ALGEBRA I FORMULAS AND FACTS FOR SEMESTER EXAM



ALGEBRA I FORMULAS AND FACTS FOR EOC

|Commutative Property: |a + b = b + a |Associative Property: |(a + b) + c = a + (b + c) |

|Order of Operations: |PEMDAS |

|Percent of Change: |[pic] or [pic] |

|Distributive Property: |Distribute (multiply) a term to each individual term inside the parentheses. Be careful of negative and positive |

| |values. |

| |-3y (5 + x - 2y) = (-3y)*5 + (-3y)*x + (-3y)*(-2y) = -15y-3yx + 36y2 |

|Combining Like Terms: |To combine like terms, the variables and exponents must be same. |

| |3x2 + 4x – 6 + 6x + 14 – 5x2 = (3 - 5)x2 + (4 + 6)x +( –6 + 14) |

| |= -2x2 + 10x + 8 |

|Solving equations: |CALCULATOR: Left Side = Y1. Right Side = Y2. [GRAPH]. |

| |Find Intersection: [2nd], [Trace], [5:Intersect], [ENTER], [ENTER], [ENTER] |

| |By HAND: Use SADMEP – cancel operations |

|Independent Variable: |x-values |Dependent Variable: |y-values |

|Domain: |All x-values used an equation, function, or graph (left and right) |

|Range: |All y-values used an equation, function, or graph (up and down) |

|Greatest Common Factor (GCF) |Largest Integer into all numbers and smallest exponent of variable |

| |Exp: 10x2yz4 and 15x5y3; GCF = 5x2y |

|Least Common Multiple (LCM) |Smallest integer multiply to equal, largest exponent each variable |

| |Exp: 10x2yz4 and 15x5y3; LCM = 30x5y3z4 |

|Slope: |[pic]or [pic] |

|between (x1, y1) and (x2, y2) | |

|Parallel Lines: |Same slope |

|Perpendicular Lines: |Opposite (Negative) reciprocal slope. |

|Slope – Intercept form: |y = mx + b |

| |m = slope and b = y-intercept |

|Point-Slope Form: |y – y1 = m (x – x1) |

| |point: (x1, y1) and m = slope |

|Standard Form: |Ax + By = C |

| |GCF of A, B, and C = 1 |

| |NO FRACTIONS |

| |A is positive |

|Horizontal Line: |Equation: y = #; Zero (No) Slope |

|Vertical Line: |Equation: x = #; Undefined Slope |

|Direct Variation: |y = kx ; y varies directly with x; multiply from X to Y or vice versa |

|Midpoint Formula: between (x1, y1) and (x2, y2)|[pic] |

|Matrices: |CALCULATOR: Create - [2nd], [Matrix], EDIT, Select a Matrix to use ([A], [B], etc), Input Rows and Columns, Input |

|rows go across, |Elements |

|columns go down. |Use Operations – [2nd], [Matrix], NAMES, Select a Matrix ([A], [B],…) |

|Solving a system of equations: |GRAPHING: Solve for slope intercept form |

| |Plug into y = and find the intersection of the lines |

| |MATRIX: Write both equations in STANDARD FORM of lines |

| |[pic] MATRIX = [pic] |

| |Plug coefficients and constants matrix and perform RREF operation. |

| |If bottom row is 0 1 #, the last column is your answer. |

| |If bottom row is 0 0 1, there is no solution. |

| |If bottom row is 0 0 0, there are infinitely many solutions. |

|Graph Inequalities: |Step #1: Solve for slope intercept form. If you multiply or divide by a negative, then flip direction of inequality. |

| |Step #2: Solid Line when ≥ or ≤ AND Dotted Line when > or < |

| |Step #3: Shade Up (right) when ≥ or > AND Shade Down (left) when ≤ or < |

|Distance Formula: |DISTANCE =[pic]; Draw a right triangle |

|between (x1, y1) and (x2, y2) | |

|Quadratic Formula: | [pic] |

|ax2 + bx + c = 0 | |

|Pythagorean Theorem: |a2 + b2 = c2 |

|Exponential Functions: |y = abx |

| |b = base or pattern of multiplication, a = initial value |

|Exponential Growth: (Increases/ Appreciates) |y = a(1 + r)x |

| |a = initial value, r = rate of percent increase (4.5%; r = 0.045) |

|Exponential Decay: (Decrease/ Depreciates) |y = a(1 - r)x |

| |a = initial value, r = rate of percent decrease (5.7%’ r = 0.057) |

Common Shapes’ Area and/or Perimeter Formulas

|Perimeter of a Figure: add up all sides |

|Area of a Circle: A = (r2 |[pic] |Area of a Rectangle: A = l*w |[pic] |

|Circumference of Circle: | |Perimeter of a rectangle: | |

|C = 2(r | |P = 2l + 2w | |

|Area of Triangle: |[pic] |Area of a Square: |[pic] |

|A = ½ bh | |A = s2 | |

|Perimeter of a triangle: | |Perimeter of a square: | |

|P = s1 + s2 + s3 | |P = 4s | |

| |[pic] | | |

|Area of a Trapezoid: | |Volume of a Cylinder: | |

|A = ½ (b1 + b2)h | |V = (r2h | |

CALCULATOR COMMANDS

Graphing: [Y=] enter in the equation, [ZOOM], [6: ZStandard]

If the graph is not shown, then the change the window by: Make sure xmin < xmax and ymin < ymax

[WINDOW] and adjust… YMAX (see farther up)

XMIN (see more left) XMAX (see more right)

YMIN (see farther down)

To find the MAXIMUM VALUE: [2nd], [TRACE], [4] (maximum)

Left Bound: move the cursor to the left of the maximum (top of hill) ENTER

Right Bound: move the cursor to the right of the maximum (top of hill) ENTER

Guess: move the cursor to the maximum (top of the hill) ENTER

To find the MINIMUM VALUE: [2nd], [TRACE], [3] (minimum)

Left Bound: move the cursor to the left of the minimum (bottom of valley) ENTER

Right Bound: move the cursor to the right of the minimum (bottom of valley) ENTER

Guess: move the cursor to the minimum (bottom of valley)ENTER

To find the ROOTS/ ZEROS/ X-INTERCEPTS:

[Y =] make Y1 = Equation and Y2 = 0 [GRAPH], [2nd], [TRACE], [5] (intersect)

Move cursor to intersection [ENTER], [ENTER], [ENTER]

To find X when Y = #:

[Y =] make Y1 = Equation and y2 = #, [GRAPH], [2nd], [TRACE], [5] (intersect)

Move cursor to intersection ENTER, ENTER, ENTER

Make sure that your window shows the intersection

To find Y when X = #.

Option #1: [2nd], [TRACE], [1] (value), X= #, [ENTER]

Option #2: [2nd], [WINDOW] let TblStart = #, [2nd], [Graph]

To find INITIAL VALUE or Y-INTERCEPT, look when x = 0.

LINES OF BEST FIT and LINEAR REGRESSION and PREDICTION EQUATIONS

INPUTTING DATA:

1) STAT -> EDIT

2) L1 –X VALUES, L2 - Y VALUES

Make sure that rows represent points/ ordered pairs

MAKE A SCATTER PLOT:

1) 2ND, STAT PLOT [Y=], ENTER

2) TURN STAT PLOT ON (ENTER)

3) TYPE: HIGHLIGHT FIRST GRAPH

4) X-LIST: L1

5) Y-LIST: L2

SEE THE ENTIRE SCATTER PLOT:

1) WINDOW

2) XMIN: # < smallest number in L1

3) XMAX: # > biggest number in L1

4) YMIN: # < smallest number in L2

5) YMAX: # > biggest number in L2

FINDING THE LINE OF FIT:

1) [STAT] -> Scroll to CALC

2) 4 [LINREG(ax + b)]

• a = slope (m) and b = y-intercept

3) VARS, Y-VARS, FUNCTION, Y1, [ENTER]

4) GO TO [Y =];

You can now see the equation of the line

5) GRAPH;

You can see the stat plot and the line together

Now have an equation to help predict values by

1) Searching the TABLE:

2ND, WINDOW, TblStart = value you want

2ND, GRAPH [TABLE]

2) Finding a specific VALUE:

2ND – TRACE [CALC] – 1 - ENTER

TOUGH VERBAL TRANSLATIONS

|1) 4 less a number: n – 4 |6) 5 more a number: 5 + n |

|2) 4 less than a number: 4 – n |7) 5 is more than a number: 5 > n |

|3) 4 is less than a number: 4 < n |8) 5 more than a number: n + 5 |

|4) a number is at least 10: n ≥10 |9) a number is at most 10: n ≤ 10 |

|5) 4 less ½ of the difference of a number and 5: |10) 2 more than ½ the sum of a number and 12: |

|4 – ½ (n – 5) |½ (n + 12) + 2 |

Chapter 9 Factoring Review

|Section 9.1: Is a number prime or composite? |

|Find the Prime Factorization: All Prime numbers that multiply together to equal a number. Draw your tree. |

|Example: 180 = 2*2*3*3*5 |

|Find the Greatest Common Factor (GCF) between numbers: pair up all common prime |Find the Greatest Common Factor (GCF) between monomials: find the GCF of the |

|factors of the numbers and multiply together |coefficients and then for any common variable pick the smallest exponent for each|

|Example: GCF = 2*3 or 6 |variable |

|72 = 2 * 2 *2 * 3* 3 |Example: GCF = 2*x2*y3 = 2x2y3 |

|36 = 2 * 2 * 3* 3 |6x2y6 = 2*3*x2*y6 |

|42 = 2 * 3 * 7 |32x3y4 = 2*2*2*2*x3*y4 |

| |10x5y3 = 2*5*x5*y3 |

| |FACTORING TECHNIQUE |EXAMPLES |

|2 or more |GREATEST COMMON FACTOR (GCF): |3x3 + 6x2 + 15x |

|terms |(1) Find the GCF of all terms of the polynomial, |(1) GCF = 3x |

| |(2) Divide the polynomial by the GCF, and |(2) (3x3 + 6x2 + 15x)/(3x) = x2 + 2x + 5 |

| |(3) Write the factors as the GCF times polynomial from step 2. |(3) = 3x(x2 + 2x + 5) |

|2 terms |Special Case: DIFFERENCE OF SQUARES: |4x2 – 25 |

| |a2 – b2 = (a + b) (a – b) |(2x)2 – (5)2 |

| | |(2x – 5) (2x + 5) |

|3 terms |ax2 + bx + c |x2 + 11x + 24 |

| |Step #1:Greatest Common Factor of a, b, and c |3*8 = 24 and 3 + 8 = 11 |

| | |x2 + 3x + 8x + 24 |

| |Step #2: Factor-Sum Tree to find a pair M and N |= (x + 3)(x + 8) |

| |Multiply to equal = product of a and c | |

| |Add to equal = b | |

| | | |

| |Step #3: Split the middle term, bx = Mx + Nx | |

| | | |

| |Step #4: Factor by Grouping (See below) | |

| | |6x2 – x – 2 |

| | |3*-4 = -12 and 3 + - 4 = -1 |

| | |= 6x2 + 3x – 4x – 2 |

| | |= 3x(2x + 1) – 2(2x + 1) |

| | |= (3x – 2) (2x + 1) |

|4 terms |Factor by Grouping: |3x2– 6x + 5x – 10 |

| |Factor the 1st 2 terms and the last 2 terms separately by their respective GCFs. |= (3x2 – 6x) + (5x – 10) |

| |ax + bx + ay + by |GCF = 3x GCF = 5 |

| | |= 3x(x – 2) + 5(x – 2) |

| | |= (3x + 5) (x – 2) |

| | | |

| | |Hint: If x (a - b) + y(-a + b), then make a subtraction for y |

| | |x (a - b) – y(a - b) |

| |GCF of ax + bx is x |GCF of ay + by is y | |

| |x (a + b) + y(a + b) | |

| |(x + y)(a + b) | |

CHAPTER 8 MONOMIALS and POLYNOMIALS REVIEW

|MULTIPLYING MONOMIALS |DIVIDING MONOMIALS |

|When multiplying monomials of the same base, we ADD the exponents and MULTIPLY |When dividing monomials of the same base, we SUBTRACT the exponents and SIMPLIFY the |

|the coefficients. |coefficients. |

|See the base more than once (Left to Right) and combine the powers to make one |Subtract the bigger MINUS smaller power and place the new base in the location of the|

|base. |bigger. |

| | |

|Example: (4m3n4)(5m5n3) = (4*5) m3+5 n4+3 Solution: 20m8n7 |Example: [pic]; Solution: [pic] |

| |[pic]; x: 5 – 3 = 2 in top; y: 8 – 2 = 6 in bottom |

|POWERS OF MONOMIALS |

|When we raise a monomial to an exponent, we MULTIPLY the exponents in the monomial and COEFFICIENTS gains the exponent. |

|Example 1: (-2m2n5)3 = (-2)3 m2*3 n5*3 |EXAMPLE 2: [pic] |

| | |

|SOLUITION: -8 m6 n15 | |

|SPECIAL MONOMIAL CASES: |POLYNOMIAL INTRODUCTION: |

|A ZERO EXPONENT: CANCELS THE BASE |POLYNOMIAL: A monomial or the sum (addition or subtraction) of monomials. |

| | |

|Algebraically: If a = base, then a0 = 1 |Degree of a monomial: the sum of the exponents of all its variables. |

|Example: [pic] |Degree of 5mn2 is 1 + 2 = 3 |

| | |

|NEGATIVE EXPONENTS: Change the exponent to a positive and switch it’s location |Degree of a polynomial: the greatest degree of any term in the polynomial. |

|(OR bottom to top) |Degree of -4x2y2 + 3x2 + 5y is 4. |

|(Top to Bottom): If a = base, [pic] | |

|(Bottom to Top): If a = base,[pic] |ADDING AND SUBTRACTING POLYNOMIALS: |

|Examples: |Combine Like Terms between polynomials. |

|#1: [pic] #2: [pic] |Do not change the exponents of your variable only the coefficients from your addition|

| |or subtraction. |

|#3: [pic] | |

| |Examples: |

| |1. (3x2 – 4x + 8)+(2x – 7x2 – 5) = - 4x2 - 2x + 3 |

| |3x2 + -7x2 + -4x + 2x + 8 + -5 |

| | |

| |2. (3n2 + 13n3 + 5n)–(7n + 4n3) = 9n3 + 3n2 –2n |

| |3n2 + 13n3 – 4n3 + 5n – 7n |

|DISTRIBUTIVE PROPERTY: |FOIL |

|Circle the ENTIRE term in front of parentheses |(Arrow or Box Method) |

|Draw Arrows to each term in polynomial include plus or minus sign |Circle the EACH term in FIRST Binomial include plus or minus sign |

|Multiply to get new terms |Draw Arrows from each circled term to each term in 2nd Binomial |

|Add all terms |Multiply to get 4 new terms |

| |Combine any Like Terms |

|-3x2 (6xy2 – 3x + 5x2y + 2y - 8) | |

| |(5x – 3) (2x + 7) |

|(-3x2)(6xy2) = -18x3y2 | |

|(-3x2)(– 3x) = 9x3 |F: (5x)(2x) = 10x2 |

|(-3x2)( 5x2y) = -15x4y |O: (5x)(7) = 35x |

|(-3x2)( 2y) = -6x2y |I: (-3)(2x) = -6x |

|(-3x2)(- 8) = 24x2 |L: (-3)(7) = -21 |

| | |

|Solution: |Solution: 10x2 + 29x - 21 |

|-18x3y2 + 9x3 - 15x4y - 6x2y + 24x2 | |

| | |

|SPECIAL PRODUCTS |POLYNOMIAL times POLYNOMIAL |

|Special products are shortcuts for FOIL |Circle the EACH term in FIRST polynomial include plus or minus sign |

| |Draw Arrows from each circled term to each term in polynomial |

|SQUARE OF A SUM: The square of a + b is the square of a plus twice the product of a|Multiply to get new terms |

|and b plus the square of b. |Combine Like Terms |

|[pic] | |

| |(3x – 2) (6x2 + 7x - 8) |

|SQUARE OF A DIFFERENCE: The square of a + b is the square of a minus twice the | |

|product of a and b plus the square of b. | |

|[pic] |(3x)(6x2) = 18x3 (-2)(6x2) = -12x2 |

| |(3x)(7x) = 21x2 (-2)(7x) = -14x |

|PRODUCT OF SUM AND DIFFERENCE: The product of a + b and a – b is the square of a |(3x)(-8) = -24x (-2)(-8) = 16 |

|minus the square of b. | |

|[pic] |Solution: 18x3+ 9x2 - 38x +16 |

REMINDER for BOX METHOD: If the arrows don’t work for you, use the BOX METHOD to organize your multiplications. After filling in your boxes, you will add together all the terms by combining like terms.

-----------------------

r

l

w

h

b

s

s

b1

b2

h

MULTIPLYING POLYNOMIALS

Like Terms

| |2x |+3 |

|x |x(2x) = 2x2 |x(3x) = +3x |

|-7 |-7(2x) = -14x |-7(3) = -21 |

(x – 7) (2x + 3) = 2x2 – 11x – 21

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