Thinking Mathematically - Marquette University High School

Thinking Mathematically - Marquette University High School

Thinking Mathematically Algebra 1 By: A.J. Mueller Properties Proprieties Addition Property (of Equality) 4+5=9 Multiplication Property (of Equality)

58=40 Reflexive Property (of Equality) 12=12 Symmetric Property (of Equality) If a=b then b=a Proprieties

Transitive Property (of Equality) Associative Property of Addition (0.6+5.3)+4.7=0.6+(5.3+4.7) Associative Property of Multiplication If a=b and b=c then a=c

(-57) 3=-5(73) Commutative Property of Addition 2+x=x+2 Proprieties Communicative Property of Multiplication Distributive Property

5(2x+7)= 10x+35 Prop. of Opposites or Inverse Property of Addition b3a2=a2b3 a+(-a)=0 and (-a)+a=0 Prop. of Reciprocals or Inverse Prop. of Multiplication

x2/77/x2=1 Proprieties Identity Property of Addition Identity Property of Multiplication x1=x

Multiplicative Property of Zero -5+0=-5 50=0 Closure Property of Addition For real a and b, a+b is a real number Proprieties Closure Property of Multiplication

Product of Powers Property x3+x4=x7 Power of a Product Property ab = ba

(pq)7=p7q7 Power of a Power Property (n2) 3 Proprieties Quotient of Powers Property Power of a Quotient Property

(a/b) 2 Zero Power Property X5/x3=x2 (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 20= 45 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 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

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 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} 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, pointslope 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 rise Slope- run

Standard/General form- ax+bx=c y y Point-slope form-x x Use point-slope formula when you know 4 points on 2 lines. b Vertex- 2a X-intercepts- set f(x) to 0 then solve Y-intercepts- set the x in the f(x) to 0 and then solve 1 2 1 2

Examples of Linear Equations Example 1 y=-3/4x-1 4 2 -5 5 -2

-4 Examples of Linear Equations Example 2 3x-2y=6 (Put into standard form) 2y=-3x+6 (Divide by 2) y=-3/2x+6 (Then graph) 4 2 -5

5 -2 -4 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!) 3x y 1 7 x 2 y 4

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 3(9x4+25y4) 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 (x+8)(x-4)

Example 2 6y2-15y+12 (3y-4)(2y-3) 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)-(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 MethodBinomials

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 MethodBinomials

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 MethodTrinomials 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 2 x 2 12 x 4 0 (Factor

2(x2-6x-2)=0 out the GCF) 2 (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 sides) (Then take the square root of both (x-3)= 11 ( x 3) 11

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! 2 - 4ac -b b QUADRATIC FORMULA: x = 2a

Quadratic Formula Examples Example 1 3x2-6=x2+12x Put this in standard form: 2x212x-6=0 Put into quadratic formula -(-12) (-12)2 - 4(2)(-6) x = 2(2) Quadratic Formula

Examples 12 144+48 12 192 x = 4 4 12 64 3 x = 4 128 3 x = 3 2 3

4 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 4. b2 4ac zero one rational double root negative no real roots The Discriminant Making Predictions Use the discriminant to predict how many roots each equation will have. 1. x2 7x 2 = 0 494(1)(-2)=57 2 irrational roots

2. 0 = 2x2 3x + 1 94(2)(1)=1 2 rational roots 3. 0 = 5x2 2x + 3 44(5)(3)=-56 no real roots 4. x2 10x + 25=0 1004(1)(25)=0 1 rational double root The Discriminant Making Predictions The zeros of a function are the x-intercepts on its graph. Use the discriminant to predict how many x-intercepts each parabola will have and where the vertex is2 located. 14(2)(-6)=49 2 rational zeros 1. y = 2x x - 6

opens up/vertex below x-axis/2 x-intercept no real zeros 2. f(x) = 2x2 x + 6 14(2)(6)=-47 opens up/vertex above x-axis/No x-interce The Discriminant Making Predictions 2 irrational zeros 3. y = -2x2 9x + 6814(-2)(6)=129 opens down/vertex above x-axis/2 x-intercept 4. f(x) = x2 6x + 9 364(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. Hardtkes 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)= x 2 -2x-8 First find the vertex.

( 2ba ) 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/modul

es/simpexpo.htm Radicals Example 1 12 50 18 (Simplify) 3 4 2 25 2 9 2 3 5 2 3 2 (Now you can cancel the 2s)

Radicals Example 2 2 2 (Multiply by 2 2 2

That 2 2 2 2 ) 2 2 2 2 equals 4 Cancel out the 2s and the final answer 2 is

Radicals Example 3 x16 Take the square root of that. 4 Final answer isx 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/ho mework/quadratic/word/02-quadrat ic.wpm Word Problems Here is another link to a word problem about time and travel. http://www.algebra.com/algebra/ho mework/word/travel/07-cockroach. wpm

Word Problems This word problem is about geometry. http://www.algebra.com/algebra/ho mework/word/geometry/02-rectang le.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/ActivityD etail.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.

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