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