FORMULAS & FACTS I NEED TO KNOW FOR THE REGENTS Flashcards by Katharine Colgan

Positive Perfect Square

Real, rational, unequal

2 x-intercepts

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Positive Nonperfect Square

Real, irrational, unequal

2 x-intercepts

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Zero

Real, rational, equal

1 x-intercept

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Negative

Imaginary

0 x-intercepts

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i^0

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i^1

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i^2

-1

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i^3

-i

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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

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

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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

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

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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

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

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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

y= Pe^rt

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

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

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“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.*

y= a(b)^t/H

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EXPONENTIAL AND LOGARITHM RULE

Product

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

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EXPONENTIAL AND LOGARITHM RULE

Quotient

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

EXPONENTIAL AND LOGARITHM RULE

Power

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

MORE EXPONENT RULES

Look to green packet for answers

Discriminant

b^2-4ac

Trig ratios

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

DEGREES TO RADIANS AND RADIANS TO DEGREES

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

ARC LENGTH OF A CIRCLE

S=(phata)(r)

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

ON A UNIT CIRCLE

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

CAST

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

FINDING COTERMINAL ANGLES

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

RULES FOR FINDING REFERENCE ANGLES

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

SPECIAL ANGLES CHART

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

TRIGONOMETRIC GRAPHS

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

ADDITIONAL TRIG GRAPH INFORMATION

- 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"

ODD, EVEN, OR NEITHER FUNCTIONS

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

GRAPHING POLYNOMIALS

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

AVERAGE RATE OF CHANGE

Average rate of change= find the slope Slope formula= y2-y1/ x2-x1

FINDING THE INVERSE

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.