pemdas, functions, graphs, summation and factorials
DESCRIPTION
PEMDAS Without PEMDAS, two different answers: x x 3 = (3 - 2) x 3 = 1 x 3 = x 3 = 3 - (2 x 3) = = -3TRANSCRIPT
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 - 2 x 3 3 - 2 x 3 = (3 - 2) x 3 = 1 x 3 = 33 - 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 inequalitiesif 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 inequalitiesif 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 xy 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 Y -1 -3
0 -1 1 1
2 3 3 5
LINEAR FUNCTIONTry: x - 3
m = ___, b = ___
3x - 3m = ___, b = ___
LINEAR FUNCTIONTry: x - 3
m = ___, b = ___
-2x + 3m = ___, b = ___
LINEAR FUNCTIONY = body weight, x = heightIdeal body weight for males:
y = 106 + 6(x - 60)m = ___, b = ___
Ideal body weight for females:y = 100 + 5(x - 60)m = ___, b = ___ 60
100
Vertical grid by 5, horizontal by 1
FUNCTIONS AND GRAPHS
EXPONENTIAL FUNCTION: y = ex
x > 0 implies growthx < 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
Y-axis on natural scaleY-axis on log10 scale
0 1 2 3 4 5 6 7 8 90
20000000
40000000
60000000
80000000
100000000
120000000
Series1
0 1 2 3 4 5 6 7 8 91
10
100
1000
10000
100000
1000000
10000000
100000000
Series1
GRAPHS
Real earnings of young college graduates
Country A Country B
SUMMATION
Σ: summation (Greek capital letter sigma)i: indexa: beginning value of indexb: end value of index
SUMMATION
Examples:
SUMMATION
Try:
EXAMPLES IN STATISTICS
Mean:
Sample variance:
Chi-square statistics: χ2 =
SUMMATION
Properties of Summation(all summations go from i=1 to n):
axi = aΣxi
Σ(axi + byi + czi) = Σaxi + Σbyi + Σczi
=aΣxi + bΣyi + cΣzi
Σa = naNB: Σxi
2 (Σxi)2
Try: Σ(a + xi) Σ(a + xi)2
FACTORIAL
n!: product of all positive integers ≤ n0! = 1
4! = 4*3*2*1 = 24
Try: