P → Parentheses (Grouping Symbols) E → Exponents (Powers) 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) Example: 3 – 5(2 + 7) = 3 – 5(9) Unit 2: slides 5- = 3 – 45 = -42 Order of Operations

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. 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 – 9 terms: 36x3 27x2 – 182x – 9 coefficients: 36,27, -182 constant: -9 Expressions, coefficients, constants

Exponential Equations a= initial value b = base x = time If b is a whole number then it is a growth, if b is a decimal or fraction then it is a decay. a= initial value r = rate t = time a= initial value where n is the number r = rate of times compounded t = time Unit 2: slides 5- If r is + then it is a growth, if r is – then it is a decay. Exponential Equations

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

Positive Negative Zero Undefined (Horizontal) (Vertical) Slope Formula: Positive Negative Zero Undefined (Horizontal) (Vertical) Examples: y = -1 + 2x y = 2 – x y = 3 x = 2 Slope examples

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 Slope- Intercept Form of a Line

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) Point – Slope Form of a Line

8 pints = 1 gallon 1 mile = 5280 feet 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 3 6 pints Conversions

Properties of Exponents

To solve an exponential equation, make the bases the same, then set the exponents equal to each other and solve. Example: 2nd card Exponential Equations

Properties of Operations Commutative property of addition a+ 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 multiplication ab=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: 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) Intercepts

Arithmetic Sequence 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

Geometric Sequence 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

Mode: the number that occurs most often in a set of data. 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 Unit 1: slides 1-4 Measures of Center

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 Five Number Summary and IQR and Range

1, 2, 3, 5, 5, 7, 8, 9, 12, 15, 16 Box and Whiskers plot Q1 Q2 Q3 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 = 15 IQR = 12 – 3 = 9 Box and Whiskers plot

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 3.5 3 3-5.5 2.5 5 5-5.5 0.5 6 6-5.5 0.5 8 8-5.5 2.5 9 9-5.5 3.5 MAD = 13/6 = 2.17 Mean Absolute Deviation 19

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 = 9 Q3 – Q1 = 9 – 3 = 6 (6)(1.5)=9 Q1 –9 = 3 – 9 = - 6 Q3 + 9 = 9 + 9 = 18 19 is an outlier!!! Outliers 20

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-Horizontal and Vertical Shifts 21

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-Reflections 22

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) Transformations-Rotations 23

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) Midpoint Formula 24

Distance between points in a coordinate plane: Distance of length of a segment : Example: Find the distance between (3,4) and (-2,5) Distance Formula 25