final exam equations Flashcards
vertex form of quad function
f(x)= a(x-h)²+k
completing the square
- add and subtract (b/2)² from x²+bx
- factor x²+bx+(b/2)² into (x+b/2)² and combine any constants
quadratic equation
x = -b ± √(b² - 4ac)/2a
factored form
f(x)= a(x-x1)(x-x2)
vertex form (h,k)
h= b/2a AND k=f(h)
discriminant
d= b²-4ac
d>0= 2 solutions
d<0= no solutions
d=0= 1 solution
power functions
y=kx^p
y=#x^#
ex: 2x^5
exponent rules
a^1=0
a^-p=1/p
a^p*a^r=a^p+r
a^p/a^r=a^p-r
(a^p)^r=a^pr
fractional exponents
n√a^m=a^m/n
root/factional exponent properties
n√a x b= n√a x n √b AND n√a/b= n√a/n√b
exponential functions
y=ab^t
a= initial value
b= growth factor
growth factor
b= 1+r where r is the growth factor (or decay) rate as a decimal
r= growth rate
b= growth factor
if GIVEN r, b= 1+r
if GIVEN b, r= b-1
pos= exp. growth
neg= exp. decay
doubling time T
f(t)= A(2)^t/T
half life T
f(t)= A(1/2)^t/T
for an arbitrary scaling amount c
f(t)= A*c^t/T
exponential function with base of e (continuous)
f(t)= A*e^kt OR b= e^k
k= continuous growth rate
pos= exp. growth
neg= exp. decay
standard form of quad function
y=ax²+bx+c
factor a quad function in the standard form
x² +Bx+C= (x+M) (x+N)
find 2 numbers M and N whose product is C and whose sum is B
Positive coefficient even negative integer exponent
x-² (graphs in quad 1 & 2, pointing away from each other)
factoring a difference of squares
a²-b² factors into (A+B) (A-B)
negative coefficient odd negative integer exponent
x^-1 (graphs in quad 1 & 4, pointing away from each other)
positive coefficient even positive integer exponent
x² ( U graph)
fractional exponent rules
n√a^m= a^m/n
√a= a^1/2
3√a^4= a^4/3
logarithm definition
logb(x)= p
means b^p=x
common log
log with a base of 10 (written as log)
natural log
logarithm with a base of e (written as ln)
logarithm properties
logb(b^t)=t
b^logb(t)=t
logb(x*y)= logb(x) + logb(y)
logb(x/y)= logb(x) - logb(y)
logb(x^t)= t * logb(x)
change of base
logb(x)= log(x)/log(b) OR logb(x)= ln(x)/ln(b)
anytime base is larger its a fraction
log9(3)
no real answer (CANT do logs of 0 or negative numbers)
log2(-8)
vertical shift
g(x)=f(x)+k is a vertical shift of f(x) up k units if k>0 (or positive) and down k units if k<0 (or negative)
horizontal shift
g(x)= f(x+h) is a horizontal shift of f(x) right h units if h<0 and left k units if h>0
reflections
g(x)=-f(x) is a reflection of f(x) about the horizontal axis, and g(x)= f(-x) is a reflection of f(x) about the vertical axis
vertical scaling
g(x)= a*f(x) is a vertical stretching of f(x) if |a|>1 and a vertical compression if 0<|a|<1
horizontal scaling
g(x)= f(b*x) is a horizontal compression of f if |b|>1 and a horizontal stretching if 0<|b|<1
y= -2(5(x-2))^3-6
original function= x^3
vertical shift down= 6 units
horizontal shift right= 2 units
vertically stretched by a factor of 2 (outside)
reflected over= x axis
horizontally compressed by a factor of 5
composition
(f o g)(x)= f(g(x)) is the composition of f with g
periodic function
f(t)= f(t+p)
period
the length of a cycle in a periodic function
amplitude
amplitude is half the vertical height of the function y= maxy-miny/2
midline
midline is the horizontal line through the midpoint between the maximum and minimum y- value y= maxy+miny/2
coeterminal angles
a= b+ 360k (for degrees) or a=b+2πk (for radians) for some integer k
arc length
s= p*r
for radius r and angle p in radiants
to convert from degrees to radians
multiply the degree angle by: π radius/180
D>R * π/180
to convert radians to degrees
multiply the radian angle by: 180/π radians
R>D * 180/π
for an ordered pair (x,y) on the unit circle with associated angle a we define:
x= cos a
y= sin a
transformation of sine and cosine functions
Sine: f(t)= Asin(Bt)+D
cosine: f(t)= Acos(Bt)+D
amplitude |A|= maxy-miny/2
period P= 2π/B; this implies that B= 2π/P
vertical shift D (mindline)= maxy+miny/2
sine graph
SAHALA (axis high axis low axis)
reflected sine curve (axis low axis high axis)
cosine graph
CHALA (high axis low axis high)
reflected cosine curve (low axis high axis low)
