an angle in standard position.

The measure of an angle in standard position can be

between 0 to 360 .

The symbol is often used to represent the measure

of an angle.

The vertex of the angle is located at the origin (0,0) on a

Cartesian plane.

The initial arm of the angle lies along the positive x-axis.

The terminal arm rotates in a positive direction.

The angle in standard position is measured from the

positive x-axis.

Each angle in standard position has a corresponding

reference angle.

Reference angles are positive acute angles (< 90)

measured from the terminal arm to the nearest x-

axis.

Any angle from 90 to 360 is the reflection in the

x- axis and/or the y-axis of its reference angle.

Consider a reference right angle triangle

The three primary trigonometry ratios are

To calculate a ratio given an angle using your

caluclator:

The mode of your calculator should be in degrees.

The ratios are given as decimal approximations.

4

Given any point P(x, y) on the Terminal Arm of an

angle in standard position, the Pythagorean

Theorem can be used to determine the distance the

point is from the origin.

This distance can be labeled r.

The x- and y- coordinates of the point can be used to

determine the exact values for the primary trig

ratios.

The points P(x, y), P(x, y), P(x, y) and P(x, y)

are points on the terminal sides of angles in standard

position that have the same reference angle. These

points are reflections of the point P( x, y) in the x-

axis, y-axis or both axes.

The trigonometry ratios may be positive or negative

in value depending on which quadrant the terminal

arm is in. A point on the terminal arm would have

coordinates (x, y).

A Smart Trig Class

There are two special right triangles for which you

can determine the exact values of the primary

trigonometric ratios.

TIP: the smallest angle is always opposite the

shortest side.

Determining the measure of an angle, given the

defining ratio.

Use the inverse trig ratios.

Quadrantal Angles

If the terminal arm lies on an axis, the angle is

called a Quadrantal angle (it separates the

quadrants).

The quadrants are labeled in a counterclockwise

direction.

The Quadrantal angles for one revolution are

0, 90, 180, 270, 360

5

Ratios of Quadrantal Angles

To determine, without the use of technology, the

value of sin , cos or tan , given any point

P (x, y) on the terminal arm of angle , where = 0,

90, 180, 270 or 360 use the defining ratios and

consider any point on the axis.

Plot the point, determine the values of x, y, and r and

calculate the ratio.

The Sine Law

An angle and its opposite side create a defining ratio

that can be used to calculate the other

measurements.

Notice you are given the measurements of an angle

and its opposite side.

You may be asked to calculate the measure of the

remaining sides or angles.

sin sin sin

a b c

A B C

The Ambiguous Case of the Sine Law

Used when you are finding an Angle.

Given: two angles and one side or

two sides and an angle opposite one of the given

sides

An angle and its opposite side are given to define

one ratio sin

a

A of the Sine Law.

Since angle B could be obtuse or acute, the

ambiguous case must be considered.

There are two possible measure for angle B.

The Cosine Law

Describes the relationship between the cosine of an

angle and the lengths of the three sides of any

triangle.

2 2 22 2 2 2 cos or cos

2

b c aa b c bc A A

bc

To determine side c

20

sin31 sin104

c

20(sin104 )

sin31c

c= 37.7

6

Math 2200 Chapter 3 Quadratic Functions Review

Key Ideas Description or Example

Quadratic Functions

polynomial of degree two Standard Form: f(x) = ax

2 + bx + c, a 0

Vertex Form: f(x) = a(x p)2 + q , a 0

Factored Form: f(x) = (x + c)( x + d)

For a quadratic function, the graph is

in the shape of a parabola

Parent Graph is y = x2

Characteristics of a Quadratic

Function from the vertex form of the

equation

f(x) = a(x - p)2 + q

The coordinates of the vertex are at (p, q). Note that the negative symbol from (x - p) does not transfer to the value of p.

f(x) = 2(x - 3)2 + 4 vertex at (3, 4)

f(x) = 2(x + 3)2 + 4 vertex at (-3, 4)

Horizontal Translations:

When p > 0 the graph moves (translated) to the right.

When p < 0 is translated or shifts to the left.

Vertical Translations:

When q > 0 the parabola shifts up.

When q < 0 the parabola shifts down.

The parameter a indicates the direction of opening as well as how narrow or wide the graph is in relation to the parent graph.

When a > 0 , the graph opens up and the vertex is a minimum

min max

When a < 0, the graph opens downward and the vertex is a maximum.

When -1 < a < 1, the parabola is wider than the parent graph.

When a > 1 or a < -1, the parabola is narrower than the parent graph.

The Axis of Symmetry is an imaginary vertical line through the vertex that divides the function graph into two symmetrical parts. The

equation for the axis of symmetry is represented by x p = 0 or x = p

The domain of a quadratic function is all real numbers |x x . The only exception is for real life models.

The range of a quadratic function depends on the value of the parameters a and q.

When a is positive, the range is |y y q .

When a is negative, the range is |y y q .

x- and y- intercepts Calculate the x-intercept pt (x, 0) by replacing y with 0 in either form of

the function equation. ax2 + bx + c = 0 or a(x p)

2 + q = 0.

Calculate the y-intercept (0, y) by replacing x with 0 and solving for the

value of y.

Summary of Characteristics.

Write a quadratic function in the form

y = a(x p)2 + q for a given graph or

a set of characteristics of a graph.