FORMULAS & FACTS I NEED TO KNOW FOR THE REGENTS

Question 1

Positive Perfect Square

Answer 1

Real, rational, unequal

2 x-intercepts

Question 2

Positive Nonperfect Square

Answer 2

Real, irrational, unequal

2 x-intercepts

Question 3

Zero

Answer 3

Real, rational, equal

1 x-intercept

Question 4

Negative

Answer 4

Imaginary

0 x-intercepts

Question 5

i^0

Answer 5

1

Question 6

i^1

Answer 6

i

Question 7

i^2

Answer 7

-1

Question 8

i^3

Answer 8

-i

Question 9

Standard Form of an Equation of a Parabola with vertex (h,k)
p= distance from vertex to focus (or vertex to directrix)
USE THIS EQUATION FOR FOCUS/DIRECTRIX PROBLEMS

Answer 9

(x-h)^2 = 4p(y-k)

Question 10

Exponential Growth Model
y= ending amount
a= initial amount
r= rate of change
* percent rate of change (% increase/decrease)= r x 100%
^ To interpret percent rate change, the coefficient of t must be 1

Answer 10

y= a(1+r)^t
Growth factor = b= 1 + r
For exponential GROWTH, b> 1

Question 11

Exponential Decay Model
y= ending amount
a= initial amount
r= rate of change
* percent rate of change (% increase/decrease)= r x 100%
^ To interpret percent rate change, the coefficient of t must be 1

Answer 11

y= a(1-r)^t
Decay factor= b= 1-r
For exponential GROWTH, 0<b></b>

Question 12

CONTINUOUS GROWTH/DECAY 
where y= ending amount 
t= time 
P= initial amount 
r= rate of change
* NOTE: r is positive for continuous growth, r is negative for continuous decay

Answer 12

y= Pe^rt

Question 13

COMPOUNDING "n" TIMES 
where y= ending amount 
t= time 
P= initial amount 
r= rate of change
n= number of times compounded in a year

Answer 13

y= a(1+r/n)^nt

Question 14

“HALF LIFE” FORMULA
y= ending amount
a= initial amount
b= growth/decay factor (ex: b=1/2 for half life, b=2 for doubling, etc.)
t= time (usually in years)
H= “half life” (# of units of time it takes for substance to grow/decay)
* NOTE: This formula can be modified for any type of exponential growth/ decay that grows every # units of time.*

Answer 14

y= a(b)^t/H

Question 15

EXPONENTIAL AND LOGARITHM RULE

Product

Answer 15

Exponential
x^m x x^n=x^m+n
Logarithm
Log_bmn=Log_bm + Log_bn

Question 16

EXPONENTIAL AND LOGARITHM RULE

Quotient

Answer 16

Exponential
x^m/x^n=x^m-n
Logarithm
Log_bm/n=Log_bm-Log_bn

Question 17

EXPONENTIAL AND LOGARITHM RULE

Power

Answer 17

Exponential
(x^m)^n=x^mn
Logarithm
Log_bm^n=nLog_bm

Question 18

MORE EXPONENT RULES

Answer 18

Look to green packet for answers

Question 19

Discriminant

Answer 19

b^2-4ac

Question 20

Trig ratios

Answer 20

SINE 
Opposite over hypotenuse 
COSINE 
Adjacent over hypotenuse
TANGENT 
Opposite over adjacent 
COSECANT
Hypotenuse over opposite
SECANT 
Hypotenuse over adjacent 
COTANGENT
Adjacent over opposite

Question 21

DEGREES TO RADIANS AND RADIANS TO DEGREES

Answer 21

DEGREES TO RADIANS
Multiply by pie over 180 degrees
RADIANS TO DEGREES
Multiply by 180 degrees over pie

Question 22

ARC LENGTH OF A CIRCLE

Answer 22

S=(phata)(r)

S= arc length
Phata= central angle intercepting the arc, in RADIANS 
r= radius

Question 23

ON A UNIT CIRCLE

Answer 23

x= cosine 
y= sine 
(x,y)= (cosine, sine) 
tangent= y over x= sine over cosine 
PYTHAGOREAN TRIG IDENTITY 
sine^2 + cosine^2= 1

Question 24

CAST

Answer 24

Quadrant 1- ALL trig functions are POSITIVE
Quadrant 2- SINE IS POSITIVE
Quadrant 3- TANGENT IS POSITIVE
Quadrant 4- COSINE IS POSITIVE

Question 25

FINDING COTERMINAL ANGLES

Answer 25

Add or subtract 360 degrees (can do this as many times as needed)

Question 26

RULES FOR FINDING REFERENCE ANGLES

Answer 26

Quadrant 1- phata
Quadrant 2- 180-phata
Quadrant 3- phata-180
Quadrant 4- 360- phata

Question 27

SPECIAL ANGLES CHART

Answer 27

Phata(in degrees) 30 degrees 45 degrees 60 degrees
Sine 1/2 Square root of 2/2 Square root of 3/2
Cosine Square root of 3/2 Square root of 2/2 1/2
Tangent Square root of 3/3 1 Square root of 3

Question 28

TRIGONOMETRIC GRAPHS

Answer 28

y= Asin(B(x-C)) + D
y= Acos(B(x-C)) + D
|A|= amplitude (height of the curve from the midline to max or midline to min)
B= frequency (# of cycles in 2 pie interval)
C= phase shift (horizontal shift)
D= midline (vertical shift)
Period= 2 pie/B (Period is the length of one complete cycle)
Note: B= 2 pie/ period

Question 29

ADDITIONAL TRIG GRAPH INFORMATION

Answer 29

• Formula for Magic # = period/4. This tells you how to mark out your x-axis
• to find the maximum value: midline + amplitude
• to find the minimum value: midline - amplitude
• sine functions start at “origin” (midline when translated) - “OMOMO”
• cosine functions start at maximum (minimum when reflected)- “MOMOM”

Question 30

ODD, EVEN, OR NEITHER FUNCTIONS

Answer 30

Odd: symmetric to the origin
F(-x)= -F(x)
Even: symmetric to the y-axis
F(x)= F(-x)

Both are unequal if its neither

Question 31

GRAPHING POLYNOMIALS

Answer 31

Zeroes= x-intercepts
Zeros of ODD multiplicities CROSS the x-axis, zeros of EVEN multiplicities BOUNCE
Odd-degree polynomials have end behavior like x^3, even-degree polynomials have end behavior like x^2

Question 32

AVERAGE RATE OF CHANGE

Answer 32

Average rate of change= find the slope

Slope formula= y2-y1/ x2-x1

Question 33

FINDING THE INVERSE

Answer 33

The inverse is a reflection in the line y=x
(x,y)- (y,x)
To find the inverse algebraically, simply switch the x and y values in the equation and then get the equation in y= form.