final exam equations

Question 1

vertex form of quad function

Answer 1

f(x)= a(x-h)²+k

Question 2

completing the square

Answer 2

1. add and subtract (b/2)² from x²+bx
2. factor x²+bx+(b/2)² into (x+b/2)² and combine any constants

Question 3

quadratic equation

Answer 3

x = -b ± √(b² - 4ac)/2a

Question 4

factored form

Answer 4

f(x)= a(x-x1)(x-x2)

Question 5

vertex form (h,k)

Answer 5

h= b/2a AND k=f(h)

Question 6

discriminant

Answer 6

d= b²-4ac
d>0= 2 solutions
d<0= no solutions
d=0= 1 solution

Question 7

power functions

Answer 7

y=kx^p
y=#x^#
ex: 2x^5

Question 8

exponent rules

Answer 8

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

Question 9

fractional exponents

Answer 9

n√a^m=a^m/n

Question 10

root/factional exponent properties

Answer 10

n√a x b= n√a x n √b AND n√a/b= n√a/n√b

Question 11

exponential functions

Answer 11

y=ab^t
a= initial value
b= growth factor

Question 12

growth factor

Answer 12

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

Question 13

doubling time T

Answer 13

f(t)= A(2)^t/T

Question 14

half life T

Answer 14

f(t)= A(1/2)^t/T

Question 15

for an arbitrary scaling amount c

Answer 15

f(t)= A*c^t/T

Question 16

exponential function with base of e (continuous)

Answer 16

f(t)= A*e^kt OR b= e^k

k= continuous growth rate
pos= exp. growth
neg= exp. decay

Question 17

standard form of quad function

Answer 17

y=ax²+bx+c

Question 18

factor a quad function in the standard form

Answer 18

x² +Bx+C= (x+M) (x+N)
find 2 numbers M and N whose product is C and whose sum is B

Question 19

Positive coefficient even negative integer exponent

Answer 19

x-² (graphs in quad 1 & 2, pointing away from each other)

Question 19

factoring a difference of squares

Answer 19

a²-b² factors into (A+B) (A-B)

Question 20

negative coefficient odd negative integer exponent

Answer 20

x^-1 (graphs in quad 1 & 4, pointing away from each other)

Question 21

positive coefficient even positive integer exponent

Answer 21

x² ( U graph)

Question 22

fractional exponent rules

Answer 22

n√a^m= a^m/n

√a= a^1/2

3√a^4= a^4/3

Question 23

logarithm definition

Answer 23

logb(x)= p

means b^p=x

Question 24

common log

Answer 24

log with a base of 10 (written as log)

Question 25

natural log

Answer 25

logarithm with a base of e (written as ln)

Question 26

logarithm properties

Answer 26

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)

Question 27

change of base

Answer 27

logb(x)= log(x)/log(b) OR logb(x)= ln(x)/ln(b)

Question 28

anytime base is larger its a fraction

Answer 28

log9(3)

Question 29

no real answer (CANT do logs of 0 or negative numbers)

Answer 29

log2(-8)

Question 30

vertical shift

Answer 30

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)

Question 31

horizontal shift

Answer 31

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

Question 32

reflections

Answer 32

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

Question 33

vertical scaling

Answer 33

g(x)= a*f(x) is a vertical stretching of f(x) if |a|>1 and a vertical compression if 0<|a|<1

Question 34

horizontal scaling

Answer 34

g(x)= f(b*x) is a horizontal compression of f if |b|>1 and a horizontal stretching if 0<|b|<1

Question 35

y= -2(5(x-2))^3-6

Answer 35

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

Question 36

composition

Answer 36

(f o g)(x)= f(g(x)) is the composition of f with g

Question 37

periodic function

Answer 37

f(t)= f(t+p)

Question 38

period

Answer 38

the length of a cycle in a periodic function

Question 39

amplitude

Answer 39

amplitude is half the vertical height of the function y= maxy-miny/2

Question 40

midline

Answer 40

midline is the horizontal line through the midpoint between the maximum and minimum y- value y= maxy+miny/2

Question 41

coeterminal angles

Answer 41

a= b+ 360k (for degrees) or a=b+2πk (for radians) for some integer k

Question 42

arc length

Answer 42

s= p*r for radius r and angle p in radiants

Question 43

to convert from degrees to radians

Answer 43

multiply the degree angle by: π radius/180 D>R * π/180

Question 44

to convert radians to degrees

Answer 44

multiply the radian angle by: 180/π radians R>D * 180/π

Question 45

for an ordered pair (x,y) on the unit circle with associated angle a we define:

Answer 45

x= cos a y= sin a

Question 46

transformation of sine and cosine functions

Answer 46

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

Question 47

sine graph

Answer 47

SAHALA (axis high axis low axis) reflected sine curve (axis low axis high axis)

Question 48

cosine graph

Answer 48

CHALA (high axis low axis high) reflected cosine curve (low axis high axis low)