Thinking Mathematically A review and summary of Algebra 1 Anthony Treviso Darius DeLoach

In the next slides you will review: Review all the Properties and then take a Quiz on identifying the Property Names

Addition Property (of Equality) Multiplication Property (of Equality) a=b, then a+c=b+c Example: 3=y, then 3+2=y+2 a=b, then ac=cb Example: 3=b, then 3c=cb

Reflexive Property (of Equality) Symmetric Property (of Equality) Transitive Property (of Equality) a=a Example: 10=10 a=a Example: 10=10 a=b, then b=a Example: 10=b, then b=10 a=b, then b=a Example: 10=b, then b=10 a=b, c=b, so a=c Example: 10=b, c=b, so 10=c a=b, c=b, so a=c Example: 10=b, c=b, so 10=c

Associative Property of Addition Associative Property of Multiplication a+(b+c)=(a+b)+c Example: 7+(b+3)=(7+b)+3 (axb)c=(a+b Example:

Commutative Property of Addition Commutative Property of Multiplication a+b=b+a Example: 7+b=b+7 ab=ba Example: 7b=b7

Distributive Property (of Multiplication over Addition a(b+c)=ab+ac Example: 10(b+c)=10b+10c

Prop of Opposites or Inverse Property of Addition Prop of Reciprocals or Inverse Prop. of Multiplication a+(-a)=0 Example: 4+(-4)=0 (b)1/b=1 Example: (3)1/3=1

Identity Property of Addition Identity Property of Multiplication a+0=a Example: 13+0=13 (a)1=a Example: (121)1=121

Multiplicative Property of Zero Closure Property of Addition Closure Property of Multiplication (a)0=0 Example: (999)0=0 a+c=b+c, then a=b Example: 88+c=b+c, then 88=b ac=bc, so a=b Example: 4c=bc, so 4=b

Product of Powers Property Power of a Product Property Power of a Power Property (a b )(a c )=a (b+c) Example: (a 6 )(a 3 )=a (6+3) (a b )(b a )=(ab) b Example: (10 b )(b 10 )=(10b) b (a b ) c =a bc Example: (5 2 ) 6 =5 8

Quotient of Powers Property Power of a Quotient Property a b /a c =a b-c Example: 5 2 /5 4 =5 2-4 a c /b c =(a/b) c Example: 5 4 /10 4 =(5/10) 4

Zero Power Property Negative Power Property a 0 =a Example: 4 0 =4 a -b =1/a Example: 4 -2 =1/16

Zero Product Property ab=0, then a=0 or b=0 Example: 7b=0, then b=0

Product of Roots Property Quotient of Roots Property

Root of a Power Property Power of a Root Property

3x-3<2x+1 3x-2x1+31 st or 2 nd x<4 Get like terms on the same side Rearrange the prob. 3 rd Plot Solving inequalities with one sign

Linear Equations Standard form: ax+by=c Point slope form: y=mx+c C=y-intercept m=slope X=x point value Y=y point value Vertex: -b/2a

How to Graph Linear Equations have straight lines Parabolas have curved lines

Substitution Method 1 st take one equality to the variable and substitute it into the other equation 2 nd solve other equation 3 rd take variable that was just found and put it in the first equation

Substitution Method 2y + x = 3 4y – 3x = 1 2y+x=3 -2y X=3-2y4y-3(3-2y)=14y-9+6y= y=10 10y/10=10/10 y=1 X=3-2(1) X=1

Elimination Method 1.Add both equations 2. Solve 3. Now substitute the variable

In the next slides you will review: Factoring

Greatest Common Factor(GCF) 1. 50x+100x+40x 2. 10x(5+10+4) Find the greatest common factor and put it in front of the parentheses because of the Distributive property

Grouping 3x1 Find a GCF if there is one Find a GCF if there is one Solve the PST Solve the PST Find the perfect square of the one number that was left out. Find the perfect square of the one number that was left out. Put the answer into two trinomials Put the answer into two trinomials

Grouping 3x1 1. 4x+20x+25-y2 2. (2x+5)-y2 3. [(2x+5)-y][(2x+5)+y] 4. (2x+5-y)(2x+5+y)

Difference of Squares t 2 2. (9-t)(9+t) t-9t-t 2 81-t 2 1 st find a GFC if there is one 2 nd find the square root of both variables 3 rd reverse FOIL 4 th check work by FOILing

Perfect Square Trinomial (PST) x 2 -x+1 1 st find the GCF if there is one 2 nd find the square root of the first and last term (x-1) 2

Sum & Difference in Squares x 3 -y 3 1 st find the cubed root of both variables 2 nd square the first variable 3 rd add up the two variables then find the opposite of that 4 th square the last variable (x-y)(x 2 +xy+y 2 )

Grouping 2x2 2x 3 +16x 2 +x 2 +8x 2x 2 (x+8)+x(x+8) 1 st 2 nd 3 rd (x+8)(2x2+x)

Simplify by factor and cancel 1 st 2 nd 3rdThen the simplified form is:

Addition and subtraction of rational expressions (2)(4) = --- = = (4)(5) 5

Multiplication and division of rational expressions 3x2 - 4x x(3x - 4) 3x = = x2 - x x(2x - 1) 2x -1 3x2 - 4x x(3x - 4) 3x = = x2 - x x(2x - 1) 2x -1

What does f(x) mean? Are all relations function? F(x) also knows as F of X also means the Y variable F(x) also knows as F of X also means the Y variable

Find the domain and range of a function. {(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)} {(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)} Domain ={2, 3, 4, 6} Domain ={2, 3, 4, 6} Range ={–3, –1, 3, 6} Range ={–3, –1, 3, 6}

Quadratic functions – Quadratic formula = X-coordinate = Y-coordinate = Vertex = Set x values to zero Set the y values to zero

Simplifying expressions with exponents Simplify the following expressions: Simplify the following expressions: 1 st 2 nd (–46x2y3z)0 1 st 2 nd This is simple enough: anything to the zero power is just 1. (–46x2y3z)0 = 1

Simplifying expressions with radicals Simplify = =

In the next slides you will review: Minimum of four word problems of various types. You can mix these in among the topics above or put them all together in one section. (Think what types you expect to see on your final exam.)

Word prob tickets were sold. Adult tickets cost $8.50, children's cost $4.50, and a total of $7300 was collected. How many tickets of each kind were sold? 2. Mrs. B. invested $30,000; part at 5%, and part at 8%. The total interest on the investment was $2,100. How much did she invest at each rate? 3. A saline solution is 20% salt. How much water must you add to how much saline solution, in order to dilute it to 8 gallons of 15% solution? 4.It takes 3 hours for a boat to travel 27 miles upstream. The same boat can travel 30 miles downstream in 2 hours. Find the speeds of the boat and the current.

Line of Best Fit A line on a scatter plot that best defines or expresses the trend shown in the plotted points. It is chosen so that the sum of the squares of the distances from the points to the line is a minimum. A line on a scatter plot that best defines or expresses the trend shown in the plotted points. It is chosen so that the sum of the squares of the distances from the points to the line is a minimum. Your calculator helps because it automatically finds a line that would accurately go through most of the coordinates. Your calculator helps because it automatically finds a line that would accurately go through most of the coordinates.

Line of Best Fit This graph shows multiple SAT scores This graph shows multiple SAT scores

Line of Best Fit This graph shows a line of best fit This graph shows a line of best fit

Line of best fit problem Age Height in Inches

Line of Best Fit