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Math 30-1 Review Sheet
Function
Type
General Function
Form
Original
Function
Horizontal
Stretch
y  f ( x)
y  f (bx)
Vertical
Stretch
y  af ( x)
1
b
Mapped To ( x, y )  ( x, y )
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
( x, y )  ( x  h, y )
( x, y )  ( x, y  k )
By Factor a
Wider: 0  b  1
No Effect
Effect
Horizontal
Translation
Vertical Stretch
Horizontal Stretch
By Factor
Study Smarter, Do Better
Shorter: 0  a  1
Narrower: b  1
Taller: a  1
Reflect y  axis : b  0
Reflect x  axis : a  0
1

( x, y )   x, y 
b

( x, y )  ( x, ay )
O rde r o f O pe ra t io ns
1)Ho riz. stretch and reflectio n
2 )Vertical strech and reflectio n
3 )Ho riz. and vertical translatio ns
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 Multiplicity of Zero
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
Maximum or Minimum
No max or min
an  0
Qua dra nt II to I
P( x)
R
 Q( x) 
xa
xa
Polynomial has factor x - a
x  2x  4
x  1 x3  3x 2  2 x  5
of multiplicity n
ex) y  ( x  2) 2 x  2 is a zero of
repeated n times, x  a is zero
2
x  x
3
2

2 x2  2 x
  2x  2x
an  0
an  0
an  0
Qua dra nt III to IV Qua dra nt III to I
y  a log  b( x  h)   k
multiplicty 2
Even Multiplicity
Odd Multiplicity
Touches the x-axis
Cuts the x-axis
ex) y  ( x - 2) 2
ex) y  ( x - 2)3
2
Q( x)
  4 x  4 
 x2  2x  4 
9
Radical Functions
Inverse of Function
Domain of y 
Denoted by:
Range of y 
f ( x) any value f ( x)  0
f ( x) any value
f ( x)  0
f ( x)  0
y
f ( x)
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)
Solve Radical Functions Algebraically

y  f 1 ( x)
( x, y )  ( y , x )
y  f ( x)  x  f ( y )
f ( x)  1
y
f ( x) and
f ( x)  1
y
y  f ( x) intersect
f ( x) below
y  f ( x)
at f ( x)  1
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)  0 then x - a is a factor of P( x)
3)Find Roots of Equation (Solve for x)
4)Check solutions in original equation
Integral Zero Theorem
9
x 1
xa
Logarithmic and Exponential Modeling
Model Real Life Problems Using
Final Quantity=initial quantity  (change factor) number of changes
y  Final Quantity
y  a  bx
a  initial quantity
b  change factor
x  number of changes
Pythagorean Theorem
a 2  b2  c2
Right Angle
B
Triangles Only
a
c
If x - a is a factor of P( x)
Then x  a is an integral zero of P( x)
R
 4x  5
Qua dra nt II to IV
A
b
C
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 A
h( x )  f ( x )  g ( x )
1
opposite
csc  
sin 
hypotenuse
adjacent
1
cos  
sec  
hypotenuse
cos 
opposite
1
tan  
cot  
adjacent
tan 
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)
Pythagorean Identity
cos  A  B   cos A cos B  sin A sin B
sin 2   cos 2   1
tan  A  B  
sin  
1  tan 2   sec 2 
2
Difference
sin  A  B   sin A cos B  cos A sin B
tan A  tan B
1  tan A tan B
Exponent Rules
1  cot 2   csc 2 
a a a
b
c
bc
a 
b c
a
h( x)  ( f  g )( x)
ab
 a b c
ac
bc
tan 2 A 
2 tan A
1  tan 2 A
h( x)  f ( x) g ( x)  ( f g )( x)
Log Rules
f ( x)
log c MN  log c M  log c N h( x)  g ( x)
M
log c
 log c M  log c N h( x)   f  ( x) g ( x)  0
N
g
log c M 
log a M
log a C
h( x)  f ( g ( x))  ( f g )( x)
y  sin 
Max is  1 and Min is -1
Amplitude is 1
𝑦 = sin 𝑥
Properties
Trigonometry and Unit Circle
Period is 2
y -intercept is 0
 intercepts at - ,0, , 2
Domain  |   R
Range  y | 1  y  1, y  R
y  cos 
1
ab
Coterminal Angle
a  r
CAST rule
Period 
  360 n, n  N
  2 n, n  N
Amplitude 
b of second kind that are identical
n!
can be arranged in
ways
a !b !
11/26/13
  3 5
 intercepts at - , , ,
2 2 2
Domain  |   R
2
Range  y | 1  y  1, y  R
Binomial Theorem
# of Coefficients  (n  1) row of Pascal's Triangle

+'ve  x
a of one kind that are identical and
n!
, n N
(n  r )!r !
y -intercept is 1
y
n objects with
Cr 
Period is 2
Angle Direction
n !  n  (n  1)  (n  2)  3  2  1
n!
, n N
n Pr 
(n  r )!
2 360
or
b
b
Amplitude is 1
Max Value - Min Value
2
Permutations and Combinations
n
Max is  1 and Min is -1
𝑦 = cos 𝑥
Properties
a b 
Pascal's Triangle
NOTE
0!  1
n objects to choose from
Combinations
1
y
1

1
1
-'ve  x
1
1
2
3
4
5
0C0
1
1C0
1
3
6
10
2C0
1
4
10
3C0
1
5
4C0
1 5C0
3C1
4C1
5C1
1C1
2C1
2C2
3C2
4C2
5C2
3C3
4C3
5C3
4C4
5C4
5C5
r number of items taken
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
n
General Term Number
tk 1  n Ck ( x) n  k ( y ) k
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