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Function
Type
General Function
Form
Original
Function
Horizontal
Stretch
y  f ( x)
y  f (bx)
Vertical
Stretch
Horizontal
Translation
Vertical
Translation
Combined
Transformation
y  f ( x  h)
y  f ( x)  k
y  af  b( x  h)   k
h  0 Shift Right
k  0 Shift Up
h  0 Shift Left
k  0 Shift Down
Apply Transformations in
BEDMAS Order
(Stretches and reflections,
then translations)
( x, y )  ( x, ay ) ( x, y )  ( x  h, y )
( x, y )  ( x, y  k )
y  af ( x)
Vertical Stretch
Horizontal Stretch
By Factor
1
b
By Factor a
Wider: 0  b  1
No Effect
Effect
Study Smarter, Do Better
Shorter: 0  a  1
Narrower: b  1
Taller: a  1
Reflect y  axis : b  0
Reflect x  axis : a  0


Mapped To ( x, y )  ( x, y ) ( x, y )   b x, y 
1


1

( x, y )   x  h, ay  k 
b

y x
y  bx
ya x
y  xh
y  x k
y  a b( x  h)  k
y  sin 
y  sin b
y  a sin 
y  sin   h 
y  sin   k
y  a sin  b(  h)   k
y  cos 
y  cos b
y  a cos 
y  cos   h 
y  cos   k
y  a cos  b(  h)   k
Exponential
y  cx
y  c bx
y  ac x
y  c xh
y  cx  k
y  ac b ( x  h )  k
Logarithmic
y  log x
y  log bx
y  a log x
y  log  x  h 
y  log x  k
Radical
Sinusiodal
Polynomial Functions
Remainder Theorem
Form: f ( x)  an x n  an 1 x n 1  an  2 x n  2    a2 x 2  a1 x1  a0
Even Degree
Odd Degree
At least 1 to maximum of n x  intercepts
0 to maximum of n x  intercepts
Domain x | x  R
Domain x | x  R
Range depends on max and min
Range  y | y  R
y -intercept ao
y -intercept ao
an  0
Qua dra nt II to I
Polynomial has factor x - a
x  1 into x  3 x  2 x  5
of multiplicity n
3
2


1 2 4
an  0
an  0
an  0
Qua dra nt III to IV Qua dra nt III to I
Qua dra nt II to IV
Multiplicity of Zero
P( x)
R
 Q( x) 
xa
xa
1 1 3 2 5
 1 2 4
No max or min
Maximum and Minimum
y  a log  b( x  h)   k
9

x3  3x 2  2 x  5
x 1
9
2
 x  2x  4 
x 1
repeated n times, x  a is zero
ex)
y  ( x  2) 2
x  2 is a zero
multiplicty of 2
Angle Direction
-'ve
+'ve
Logarithmic and Exponential Modeling
Inverse of Function
Radical Functions
Domain of y 
Range of y 
f ( x) any value
f ( x)  0
f ( x)  0
y
f ( x)
Denoted by:
f ( x) any value f ( x)  0
y
f ( x) and
y  f ( x) intersect
Undefined
at x  0
Swap x
and y
f ( x) is defined
0  f ( x)  1
y
f ( x) above
y  f ( x)

( x, y )  ( y , x )
y  f ( x)  x  f ( y )
f ( x)  1
y
y  f 1 ( x)
f ( x) and
y  f ( x) intersect
f ( x)  1
y
Model Real Life Problems Using
Final Quantity=initial quantity  (change factor) number of changes
y  Final Quantity
a  initial quantity
b  change factor
y  a  bx
x  number of changes
f ( x) below
y  f ( x)
at f ( x)  1
Solving Radical Functions Algebraically
Factor Theorem
1) List Restrictions
If x - a is a factor then P( a)  0
2)Isolate Radical and Square Both Sides
If P(a) then x - a is a factor of P( x)
Sine Law
Pythagorean Theorem
sin A sin B sin C


a
b
c
a 2  b2  c2
B
Cosine Law
a 2  b 2  c 2  2ab cos A
3)Find Roots of Equation (Solve for x)
NOTE: Sine and Cosine Law
4)Check solutions in original equation
True For Any Triangle
Right Angle
Triangle Only
a
c
C
A
b
Trig Identities
Sum
Double Angle
Function Notation
SOH CAH TOA
sin  A  B   sin A cos B  cos A sin B
sin 2 A  2sin A cos B
h( x )  f ( x )  g ( x )
cos  A  B   cos A cos B  sin A sin B
cos 2 A  cos 2 A  sin 2 A
tan A  tan B
tan  A  B  
1  tan A tan B
cos 2 A  2 cos 2 A  1
h( x )  f ( x )  g ( x )
cos 2 A  1  2sin A
h( x)  ( f  g )( x)
opposite
hypotenuse
adjacent
cos  
hypotenuse
sin  
tan  
opposite
adjacent
1
csc  
sin 
1
sec  
cos 
1
cot  
tan 
2
Difference
sin  A  B   sin A cos B  cos A sin B
cos  A  B   cos A cos B  sin A sin B
tan  A  B  
tan A  tan B
1  tan A tan B
Exponent Rules
a a a
b
c
bc
a 
b c
a
bc
tan 2 A 
2 tan A
1  tan 2 A
f ( x)
g ( x)
log c MN  log c M  log c N
h( x ) 
M
 log c M  log c N
N
log a M
log c M 
log a C
 f 
h( x )    ( x ) g ( x )  0
g
log c
ab
 a b c
ac
h( x)  f ( x) g ( x)  ( f g )( x)
Log Rules
y  sin 
Trigonometry and Unit Circle
h( x)  ( f  g )( x)
h( x)  f ( g ( x))  ( f g )( x)
Max is  1 and Min is -1
Amplitude is 1
Period is 2
Period 
y -intercept is 0
2 360
or
b
b
 intercepts at - ,0, , 2
Domain  |   R
Range  y | 1  y  1, y  R
a b 
y  cos 
1
ab
Max is  1 and Min is -1
Amplitude is 1
Period is 2
y -intercept is 1
Amplitude 
Max Value - Min Value
2
  3 5
 intercepts at - , , ,
2 2 2
Domain  |   R
Coterminal Angle
a  r
2
Range  y | 1  y  1, y  R
CAST rule
Binomial Theorem
  360 n, n  N
  2 n, n  N
Coefficients of terms  (n  1) row of Pascal's Triangle
Permutations and Combinations
n !  n  (n  1)  (n  2)  3  2  1
n objects with
a of one kind that are identical and
NOTE
b of second kind that are identical
n!
can be arranged in
ways
a !b !
0!  1
n Pr 
n
Cr 
n!
, n N
(n  r )!
n!
, n N
(n  r )!r !
n objects to choose from
r number of items taken
n
Properties of Binomial Expansion of ( x  y ) n  N
Order MATTERS
Write Expansion in descending order of exponent of first term (In this case x)
n objects to choose from
Term number is k  1
r number of items taken
Expansion contains n  1 terms
Order DOESN'T matter
Sum of exponents in any term n
General Term Number
tk 1  n Ck ( x) n  k ( y ) k
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9/12/13