FORMULAS & FACTS I NEED TO KNOW FOR THE REGENTS Flashcards
Positive Perfect Square
Real, rational, unequal
2 x-intercepts
Positive Nonperfect Square
Real, irrational, unequal
2 x-intercepts
Zero
Real, rational, equal
1 x-intercept
Negative
Imaginary
0 x-intercepts
i^0
1
i^1
i
i^2
-1
i^3
-i
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)
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
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>
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
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
“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
EXPONENTIAL AND LOGARITHM RULE
Product
Exponential
x^m x x^n=x^m+n
Logarithm
Log_bmn=Log_bm + Log_bn
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.
