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Thinking Mathematically: Review and Summary of Algebra 1

This presentation provides a review and summary of Algebra 1, covering properties of equality, linear equations, factoring, rational expressions, and quadratic functions.

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Thinking Mathematically: Review and Summary of Algebra 1

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  1. Thinking MathematicallyA review and summary of Algebra 1Anthony TrevisoDarius DeLoach

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

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

  4. a=aExample: 10=10 Reflexive Property (of Equality) Symmetric Property (of Equality) a=b, then b=aExample: 10=b, then b=10 Transitive Property (of Equality) a=b, c=b, so a=cExample: 10=b, c=b, so 10=c

  5. a+(b+c)=(a+b)+cExample: 7+(b+3)=(7+b)+3 Associative Property of Addition Associative Property of Multiplication (axb)c=(a+bExample:

  6. a+b=b+aExample:7+b=b+7 Commutative Property of Addition Commutative Property of Multiplication ab=baExample:7b=b7

  7. Distributive Property (of Multiplication over Addition a(b+c)=ab+acExample: 10(b+c)=10b+10c

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

  9. a+0=aExample: 13+0=13 Identity Property of Addition Identity Property of Multiplication (a)1=aExample: (121)1=121

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

  11. (ab)(ac)=a(b+c)Example: (a6)(a3)=a(6+3) Product of Powers Property Power of a Product Property (ab)(ba)=(ab)bExample: (10b)(b10)=(10b)b Power of a Power Property (ab)c=abcExample: (52)6=58

  12. ab/ac=ab-cExample: 52/54=52-4 Quotient of Powers Property Power of a Quotient Property ac/bc=(a/b)cExample: 54/104=(5/10)4

  13. a0=aExample: 40=4 Zero Power Property Negative Power Property a-b=1/aExample: 4-2=1/16

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

  15. Product of Roots Property Quotient of Roots Property

  16. Root of a Power Property Power of a Root Property

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

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

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

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

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

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

  23. In the next slides you will review:Factoring

  24. 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

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

  26. 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)

  27. Difference of Squares 1st find a GFC if there is one 2nd find the square root of both variables 3rd reverse FOIL 4th check work by FOILing 1. 81-t2 2. (9-t)(9+t) 3. 81+9t-9t-t2 81-t2

  28. Perfect Square Trinomial (PST) 1st find the GCF if there is one 2nd find the square root of the first and last term x2-x+1 (x-1)2

  29. Sum & Difference in Squares 1st find the cubed root of both variables 2nd square the first variable 3rd add up the two variables then find the opposite of that 4th square the last variable x3-y3 (x-y)(x2+xy+y2)

  30. Grouping 2x2 2x3+16x2+x2+8x 1st 2x2(x+8)+x(x+8) 2nd 3rd (x+8)(2x2+x)

  31. Simplify by factor and cancel 1st 2nd 3rd Then the simplified form is: 

  32. Addition and subtraction of rational expressions           3           5         8      (2)(4)            2         ---      + ---   = ---    =--------      = ---                  20         20       20      (4)(5)            5

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

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

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

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

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

  38. Simplifying expressions with radicals Simplify Simplify = =

  39. 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.)

  40. Word prob. 1. 1000 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.

  41. 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. • Your calculator helps because it automatically finds a line that would accurately go through most of the coordinates.

  42. Line of Best Fit • This graph shows multiple SAT scores

  43. Line of Best Fit • This graph shows a line of best fit

  44. Line of best fit problem • Age Height in Inches

  45. Line of Best Fit

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