Thinking Mathematically Algebra 1 By: A.J. Mueller

Properties

Proprieties Addition Property (of Equality) 4+5=9 Multiplication Property (of Equality) 5●8=40 Reflexive Property (of Equality) 12=12 Symmetric Property (of Equality) If a=b then b=a

Proprieties Transitive Property (of Equality) If a=b and b=c then a=c Associative Property of Addition (0.6+5.3)+4.7=0.6+(5.3+4.7) Associative Property of Multiplication (-5●7) 3=-5(7●3) Commutative Property of Addition 2+x=x+2

Proprieties Communicative Property of Multiplication b3a2=a2b3 Distributive Property 5(2x+7)= 10x+35 Prop. of Opposites or Inverse Property of Addition a+(-a)=0 and (-a)+a=0 Prop. of Reciprocals or Inverse Prop. of Multiplication x2/7•7/x2=1

Proprieties Identity Property of Addition -5+0=-5 Identity Property of Multiplication x●1=x Multiplicative Property of Zero 5●0=0 Closure Property of Addition For real a and b, a+b is a real number

Proprieties Closure Property of Multiplication ab = ba Product of Powers Property x3+x4=x7 Power of a Product Property (pq)7=p7q7 Power of a Power Property (n2) 3

Proprieties Quotient of Powers Property Power of a Quotient Property X5/x3=x2 Power of a Quotient Property (a/b) 2 Zero Power Property (9ab)0=1 Negative Power Property h-2=1/h2

Proprieties Zero Product Property Product of Roots Property ab=0, then a=0 or b=0 Product of Roots Property √20= √4•√5 Power of a Root Property (√7) 2=7

Solving 1st Power Inequalities in One Variable

Solving 1st Power Inequalities in One Variable With only one inequality sign x > -5 Solution Set: {x: x > -5} Graph of the Solution: -5

Conjunctions {x: -4 < x ≤ 9} Open endpoint for these symbols: > < Closed endpoint for these symbols: ≥ or ≤ Conjunction must satisfy both conditions Conjunction = “AND” {x: -4 < x ≤ 9} -4 9

Disjunctions {x: x < -4 or x ≥ 7} 7 -4 Open endpoint for these symbols: > < Closed endpoint for these symbols: ≥ or ≤ Disjunction must satisfy either one or both of the conditions Disjunction = “OR” {x: x < -4 or x ≥ 7} -4 7

Special Cases That = {All Reals} Watch for special cases No solutions that work: Answer is Ø Every number works: Answer is {reals} Disjunction in same direction: answer is one arrow {x: x > -5 or x ≥ 1} -5 1

Special Cases That = {x: -x < -2 and -5x ≥ 15} Ø

Linear equations in two variables

Linear equations in two variables Lots to cover here: slopes of all types of lines; equations of all types of lines, standard/general form, point-slope form, how to graph, how to find intercepts, how and when to use the point-slope formula, etc. Remember you can make lovely graphs in Geometer's Sketchpad and copy and paste them into PPT.

Important Formulas Slope- Standard/General form- ax+bx=c Point-slope form- Use point-slope formula when you know 4 points on 2 lines. Vertex- X-intercepts- set f(x) to 0 then solve Y-intercepts- set the x in the f(x) to 0 and then solve

Examples of Linear Equations y=-3/4x-1

Examples of Linear Equations 3x-2y=6 (Put into standard form) 2y=-3x+6 (Divide by 2) y=-3/2x+6 (Then graph)

Linear Systems

Substitution Method Goal: replace one variable with an equal expression Step 1: Look for a variable with a coefficient of one. Step 2: Isolate that variable Equation A now becomes: y = 3x + 1 Step 3: SUBSTITUTE this expression into that variable in Equation B Equation B now becomes 7x – 2( 3x + 1 ) = - 4 Step 4: Solve for the remaining variable Step 5: Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

Addition/ Subtraction (Elimination) Method Goal: Combine equations to cancel out one variable. Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are recommended to avoid errors.) Here: -3y and +2y could be turned into -6y and +6y Step 2: Multiply each equation by the necessary factor. Equation A now becomes: 10x – 6y = 10 Equation B now becomes: 9x + 6y = -48 Step 3: ADD the two equations if using opposite signs (if not, subtract) Step 4: Solve for the remaining variable Step 5: Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.)

Factoring

Types of Factoring Greatest Common Factor (GFC) Difference of Squares Sun and Difference of Cubes Reverse FOIL Perfect Square Trinomial Factoring by Grouping (3x1 and 2x2)

GFC To find the GCF, you just look for the variable or number each of the numbers have in common. Example 1 x+25x+15 x(25+15)

Difference of Squares Example 1 27x4+75y4 Example 2 45x6-81y4 3(3x2+5y2)(3x2-5y2) Example 2 45x6-81y4 9(5x4-9y4)

Sun and Difference of Cubes Example 1 (8x3+27) (2x+3) (4x2-6x+9) Example 2 (p3-q3) (p-q) (p2+pq+q2)

Reverse FOIL Example 1 x2-19x-32 Example 2 6y2-15y+12 (x+8)(x-4)

Perfect Square Trinomial Example 1 4y2+30y+25 (2y+5) 2 Example 2 x2-10x+25 (x-5) 2

Factoring By Grouping 3x1 Example 1 a2+4a+4-b2 (a+4a+4)-(b2) (a+2-b)(a+2+b)

Factoring by Grouping 2x2 Example 1 2x+y2+4x+4y [x+y][2+y]+4[x+y]

Quadratic Equations

Factoring Method Set equal to zero Factor Use the Zero Product Property to solve. Each variable equal to zero.

Factoring Method Examples Any # of terms- look for GCF first Example 1 2x2=8x (subtract 8x to set equation equal to zero) 2x2-8x=0 (now factor out the GCF) 2x(x-4)=0

Factoring Method Examples Set 2x=0, divide 2 on both sides and x=0 Set x-4=0, add 4 to both sides and x=4 x is equal to 0 or 4 The answer is {0,4}

Factoring Method- Binomials Binomials – Look for Difference of Squares Example 1 x2=81 (subtract 81 from both sides) x2-81=0 (factoring equation into conjugates) (x+9)(x-9)=0 x+9=0 or x-9=0

Factoring Method- Binomials x+9=0 (subtract 9 from both sides) x=-9 x-9=0 (add 9 to both sides) x=9 The answer is {-9,9}

Factoring Method-Trinomials Trinomials – Look for PST Example 1 x2-9x=-18 (add 18 to both sides) x2-9x+18=0 (x2-9x+18 is a PST) (x-9)(x-9)=0 x-9=0 (add 9 to both sides) x=9 The answer is {9d.r.} d.r.- double root

Square Roots of Both Sides Reorder terms IF needed Works whenever form is (glob)2 = c Take square roots of both sides Simplify the square root if needed Solve for x, or in other words isolate x.

Square Roots Of Both Sides Example 1 (Factor out the GCF) 2(x2-6x-2)=0 (You can get rid of the 2 because it does not play a role in this type of equation) x2-6-2x=0 (Add the 2 to both sides) x2-6x__=2__ (Take half of the middle number which right now is 6) x2-6x+9=2+9 (Simplify)

Square Roots Of Both Sides (x-3)=11 (Then take the square root of both sides) (x-3)= 11 (Continue to simplifying) (Add the 3 to both sides) (Final Answer)

Completing the Square Example 1 2x2-12x-4=0 (Factor out the GCF) 2(x2-6x-2)=0 (You can get rid of the 2 because it does not play a role in this type of equation) x2-6-2x=0 (Add the 2 to both sides) x2-6x__=2__ (Take half of the middle number which right now is 6) x2-6x+9=2+9 (Simplify)

Completing the Square (x-3)=11 (Then take the square root of both sides) √(x-3)= +/-√11 (Continue to simplifying) (x-3)=+/- √11 (Add the 3 to both sides) x=3+/- √11 (Final Answer)

Quadratic Formula This is a formula you will need to memorize! Works to solve all quadratic equations Rewrite in standard form in order to identify the values of a, b and c. Plug a, b & c into the formula and simplify! QUADRATIC FORMULA:

Quadratic Formula Examples 3x2-6=x2+12x Put this in standard form: 2x2-12x-6=0 Put into quadratic formula

Quadratic Formula Examples

The Discriminant – Making Predictions b2-4ac2 is called the discriminant Four Cases 1. b2 – 4ac positive non-square two irrational roots 2. b2 – 4ac positive square two rational roots 3. b2 – 4ac zero one rational double root 4. b2 – 4ac negative no real roots

The Discriminant – Making Predictions Use the discriminant to predict how many “roots” each equation will have. 1. x2 – 7x – 2 = 0 49–4(1)(-2)=57 2 irrational roots 2. 0 = 2x2– 3x + 1 9–4(2)(1)=1  2 rational roots 3. 0 = 5x2 – 2x + 3 4–4(5)(3)=-56  no real roots 4. x2 – 10x + 25=0 100–4(1)(25)=0  1 rational double root

The Discriminant – Making Predictions The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have and where the vertex is located. 1. y = 2x2 – x - 6 1–4(2)(-6)=49  2 rational zeros opens up/vertex below x-axis/2 x-intercepts 2. f(x) = 2x2 – x + 6 1–4(2)(6)=-47  no real zeros opens up/vertex above x-axis/No x-intercepts

The Discriminant – Making Predictions 81–4(-2)(6)=129 2 irrational zeros opens down/vertex above x-axis/2 x-intercepts 3. y = -2x2– 9x + 6 4. f(x) = x2 – 6x + 9 36–4(1)(9)=0  one rational zero opens up/vertex ON the x-axis/1 x-intercept I (A.J. Mueller) got these last four slides from Ms. Hardtke’s Power Point of the Quadratic Methods.

Functions

About Functions Think of f(x) like y=, they are really the same thing. The domain is the x line of the graph The Range is the y line of the graph

Functions f(x)= -2x-8 First find the vertex. ( ) The vertex of this equation is (1,-9) Find the x-intercepts by setting f(x) to 0. The x-intercepts are {-2,4} Find the y-intercept by setting the x in the f(x) to 0. You would get -8. The graph the equation.

Simplifying expressions with exponents This site will example how to simplify expressions with exponents very well. http://www.purplemath.com/modules/simpexpo.htm

Radicals Example 1 (Simplify) (Now you can cancel the √2s)

Radicals Example 2 (Multiply by ) That equals Cancel out the 2s and the final answer is

Radicals Example 3 Take the square root of that. Final answer is

Word Problems Example 1 If Tom weighs 180 on the 3th day of his diet and 166 on the 21st day of his diet, write an equation you could use to predict his weight on any future day. (day, weight) (3,180) 21,166)

Word Problems Point Slope: m=166-180/21-31 That can be simplified to -14/18 and then -7/9. 4-180=-7/9(x-3) 4-180=-7/9+21/9 Answer: y=-7/9x+182 1/3

Word Problems Click to open the hyperlink. Then try out this quadratic word problem, it will walk you through the process of finding the answer. http://www.algebra.com/algebra/homework/quadratic/word/02-quadratic.wpm

Word Problems Here is another link to a word problem about time and travel. http://www.algebra.com/algebra/homework/word/travel/07-cockroach.wpm

Word Problems This word problem is about geometry. http://www.algebra.com/algebra/homework/word/geometry/02-rectangle.wpm This site is good study tool for word problems.

Line of Best Fit The Line of Best Fit is your guess where the middle of all the points are. http://illuminations.nctm.org/ActivityDetail.aspx?id=146 This URL is a good site to example Line of Best Fit. Plot your points, guess your line of best fit, then the computer will give the real line of best fit.

Line of Best Fit Your can use a Texas Instruments TI-84 to graph your line of best fit and also all other types of graphs.