SEQUENCE a list of numbers in a particular order.
A recursive formula gives an+1 in terms of an:
cfw_an = cfw_a1, a2, a3, an,
an+1 = an + C
An explicit formula gives an as a function of n:
terms of t
Fundamental Counting Principle.
1. How many odd 3-digit positive integers can be written using the digits 2, 3, 4, 5 and 6?
2. A student council has 5 seniors, 4 juniors, 3 sophomores and 2 freshmen a
1. Summation Notation
Evaluate the following summations.
PARTIAL SUMS and SUMMATION (SIGMA) NOTATION
upper limit
4
(a)
(2i 1)
i
1
5
(b)
(1 r
2
lower limit
)
index of summation
r
3
2.
Little Gauss Pr
1.
An Epidemic Model
Rudy is one of 1,200 employees at the Saline Visteon Plant, which operates seven days a week. He arrives at the plant on Day 1 (a
Monday) with a slight fever. By noon, Rudy became
GEOMETRIC SEQUENCE a sequence characterized by a common ratio (r).
Explicit formula : an a1 r n 1
common ratio: r = an+1 an
Recursive formula : S n 1 ran
st
1 term: a = a1
NOTE: a geometric sequence i
Define: Random Variable
Binomial Experiment
Example 1. Probability Distribution
Probability
Probability Distribution
3
/8
1
/4
1
Let X be a random variable that represents the number of heads
when 4 c
Probability the likelihood that an event will occur (0 < p < 1)
Theoretical Probability
What should happen.
P ( A)
number of favorable outcomes
total number outcomes
Odds: The odds in favor of event
Example 1. Counting Principle.
(a) A customer in a computer store can choose 1 of 3 monitors, 1 of 2 keyboards and
1 of 4 printers. Assuming all components are compatible, how many different
systems c
Example 1
1. If you draw two cards from a standard deck of 52 cards find each
probability:
Probability of A and B
If A and B are any two independent events,
then the probability of A and B is:
P(A & B
Example 1. Mutually Exclusive Events.
(a) A six-sided die is rolled; what is the probability that the number rolled is
less than 3 or greater than 5?
Probability of A or B
If A and B are any two event
Example 1. A Subset of elements.
How many three letter subsets can be chosen from the set cfw_a, b, c, d, e?
Combination
An r element subset of a set with n
elements. (Order does not matter)
Example 2
The Hyperbola- The locus of points in a plane whose difference in distances from two distinct points (the foci) is constant (2a).
1. The Hyperbola.
Draw and label a hyperbola with a center
2. Graphing
1. Distance and Midpoint Formulas.
The Distance Formula
Given two points in a plane (x1, y1) and (x2, y2)
the distance between the two points is:
Consider the points (2, -3) and (-4, -1)
d ( x1 x 2 )
1. Solve a Non Linear Inequality.
Solve.
(a) x 2 4 x 12 0
(b) x 3 6 x 2 9 x
2. Solve a Rational Inequality by Graphing.
Consider the function f ( x)
x2 4
x 3
(a) Sketch the graph of y = f(x)
(b) For
The Ellipse-The locus of points in a plane whose sum of the distances from two distinct points (the foci) is constant (2a).
1. The Ellipse.
Draw and label an ellipse with a center at
the origin.
2. Gr
1. Circle.
Standard Form of a Circle
Derive the standard form of the circle with a center at the origin and a radius r.
2. Graphing Circles.
Graph each circle. Identify the radius.
(a) x2 + y2 = 16
(b
1. y varies inversely as x. y = 6 when x = 8, find x when y = 20.
2. y varies directly as x and inversely as z. y = 10 when x = 8 and z = 4, find x when y = 6 and z = 12.
3. The amount of time it take
1. Solve a Rational Equation.
(a)
z2 z
1
3 6
(b)
3
1
u 2 u 2
(c)
5
2 1
2
x 2
2x
(d)
1
1
4
y 2 y 2 y 2 4
(e) 2
5
x 3
x 2 x 6 x 2
2. Application. Distance Problem.
Tim paddled his kayak 12 km upstre
1.
2.
Simplifying Rational Expressions
Simplify each of the following. Give any restrictions on your simplification.
x 2 2x
x 2 4 x 12
2 x 2 18
(a)
(b)
(c)
x 2
x2 4
3 2 x x 2
Multiplying and Dividing
Inverse Variation
Two variables xand y show an inverse variation if y
decreases as x increases.
1. Inverse Variation.
(a) If y is inversely proportional to x, and y = 8 when x = 6, find x when
y = 15.
Rational Function = The ratio of two polynomials.
R( x)
P ( x)
, where P and Q are polynomials.
Q( x)
The domain of R is cfw_x: Q(x) 0.
Vertical Asymptotes and Holes
1. If Q(c) = 0 and P(c) 0, then R
1. Horizontal Asymptotes of a Rational Function.
2
Consider the function f ( x)
x1
Horizontal Asymptotes for Rational Functions
R( x)
(a) State the domain of f.
a m x m . a 2 x 2 a1 x a o
bn x n . b
1. Adding and Subtracting Rational Expressions.
Simplify
(a)
7 3 1
3 4 6
(d)
2
3
x 2 x
(b)
7
5
12 x 12 x
(e)
5
6
t 2 t 2
(c)
2 x 1 3 x 1
2x
3x 2
(f)
3
5
2
x x 2 x x 6
2
2. Complex Fractions.
Simpli
Reference: McDougal Littell: 7.3 Use Functions Involving e. Pages 492-495 See Examples 1-5
1. The Euler Number.
1
n
The Natural Base e
The Natural Base e is an irrational number. It is
defined as:
n
G
Reference: McDougal Littell: 7.1 & 7.2 Graph Exponential Growth and Decay Functions. Pages 480-481 See Examples 4,5 Page 488 Example 4
1. A new ice cream company began in 2010 and was able to produce
1. Once upon a time a brave knight returned from battle a hero. The king, to show his appreciation, offered him his gold
plated chess set. The brave knight, knowing that his fellow countrymen are star
Reference: McDougal Littell: Graph and Solve Quadratic Inequalities. Pages 300-303. Examples 1-7
1. Quadratic Inequality in Two Variables.
Graph the solution set to each inequality or system of inequa
Reference: McDougal Littell: Use the Quadratic Formula and the Discriminant. Pages 292-295. Examples 1-5
1. Using the Quadratic Formula.
Solve using the quadratic formula.
(a) y2 4y 13 = 0
The Quadrat