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PEMDAS, FUNCTIONS, GRAPHS,
SUMMATION AND FACTORIALS
PEMDAS
1. Parantheses
2. Exponents
3. Multiplication or Division
4. Addition or Subtraction
PEMDAS
Without PEMDAS, two different answers:
3-2x3
3 - 2 x 3 = (3 - 2) x 3 = 1 x 3 = 3
3 - 2 x 3 = 3 - (2 x 3) = 3 - 6 = -3
PEMDAS
With PEMDAS:
3 - 2 x 3 = 3 - (2 x 3) = 3 - 6 = -3
Multiplication comes before subtration:
peMdaS
EXAMPLE OF PEMDAS
7 + (6 x 52 + 3)
= 7 + (6 x 25 + 3)
parenthesis first, then exponent
= 7 + (150 + 3)
multiply
= 7 + 153 = 160
add
Try:
(3+22 - 5) x (3-22)
(7 - √9) x (42 - 3 + 1)
(9 - 22 )2 + 4
INEQUALITIES
> means ‘greater than’
a > b means a is greater than b
< means ‘less than’
a < b means a is less than b
a < b < c means b is between a and c
a > 0 iff a is positive
a < 0 iff a is negative
INEQUALITIES
If a < b and b < c then a < c and similarly
if a > b and b > c then a > c
2 < 5 and 5 < 7 then 2 < 7
Adding a constant c does not change the inequalities:
if a < b then (a + c) < (b + c) {same for >}
if 2 < 5 and c = 4 then (2 + 4) < (5 + 4) or 2 < 9
INEQUALITIES
When multiplying does not change the inequalities
if c > 0: if a < b then ac < bc (and similarly for >)
2 < 5 and c = 2 then (2*2) < (5*2) or 4 < 10
When multiplying does change the inequalities
if c < 0: if a < b then ac > bc (and similarly for >)
2 < 5 and c = -2 then (2*-2) > (5*-2) or -4 > -10
EXAMPLE OF INEQUALITY
(24 < 6 - y < 32) capture y not 6 – y
≡ (24 – 6 < 6 – y – 6 < 32 – 6)
≡ (18 < -y < 26)
≡ (-18 > y > -26)
≡ (-26 < y < -18
Try: Capture e:
(-4 < -x + e < 6)
(-4 < x-e < 6) Capture e
(-4 < -x – e <6) Capture e
FUNCTIONS
Function: a relation between an input value and
an output value with the special property for
each input value there is only one output value
FUNCTIONS
f(x): ‘f’ of ‘x’
the function ‘f’ is the rule that tells you how to
compute the output for a given input ‘x’
the output is often denoted as ‘y’
y depends on x
y is the dependent value (Codomain)
x is the independent value (Domain)
FUNCTIONS
Can also be written as a set of ordered pairs:
(input, output) → (x, f(x))
Ordered pairs are also known as coordinates
Orders pairs allow for graphing (a pictorial
representation of the function)
GRAPHS
Coordinate plane (aka Cartesian plane) contains
an ‘x’ axis and a ‘y’ axis
The x-axis is always horizontal and the y-axis is
always the vertical axis
GRAPHS
Using Cartesian coordinates, the point (12,5) is
the intersection of x=12 and y=5
FUNCTIONS AND GRAPHS
LINEAR FUNCTION: the relationship between x
and y is a straight line
f(x) = y=mx+b where m is the slope and b is the intercept
m>0
m<0
LINEAR FUNCTION
Y = 2X – 1: m=2, b=-1
X
-1
0
1
2
3
Y
-3
-1
1
3
5
LINEAR FUNCTION
Try: x - 3
m = ___, b = ___
3x - 3
m = ___, b = ___
LINEAR FUNCTION
Try: x - 3
m = ___, b = ___
-2x + 3
m = ___, b = ___
LINEAR FUNCTION
Y = body weight, x = height
Ideal body weight for males:
y = 106 + 6(x - 60)
m = ___, b = ___
Ideal body weight for females:
y = 100 + 5(x - 60)
m = ___, b = ___
100
60
Vertical grid by 5, horizontal by 1
FUNCTIONS AND GRAPHS
EXPONENTIAL FUNCTION: y = ex
x > 0 implies growth
x < 0 implies decay
FUNCTIONS AND GRAPHS
LOGRITHMIC FUNCTION: y = ln x
FUNCTIONS AND GRAPHS
Comparison exponential, linear and logrithmic
functions:
GRAPHS – LOG SCALE AXIS
f(x) = 10x
100000000
120000000
10000000
100000000
1000000
80000000
100000
10000
60000000
Series1
40000000
Series1
1000
100
20000000
10
1
0
0
5
Y-axis on natural scale
10
0
2
4
6
8
Y-axis on log10 scale
10
GRAPHS
Real earnings of young college graduates
Country A
Country B
SUMMATION
b
𝑓𝑜𝑟𝑚𝑢𝑙𝑎
i=a
Σ: summation (Greek capital letter sigma)
i: index
a: beginning value of index
b: end value of index
SUMMATION
Examples:
5
i = 1 + 2 + 3 + 4 + 5 = 15
i=1
3
xi = x1 + x2 + x3
i=1
4
2i = 22 + 23 + 24 = 4 + 8 + 16 = 28
i=2
SUMMATION
Try:
4
i2
i=2
3
xi
i=1
5
ii
i=3
EXAMPLES IN STATISTICS
Mean:
1
n
n
i=1 xi
Sample variance:
1
n−1
n
i=1
Chi-square statistics: χ2 =
xi − x
2
2
O
−E
i
i
n
i=1
Ei
SUMMATION
Properties of Summation(all summations go from i=1 to n):
Σaxi = aΣxi
Σ(axi + byi + czi) = Σaxi + Σbyi + Σczi
=aΣxi + bΣyi + cΣzi
Σa = na
NB: Σxi2 ≠ (Σxi)2
Try: Σ(a + xi)
Σ(a + xi)2
FACTORIAL
n!: product of all positive integers ≤ n
0! = 1
4! = 4*3*2*1 = 24
2!
3!
=
2∙1
3∙2∙1
5!
3!(3−1)!
Try:
5!
3!
7!
6!(7−7)!
=
1
=
3
5!
3!2!
=
5∙4∙3∙2∙1
3∙2∙1∙2∙1
= 10