Download TRIG - Wantagh School

x
b  b2  4ac
2a
1
RADICALS
Simplifying
1. Make a big radical & break the number
into its prime factors
2. If variables, list them out
3. Look at the index to see what type of
groups you're making
4. Pull out the groups & leave the leftovers
under the radical
Multiplying & Dividing
1. Multiply/divide # by # & radical by
radical
2. Simplify all radicals - reduce only the
"whole" numbers
3. FOIL if there's an +/- sign
Adding & Subtracting
1. Simplify everything
2. Add/subtract the coefficients of the radicals
with the same radicand
Rationalizing the Denominator
(aka Get the Radical Out of the Denominator)
1. Multiply by a fancy one (denominator or
conjugate)
2. Simplify radicals in numerator & find perfect
square of the denominator then combine
3. Reduce only "whole" numbers at the end
Solving Radical Equations
YOU CAN DO
THIS!!!
1. Isolate the radical
2. Square both sides (may have to FOIL)
3. Solve for the variable
4. Check answers in the ORIGINAL problem
– this is MANDATORY!
Examples
1. The expression
2. What is the solution set of the equation
9 x  10  x  0 ?
2 6
is equivalent to
3 6
6
3
 6
(2)
3
(1)
(3)
(1) {-1}
(2) {9}
2
(4)  2
3. Multiply: (2  3)(6  3)
8 6
4. Simplify: 3 4 324 p r
2
(3) {10}
(4) {10, -1}
IMAGINARY & COMPLEX NUMBERS
Simplifying Negatives Under the Radical
1. Pull out your "i"
2. Simplify the radical
Simplifying "i"s
1. Divide the exponent by 4
2. Look at the decimal
3. Remember your "cheer"
i -1
-i
.25 .5
.75
1
NO
Adding/Subtracting Imaginary Radicals
1. Pull out your "i"
2. Simplify & add/subtract the coefficients
**CAN ALSO USE iPart**
Adding/Subtracting Imaginary #'s
1. Simplify all terms
2. Combine like terms
Multiplying Imaginary Radicals
1. Pull out your "i"
2. Multiply #'s and radicals
3. Simplify at the end
Multiplying Imaginary #'s
1. Multiply the coefficients
2, Add the exponents
3. Simplify the "i"
Dividing Imaginary Radicals
1. Pull out your "i"
2. Divide the #'s & radicals
3. Simplify at the end
Dividing Imaginary #'s
1. Divide the coefficients
2. Subtract the exponents
3. Simplify the "i"
Complex Numbers
- When graphing: (3 + 2i) = (3, 2)
Complex Numbers - Multiplying
1. FOIL or use the calculator!! (a + bi mode)
Add/Subtract
1. Combine like terms
2. If graphing, make a vector
**remember: i2 = -1**
Complex Numbers - Dividing
1. Multiply by a "fancy one" - conjugate
2. Simplify the numerator & denominator
Multiplicative Inverse
1. 1 over the given number
2. Multiply by a "fancy one"
3. Simplify the numerator & denominator
Examples
1. When expressed as a monomial in terms of
i, 2 32  5 8 is equivalent to

 
(1) 2 2i
(2) 2i 2
2. In a + bi form, the expression

1
is
7  4i
equivalent to
(3) 2i 2
(4) 18i 2
7 4i

65 65
7 4i

(2)
33 33
(1)
3
7 4i

65 65
7 4i

(4)
33 33
(3)
3. What is the product of (2 – 5i) and its conjugate in simplest
a + bi form?
4. The expression i3 + i(2 – i) is equivalent to
(1) -1 + 3i
(2) 1 + i
QUADRATICS
4
(3) 1 – i
(4) 1 + 3i
Examples:
1. The product of the roots for the quadratic
equation 2x2 – 5x + 9 = 0 is
9
2
(2) 9
(1)
2. What is the nature of the roots of the
quadratic whose equation is x2 = -18x + 81?
(1) imaginary
(2) real, irrational, and unequal
(3) real, rational, and unequal
(4) real, rational, and equal
5
2
(4) 5
(3)
3. What is the quadratic equation whose roots
are (2 – 4i) and (2 + 4i)?
4. Solve for x using the quadratic formula and
leave your answer in simplest radical form.
x2 = -2x + 4
5. Solve x2 – 6x + 1 = 0 by completing the
square. Express the result in simplest radical
form.
6. What is the center and radius of the
following circle?
x2 + y2 + 8x – 6y + 4 = 0
5
EQUATIONS & INEQUALITIES
Solving Quadratic Inequalities Algebraically


Solving Quadratic Inequalities Graphically

Set it equal to 0 and factor
Put it on a number line – check “0” to
see which way to shade!


Absolute Value Equations



Absolute Value Equations

Isolate the absolute value
Drop the absolute value and create 2
equations – one that’s set equal to the
positive value, one that’s set equal to the
negative value
Check all answers in your original
equation


Linear/Quadratic Systems






Examples:
1. What is the solution of the inequality
x2 – x – 6 < 0?
(3)  x 1 x  6
(2)  x  2 x  3
(4)  x  3 x  2
When solving absolute value
inequalities, treat it just like an equation
- the only difference is that the solution
is not only one number, but a series of
numbers
The solution will get graphed on a
number line
Use test points to see which way you
can shade – write the solution in
notation form too!
Linear/Circle Systems
A linear equation will have variables
with no exponents
A quadratic equation will have
variables with squared exponents
The only way to solve a linear/quadratic
system is substitution.
(1)  x  3 x  2
If the inequality has a < or > symbol,
you will need a dotted line to show that
the points on those lines are not
included in the solution set
If the inequality has a < or > symbol,
you will need a solid line to show that
the points on the line are included in
the solution set.
You need to use a test point – usually (0,
0) if one is not given – to determine
which way you are to shade (inside the
parabola or outside the parabola)
The linear equation will be the equation
without the exponents
The equation of the circle will be the
equation with x2 and y2
The only way to solve a linear/circle
system is by substitution
2. What is the solution set of the equation
2 x  x  3  9 ?
(1) {12}
(2) {2}
6
(3) {2, 12}
(4) { }
3. Solve the absolute value inequality, graph
the solution set, and write the solution set.
4. Solve the following system of equation
algebraically and leave your answer in
simplest radical form.
2x  3  5
2x2 + x + 1 = y
y=x+7
RATIONAL EXPRESSIONS
Undefined
 To see when a fraction is undefined we
set the denominator equal to 0
 Remember to factor if you are given a
squared term!
Simplifying
 Make sure that you factor first, then
reduce!!
Multiplying & Dividing Rational
Expressions
 Make sure that you factor first, and then
reduce!!
 When you divide, keep the first
fraction, change the division sign to
multiplication, & flip the second
fraction. Then keep going as if you
were multiplying!
Adding & Subtracting Rational Expressions
 When adding & subtracting fractions, you
need a common denominator
 Here’s what we always need to ask
ourselves:
CAN WE FACTOR THE DENOMINATOR?
Factor the
Multiply the
denominators & take
denominators
your bits & pieces
together
Complex Fractions
Rational Equations
 Remember that fractions are really a
 Either cross multiply or get a common
big division problem
denominator then ignore the
denominator, and solve the numerator!
 Rewrite the problem as a division
problem before factoring and reducing!
 Remember to check your answer in the
original problem!
7
Rational Inequalities
 Write the inequality as an equation and
solve.
 Determine any values that make the
denominator equal 0 (undefined).
 Make each of the critical values from
steps 1 and 2 on a number line.
 Select a test point in each interval –
check to see if the chosen test points
satisfy the inequality.
 Mark the number line to reflect the
values and intervals that work.
 Write your answer in set notation.
Try to remain
calm…we’ll get
through this
together!
Examples:
1. If the length of a rectangular garden is
x2  2x
represented by 2
and its width is
x  2 x  15
2x  6
represented by
, which expression
2x  4
represents the area of the garden?
(1) x
(2) x + 5
2. For which value of m is the expression
15m2 n
undefined?
3 m
(1) 1
(3) 3
(2) 0
(4) -3
x2  2 x
(3)
2( x  5)
x
(4)
x5
3. What is the value of x in the equation 4. Simplify and leave your answer in simplest
x x
form.
  2?
2 6
(1) 12
(2) 8
x 2  6 x  16
64  x 2
(3) 3
1
(4)
4
8
5. Simplify the following complex fraction.
6. Divide and leave your answer in simplest
form.
1 1

x y
1 1

x2 y2
y2  6 y  9 3y  9


y2  9
y 3
RELATIONS & FUNCTIONS
Topic

Good Things To Know
Functions  _____ value cannot repeat  ____________
line test
Relations & Functions

One to One Function  _____ and _____ cannot repeat 
must pass _______________ and ______________ line test

Onto Function  all values of the _____________ are used

Domain  ____ values
 Fractions  denominators  ______
Domain & Range
 Radical  cannot be ___________  > 0
 Fraction with Radical in Denominator  denominator
must be > ____

Range  _____ values
 Graph to see what values are in the range

Moving UP or DOWN
 UP  _______________
Transformations
 DOWN  _____________

Moving LEFT or RIGHT
 LEFT  _______________
 RIGHT  _______________

Reflecting in X-AXIS or Y-AXIS
 X-AXIS  ______________
 Y-AXIS  ______________
9
Function Notation &
Compositions of
Functions
Inverse Functions
Direct & Inverse
Variation

f(x), g(x), h(x), etc – another way to write “y = “

Whatever value is inside the parentheses will replace the
x in the given function

When given points, _________________________________

When given an equation, switch ______ and ______, then
solve for ________

To justify a composition  f(f-1(x)) = f-1(f(x)) = x

DIRECT VARIATION  set up a ______________________

INVERSE VARIATION  set up _______________________
Examples:
1. The function g(x) is defined as g(x) = 5 – 6x
with the domain -4 < x < 2. What is the least
element in the range?
(1) 29
(2) 5
(3) -7
(4) -4
2. What is the domain of f ( x)  x  5 ?

(2) 
(1)
, x  5
, x  0

(4) 
, x  5
(3)
, x  5
3. Which of the following relations would not
be considered a function?
4. Which equation defines a relation that is not
a function?
(1) f(x) = {(-4, 2), (1, 0), (9, 7)}
(2) g(x) = {(4, -2), (-4, 0), (-9, -7)}
(3) h(x) = {(2, -4), (2, 1), (7, 2)}
(4) j(x) = {(-2, 4), (0, -1), (-7, -9)}
(1) y = 3 – 2x
(2) x2+ y2 = 16
(3) y = x2 + 4x + 6
(4) y = -5
5. If f(x) = 3x – 4, and g(x) = x2 – 4x, what is the 6. What is the inverse of the function
value of g(f(x))?
f(x) = 5 – 2x?
7. The frequency of a radio wave is varied
inversely to the wave length. If the wave of
300 meters has a frequency of 1,500 kilocycles
per second, what is the length, in meters of a
wave with a frequency of 1,000 kilocycles per
second?
8. If y = f(x) is shifted five units right and
reflected over the x-axis, which of the
following equations would represent that
transformation?
(1) y = -f(x) + 5
(2) y = -f(x + 5)
10
(3) y = -f(x) – 5
(4) y = -f(x – 5)
CIRCLES
Center-Radius Form
Centered at the Origin
x2 + y2 = r2
Centered at (h, k)
(x – h)2 + (y – k)2 = r2
Standard Form
x2 + y2 + ax + by + c = 0  complete the square to get from standard form into center-radius
form
Examples:
1. Determine the center and radius of the
circle whose equation is:
x2 + y2 – 2x – 8y + 1= 0
2. What is the equation of the circle below that
passes through the point (0, -1)?
EXPONENTS
Name of Law
Explanation of Law
Example
Multiplication
Law
Add the exponents
3x5 5 x9  15 x 45
Division Law
Subtract the exponents
8 x8
 4 x4
4
2x
Power Law
Multiply the exponents
Negative
Exponent Law
To make the negative exponent
positive, move it from the numerator to
the denominator (or vice versa)
11
 2x 
6 4
 16 x 24
10 x5 y 2
5x2
2 7
 5x y  7
2 x3 y 9
y
Power of Zero
Law
x8
 x0  1
8
x
Anything to the 0 power equals 1
Fractional
Exponent
Law
3
7
x  7 x3
The denominator becomes the index!
5
x x
2
2
5
Steps to Solve Equations with Fractional & Negative Exponents
1. Isolate the variable with the exponent
2. To solve for the variable, raise both sides of the equation to the reciprocal power.



In order to solve an exponential equation, you must have the same base
If the bases are the same, we set the exponents equal to each other and solve for the
variable.
If the bases are not the same, force them to be the same usually look for bases of 2, 3,
5, or 7
Growth/Decay
y = abx
Compounding Interest
 r
A  P 1  
 n
Continuous Growth/Decay
nt
A  Pe rt
Examples:
1
1. Which of the following is equivalent to x ?
81
(1) 3-4x
(2) 34
(3) 3-4
(4) 34x
3. What is the value of x in the
1
equation 16 x 2    ?
8
(2) -2
2 4 2
2. The expression
2a 3b 5
is equivalent to
2a
b
(1) ab3
(3)
2b3
(2)
a
(4) 2ab13
4. Given the equation y = abx, if the equation
models exponential growth, the value of b
must be greater than
x
(1) 1
 2a b 
(1) 1
(2) -1
7
(3)
8
8
(4)
7
12
(3) 0
(4) 2
5. Solve for x:
 x  27 
3
2
6. Andrew received a $3,200 bonus at work.
He invested his money in a savings account
that was making 4.5% interest that was
compounded monthly. Using the equaition
nt
 r
A  P  1   , where A is the value of the
 n
investment after t years, P is the principal
invested, r is the interest rate, and n is the
number of times per year it was compounded,
determine, to the nearest cent, how much
money Andrew would have after 6 years.
 4  68
LOGARITHMS
Log Rules
Exponential Rule
Log Rule
Example
Product
(xm)(xn) = xm + n
Log mn = Log m + Log n
Log 5(4) = Log 5 + Log 4
Quotient
xm
 x mn
n
x
Power
(xm)n = xmn
Log
m
= Log m – Log n
n
Log mn = n Log m
Log
5
= Log 5 – Log 2
2
Log x2 = 2 Log x
Logs to Exponentials and Exponentials to Logs
ba = c
Common Logs
* Base of 10
* Log x = Log 10 x
* Use 10x to solve
“Circle the base to finish the race”
“Log = Exponent”
log b c = a
Natural Logs
* Base of e
* Ln x = Ln e x
*Use ex to solve
Change of Base Formula
log x y 
Good Logs to Know
Log 1 = 0
Log 10 = 1
Log 100 = 2
Ln e = 1
Ln 1 = 0
13
log y
log x
Examples:
1. The expression
1
log a  3log b is equivalent
3
to
(1) log
3
a
b
3
(3) log

(4) log
a
3b 3
3
(2) log
a
3b
3. The expression log
3
a  b3
x2 y3
is equivalent to
z

2. If log 3 = x and log 5 = y, which of the
following could represent log 45?
(1) 2x + y
(2) 2xy
(3) x2 + y
(4) x2y
4. Solve for x:
log (x – 1) + log (2x – 3) = 1
1
log z
2
1
(2) (2 log x + 3 log y) - log z
2
1
(3) (log 2x + log 3y) – log z
2
2 x3 y
(4)
1
z
2
(1) (2 log x + 3 log y) +
5. Mouthwash manufacturers are constantly testing various chemicals on bacteria that thrive on
human saliva. The death of the bacteria exposed to Antigen 223 can be represented by the
function P(t) = 2,000e-0.37t where P(t) represents the number of bacteria from a population of
2,000 surviving after t minutes.
a) Determine the number of bacteria surviving 3 minutes after exposure to Antigen 223.
b) Using logarithm, determine the number of minutes, to the nearest tenth of a minute,
necessary to kill 1500 bacteria.
14
TRIG. FUNCTIONS
3 Basic Trig. Functions
 Always remember SOH CAH TOA to determine the 3 basic trig. functions
sin   __________ cos   __________ tan   __________

Good Things to Know With the Coordinate Axes

y

x


Co-Terminal Angles
Add ______ if original angle
was negative
Subtract ______ if original
angle was positive
Quadrantal Angles
Angles that lie on the
quadrants
00, 900, 1800, 2700, 3600
Reciprocal Functions
P(x, y)  P(
,
)
x =_____________
csc  = _________ = _________
y = _____________
sec  = _________ = _________
tan  = __________ = __________

*00
300
cot  = _________ = _________
450
600
*900
*1800
*2700
sin 
cos 
tan 




Cofunctions
Cofunctions are equivalent
if the angles are
complementary
sin  _______
tan  _______
sec  _______
ARC LENGTH
RADIANS  DEGREES
 Substitute _____ in
for_____
DEGREES  RADIANS
 Multiply by
15

If there’s no ____,
multiply by _____
Examples:
1. A circle has a radius of 4 inches. In inches,
what is the length of the arc intercepted by a
central angle of 2 radians?
(1) 2
(2) 2 
(3) 8
(4) 8 
3. What is the number of degrees in an angle
11
whose radian measure is
?
12
(1) 1500
(3) 3300
(2) 1650
(4) 5180
2. The expression
csc 
is equivalent to
sec 
sin 
cos 
cos 
(4)
sin 
(1) sin 
(3)
(2) cos 
4. The coordinates of a point on the unit circle

3 1
,   . If the terminal side of an
are  
2
2

angle  in standard position passes through
the given point, what is the measure of  ?
(1) 2400
(2) 2330
5. What is 2350 expressed in radian measure?
(1) 235 
(2)

235
(3) 2250
(4) 2100
6. Find the exact value of (sin 2250)(cos 3000).
47
36
36
(4)
47
(3)
TRIG. GRAPHS

Each value of a trig. function represents something important to graphing:
y = a sin bx + d
amplitude



frequency
midline
“a” – the amplitude tells us how far above and below the midline the graph will reach –
a negative “a” value will reflect the graph over the x-axis
“b” – the frequency tells us how many curves we will see between 0 and 2 - the period
 2 

 is based off of the frequency and tells us how it will take before we see one full
 b 
trig. curve
“d” – the midline tells us the line of horizontal symmetry – this line cuts the graph in half
horizontally – this will move the entire trig. graph up (positive midline value) or down
(negative midline value)
16
Examples:
1. What is the amplitude of the graph whose
equation is y = -2 sin 4x?
(1) 
(2) 2
(3) -2
2. If the period of a cosine curve is  , what is
the frequency?

(1) 2
(3)
2
1
(2) 2
(4)
2
(4) 4
3. Which is an equation of the graph shown
below?
3
4. What is the minimum value of the range of
y = 3 + 2 sin x?
(1) 1
(2) 0
(3) -1
(4) -5
2
-3
1
(1) y  3cos x
2
(2) y = 3 sin 2x
(3) y = -3 sin 2x
(4) y = 3 cos 2x
3

5. As  increases from to
, the value of
2
2
sin  will
6. What is the frequency of the graph of the
1
equation y  2 cos x  7 ?
2
(1) 2 
(3) -2 
1
(2)
(4) 2
2
(1) increase only (3) increase then decrease
(2) decrease only (4) decrease then increase
TRIG. IDENTITIES
These Identities You NEED To Memorize
Reciprocals
Quotients
Pythagorean
These Identities Are On The Reference Sheet
SUM AND DIFFERENCE IDENTITIES
Identities of the SUM of 2 Angles
Identities of the DIFFERENCE of 2 Angles
1. sin (A + B) = sin A cos B + cos A sin B
1. sin (A – B) = sin A cos B – cos A sin B
2. cos (A + B) = cos A cos B – sin A sin B
3. tan  A  B  
2. cos (A – B) = cos A cos B + sin A sin B
tan A  tan B
1  tan A tan B
3. tan  A  B  
17
tan A  tan B
1  tan A tan B
DOUBLE ANGLE IDENTITIES
Cos 2A
1. cos 2A = cos2A – sin2A
Sin 2A
1. sin 2A = 2 sin A cos A
2. cos 2A = 2 cos2A – 1
Tan 2A
2 tan A
1. tan 2 A 
1  tan 2 A
3. cos 2A = 1 – 2sin2A
sin
HALF ANGLE IDENTITIES
1
1  cos A
cos A  
2
2
1
1  cos A
A
2
2
Examples:
12
3
1. If sin x  , cos y  , and x and y are acute
13
5
angles, the value of cos(x + y) is
(1)
21
65
(2)
63
65
(3) 
14
65
(4) 
7
and x is in quadrant IV, then
25
cos 2x equals
48
25
5. Prove:
(2)
527
625
(3) 
14
25
(4)
(1) sin 
2
sin 
(2) 2 cos 
(2) cos 
(3) sec 
(4) tan 
5
and angle A is in quadrant I,
3
what is the value of cos2A?
4. If cos A 
6. Cos 700 cos 400 – sin 700 sin 400 is
equivalent to
(1) cos 300
(2) cos 700
(1)
csc 
is equivalent to
cot 
134
625
csc2  1  sin 2    cot 2 
7. The expression
1
1  cos A
A
2
1  cos A
33
65
3. If sin x  
(1) 
2. The expression
tan
sin 2
is equivalent to
sin 2 
(3) cos 1100
(4) sin 700
8
and 2700 < A < 3600, what is the
17
value of tan 2A?
8. If cos A 
(3) 2 cot 
(4) 2 tan 
18
19
Examples:
1. What value of x in the interval 900 < x < 1800
satisfies the equation sin 2 x  sin x  0 ?
2. Which of the following is not a solution to
the equation 2 cos x – 1 = sec x?
(1) 900
(1) 00
(2) 1200
(3) 1350
(4) 1800
3. Solve for the exact value of x in the interval
00 < x < 3600.
4csc x  5  3csc x  4
(2) 1200
(3) 1800
(4) 2400
4. Solve the equation below for all values of x
in the interval 00 < x < 3600. Round your
answer to the nearest degree.
2sin 2 x  5cos x  4
5. Solve for the exact value of x in the interval
00 < x < 2 .
6. Solve for the exact value of x in the interval
00 < x < 3600. Round to the nearest tenth of a
degree.
2 cos 2 x   cos x
2
 5cos x  3
cos x
20
21
Examples:
1. You must cut a triangle out of a sheet of
paper. The only requirements you must follow
are that one of the angles must be 60º, the side
opposite the 60º angle must be 40
centimeters, and one of the other sides must
be 15 centimeters. How many different
triangles can you make?
(1) 1
(2) 2
2. A garden is in the shape of an equilateral
triangle and has sides whose lengths of 10
meters. What is the area of the garden?
(1) 25m2
(2) 50m2
(3) 43m2
(4) 87m2
(3) 3
(4) 0
3. In ABC , if AC = 12, BC = 11 and m  A =
300, then ABC could be
(1) an obtuse triangle only
(2) an acute triangle only
(3) a right triangle only
(4) either an obtuse triangle or an acute
triangle
4. While sailing a boat offshore, Donna sees a
lighthouse and calculates that the angle of
elevation to the top of the lighthouse is 30.
When she sails her boat 700 feet closer to the
lighthouse, she fins that the angle of elevation
is now 50. How tall, to the nearest tenth of a
foot, is the lighthouse?
22
SERIES & SEQUENCES
Arithmetic Sequences
To Find the Nth Term
*To Find the Sum of N Terms
Geometric Sequences
To Find the Nth Term
*To Find the Sum of N Terms
Examples:
1. The value of the expression 2  n 2  2n 
2. Which expression represents the sum of the
sequence 3, 5, 7, 9, 11?
(1) 12
(1)
2
n 0
(2) 22
(3) 24
(4) 26
5
 2n  1
(3)
n 1
(2)
5
 3n
n 1
3. What is the fifteenth term of the sequence 5,
-10, 20, -40, 80…?
(1) -163,840
(2) 81,920
(3) -81,920
(4) 327,680
5
 3n  1
n 1
(4)
5
 n 1
n 1
4. Find the first four terms of the recursive
sequence defined below.
a1= -4
an = 2an-1 – 2n
23
ONE VARIABLE STATISTICS



Measures of Central Tendency
Mean  average - x
Median  middle number – numbers
must be in order
Mode  number that appears most
often




Measures of Dispersion
Range  biggest # - smallest #
Interquartile Range  Q3 – Q1
Variance  (standard deviation)2
Standard Deviation  variance
**REMEMBER THAT EACH PART OF THE NORMAL CURVE
REPRESENTS .5 STANDARD DEVIATIONS FROM THE MEAN**
24
Examples:
1. A new survey is designed to collect data
based on the number of miles people drive
each month. Which of the following groups
would create the least bias in conducting the
survey?
2. The Write-O Pen company manufactures
pens that have a mean life time of 200 pages
with a standard deviation of 12 pages.
Approximately what percentage of the pens
last between 182 and 212 pages?
(1) All students who attend the local high
school
(2) People at a senior citizens center
(3) People entering the mall
(4) People riding the subway
3. In a normal distribution, x  2  50 and
x  2  10 when x is the mean and  is the
standard deviation. What is the mean?
4. On a standardized test with normal
distribution, the mean is 85 and the standard
deviation is 6. If 1400 students took the test,
approximately how many students would be
expected to score between 79 and 97?
PROBABILITY




PERMUTATIONS
Order matters
Key words  arrangement/arrange,
filling specific roles (president, vp,
secretary or 1st, 2nd, 3rd place),
ordering/order, license plates,
telephone numbers, or making
“words”
Open up spots to fill, then multiply
 totalnumber !
Repeated “letters” 
 repeatedletters !

25
COMBINATIONS
Order does not matter

Key words  committees, chose,
chosen, groups, team, gather, or
assemble/assembly

Make a have/want chart
Bernoulli Trials
 “Exactly” Probability
(haveCwant)(swant)(fhave-want)
Probability of success

Probability of failure
(1 – prob. of success)
“AT MOST” PROBABILITY
consider all the probabilities equal to

or less than the given probability
(ex.) Mrs. Pace had a class of 10 students. She
was to choose AT MOST 7 students to attend a
trip. How many students could she have sent?
7, 6, 5, 4, 3, 2, 1, or 0
“AT LEAST” PROBABILITY
consider all the probabilities equal to
or more than the given probability
(ex.) Mrs. Pace had a class of 10 students. She
was to choose AT LEAST 7 students to attend
a trip. How many students could she have
sent? 7, 8, 9, or 10
Binomial Expansion
 Used like FOIL, but with exponents greater than 2  (x + 4)6
 On the reference sheet:
Examples:
1. A family with 6 children is selected at
random for a study. What is the probability
that the family will have at most 2 girls?
2. What is the third term in the expansion of
(x – 2)7?
3. In one state, a license plate consists of three
letters followed by two digits. If no letter or
digit can be repeated, how many different
license plates are possible?
4. In a group of 8 students, three are female
and five are male. What is the probability that
two females and one male will be chosen to
work on a subcommittee?
26
TWO VARIABLE STATISTICS
Regressions
 Correlation coefficient (r)  how close to the “line” of best fit will the data lie – closest to
+1 or -1 is the “best” fit line
 Linear regressions (LinReg), exponential regressions (ExpReg), logarithmic regressions
(LnReg), power regressions (PwrReg), cubic regressions (CubicReg) and quadratic
regressions (QuadReg)
Linear
Logarithmic
Exponential
y = ax + b
y = a + blnx
y =abx
Power
y = axb
y=
ax3
Cubic
+ bx2 + cx + d
Quadratic
y = ax2 + bx + c
Example:
The accompanying table shows the enrollment of a preschool from 1980 through 2000.
a) Write a linear regression equation to model this data.
b) How many students did they have in 1998?
c) In what year did enrollment reach 50 students?
27
BRING ON THE
REGENTS
EXAM… GO
ON…BRING IT!