An easy way to remember the order of operations is to use the mnemonic device: PEMDAS.

P → Parentheses (Grouping Symbols)E → Exponents (Powers)MD → Multiplication or Division (Left to Right)AS → Addition or Subtraction (Left to Right)

= 3 – 5(9)= 3 – 45= -42

Order of Operations

Example: 3 – 5(2 + 7)

Steps to creating equations from context:1. Read the problem statement first.2. Reread the scenario and make a list or table of the known quantities.3. Read the statement again, identifying the unknown quantity or variable.4. Create expressions and inequalities from the known quantities and variables(s).5. Solve the problem.6. Interpret the solution of the equation in terms of

the context of the problem and convert units when appropriate, multiplying by a unit rate.

Solving word problems

Expressions, coefficients, constants

The number of terms are separated by a + or –.

The coefficients are the numbers that are multiplied by the variable in the expression.

The constant is the quantity that does not change.

EX. 36x3 + 27x2 – 182x – 9terms: 36x3 27x2 – 182x – 9 coefficients: 36,27, -182constant: -9

Exponential Equations

xaby

tray )1(

a= initial valueb = basex = time

a= initial valuer = ratet = time

If r is + then it is a growth, if r is – then it is a decay.

If b is a whole number then it is a growth, if b is a decimal or fraction then it is a decay.

nt

n

ray )1( a= initial value where n is the number

r = rate of times compoundedt = time

Slope is:

To find the slope of a line that passes through the points A and B where

A = (x1, y1) and B = (x2, y2) is:

m =

The slope of a horizontal line is zero.

The slope of a vertical line is undefined.

Slope

xinchange

yinchange

x

y

run

rise

12

12

xx

yy

Slope examples

Slope Formula: 12

12

xx

yy

run

risem

Positive Negative Zero Undefined(Horizontal) (Vertical)

Examples:

y = -1 + 2x y = 2 – x y = 3 x = 2

Slope- Intercept Form of a Line

Slope-Intercept Form of a Line: y = mx + b

b is the y-intercept m is the slope

Example: Write the equation of the line with slope 2 and y-intercept 3.

Answer: y = 2x + 3

Point – Slope Form of a Line

Point-Slope Form of a Line: y – y1 = m(x - x1)

(x1, y1) is the point on the line

m is the slope

Example: Write the equation of the line with slope 2 and through the point (2 -1).

Answer: y + 1 = 2(x – 2)

Conversions

10mm = 1 cm 2 pints = 1 quart

12 in. = 1 ft 4 quarts = 1 gallon

3 ft = 1 yd 1 ton = 2000 pounds

8 pints = 1 gallon 1 mile = 5280 feet

Example: 6 pints=__________quarts

36 pints quarts

sp

quart3

int2

1

1).5( 0 x

bab

a

xx

x ).3(

xx

1).4( 1 baba xxx ).1(

abba xx )().2(

nn

y

x

x

y

).6(

Properties of Exponents

To solve an exponential equation, make the bases the same, then set the exponents equal to each other and solve.

Example: 11664 xx

13 24 4x x

3 2 24 4x x3 2 2x x

2x

2nd card Exponential Equations

Commutative property of additiona+ b = b + a 3+8=8+3

Associative property of addition(a+b)+c=a+(b+c) (3+8)+2=3+(8+2)

Commutative property of multiplicationab=ba 3(8)=8(3)

Associative property of multiplication(ab)c=a(bc) (3∙8)2=3(8∙2)

Distributive property of multiplication over addition

a(b+c)=ab+ac 3(8+2)=(3)(8) + (3)(2)

Properties of Operations

Intercepts

Intercepts:

To find the x-intercept, let y = 0 and solve for x.

To find the y-intercept, let x = 0 and solve for y.

Example: Find the intercepts for 2x + 3y = 6

x-intercept: 2x + 3y = 6 y-intercept: 2x + 3y = 6

2x + 3(0) = 6 2(0) + 3y = 6 2x = 6 3y = 6

x = 3 y = 2

(3, 0) (0, 2)

Arithmetic Sequence

a1 is first term, d is common difference

Explicit Formula: Recursive Formula: an = a1 + d(n – 1)

Example: -3, 1, 5, 9, 13, . . .

Explicit formula: Recursive formula: an = -3 + 4(n – 1)

Arithmetic Sequence

daa

aa

nn 1

11

4

3

1

1

nn aa

a

Geometric Sequence

a1 is first term, r is common ratio

Explicit Formula: Recursive Formula: an = a1(r)n-1 or a0(r)n

Example: -3, 6, -12, 24, -48 . . .

Explicit formula: Recursive formula: an = -3 (-2)n-1 or an = (3/2)(-2)n

Geometric Sequence

))(( 1

11

raa

aa

nn

)2(

3

1

1

nn aa

a

Measures of Center

Mean: the sum of the numbers in a data set divided by the number of numbers in the set.

Median: the middle number of a data set when the numbers are arranged in numerical order.

Mode: the number that occurs most often in a set of data.

Ex: 1, 1, 3, 4, 6

Mean =

Median = 3

Mode = 1

35

15

Five Number Summary and IQR and Range

The Five Number Summary:

(1). The minimum value

(2). The first quartile (Q1)

(3). The second quartile (Q2 or the median)

(4). The third quartile (Q3)

(5). The maximum value

Range= Maximum value – Minimum value

IQR= Q3 – Q1

Box and Whiskers plot

1, 2, 3, 5, 5, 7, 8, 9, 12, 15, 16

Q1 Q2 Q3

Q1 Q2 Q3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Range = 16 – 1 = 15IQR = 12 – 3 = 9

Mean Absolute Deviation

To compute the mean absolute deviation (MAD),(1). Find the mean of the set.(2). Create a table to organize the data and find

each element’s absolute deviation from the mean. (3). Compute the average of these deviations.

Ex: Find the MAD for (3,2,6,9,5,8) mean = 5.5

2 2-5.5 3.53 3-5.5 2.55 5-5.5 0.56 6-5.5 0.58 8-5.5 2.59 9-5.5 3.5

MAD = 13/6 = 2.17

Outliers

To find if a data set has any outliers:(1). Find IQR IQR= Q3 – Q1

(2). Multiply (IQR)(1.5)(3). Q1 – (IQR)(1.5) any value below this is an

outlier Q3 + (IQR)(1.5) any value above this is an

outlier

Example: 2,3,5,6,8,9,19 Q1 = 3, Q3 = 9Q3 – Q1 = 9 – 3 = 6(6)(1.5)=9

Q1 –9 = 3 – 9 = - 6Q3 + 9 = 9 + 9 = 1819 is an outlier!!!

Transformations-Horizontal and Vertical Shifts

T h,k (x,y) = (x+h, y + k)

Example: P(7,-2)

Find T 3,-2 (P) =(7+3,-2-2) = (10,-4)

Transformations-Reflections

rx-axis= (x, – y) reflects image over the x-axis

ry-axis= (– x, y) reflects image over the y-axis

ry=x = ( y,x) reflects image over the y=x line

Example: P(7,-2)

Find rx-axis= (7,2 ) ry-axis=(-7,-2 ) ry=x = ( -2,7)

Transformations-Rotations

Rotation of an image counter clockwise.

R90= ( – y, x)

R180 = (– x, –y )

R270 = (y, –x )

Example: P(7,-2)Find R90 (P) = (2,7)

R180 (P) = (-7,2) R270 (P) =(-2, -7)

Midpoint Formula

Midpoint on a coordinate plane

M is the midpoint of AB

A(x1,y1) and B(x2,y2) then the midpoint is

M

Example: find the midpoint of A(3,2) and B(-2,4)

2,

22121 yyxx

3,2

1

2

6,

2

1

2

42,

2

23

Distance Formula

Distance between points in a coordinate plane:

Distance of length of a segment :

Example: Find the distance between(3,4) and (-2,5)

212

212 )()( yyxxd

22 )45()32( d

1.526125)1()5( 22 d