PEMDAS, FUNCTIONS, GRAPHS, SUMMATION AND FACTORIALS.

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 0 iff a is positive

a < 0 iff a is negative

INEQUALITIES

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 }

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 )

2 < 5 and c = 2 then (2*2) < (5*2) or 4 < 10

When multiplying does change the inequalitiesif c < 0: if a 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

EXPONENTIAL FUNCTION: y = ex

x > 0 implies growthx < 0 implies decay


LOGRITHMIC FUNCTION: y = ln x


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:

PEMDAS, FUNCTIONS, GRAPHS, SUMMATION AND FACTORIALS.

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