1. Algebra 1 A Review and Summary Gabriel Grahek

2. In the next slides you will review: Solving 1st power equations in one variable A. Special cases where variables cancel to get {all reals} or B. Equations containing fractional coefficients C. Equations with variables in the denominator –(throw out answers that cause division by zero)

3. 1st Power Equations Any type of equation that has only one variable.
x+3= (x2-2)+8=36
x+3-3=8 (4x2-8)+8=36
x= x2-8=28
4x2=36
x2=9
x=3
Note that the variable can be on both sides.

4. 1st Power Equations 3x-4=12+x 4y+16=24 3x-4+4=12+4+x 4y+16-16=24-16
3x-x=16+x-x y=2
2x=16
x=8

5. 1st Power Equations Special cases- Ø and {all reals}
x-(4-3)=x 4x+4=2(2x+2)
x-1=x x=4x
-1= x=x Ø

6. 1st Power Equations
Equations containing fractional coefficients and with variables in the denominator.

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

8. Addition Property (of Equality)
Example: If a, b, and c, are any real numbers, and a=b, then a+c=b+c and c+a=c+b If the same number is added to equal numbers, the sums are equal.
Multiplication Property (of Equality)
Example: If a, b, and c are real numbers, and a=b, then ca=cb and ac=bc. If equal numbers are multiplied by the same number, the products are equal.

9. Reflexive Property (of Equality)
Example: For all real numbers a, b, and c: a=a
Symmetric Property (of Equality)
Example: For all real numbers a, b, and c: If a=b, then b=a.
Transitive Property (of Equality)
Example: For all real numbers a, b, and c: If a=b and b=c, then a=c.

10. Associative Property of Addition
Example: For all real numbers a, b, and c: (a+b)+c=a+(b+c) Example: (5+6)+7=5+(6+7)
Associative Property of Multiplication
Example: For all real numbers a, b, and c: (ab)c=a(bc) Example:

11. Commutative Property of Addition
Example: For all real numbers a and b: a+b=b+a Example: 2+3=3+2
Commutative Property of Multiplication
Example: For all real numbers a, b, and c: ab=ba Example:

12. Distributive Property (of Multiplication over Addition)
Example: For all real numbers a, b, and c: a(b+c)=ab+ac and (b+c)a=ba+ca

13. Prop of Opposites or Inverse Property of Addition
Example: For every real number a, there is a real number -a such that a+(-a)=0 and (-a)+a=0
Prop of Reciprocals or Inverse Prop. of Multiplication
Example: If we multiply a number times its
reciprocal, it will equal one. For example:

14. Identity Property of Addition
Example: There is a unique real number 0 such that for every real number a,
a + 0 = a and 0 + a = a
Zero is called the identity element of addition.
Identity Property of Multiplication
Example: There is a unique real number 1 such that for every real number a,
a · 1 = a and 1 · a = a
One is called the identity element of multiplication.

15. Multiplicative Property of Zero
Example: For every real number a,
a · 0 = 0 and 0 · a = 0
Closure Property of Addition
Example: Closure property of real number addition states that the sum of any two real numbers equals another real number.
Closure Property of Multiplication
Example: Closure property of real number multiplication states that the product of any two real numbers equals another real number.

16. Product of Powers Property
Example: This property states that to multiply powers having the same base, add the exponents. That is, for a real number non-zero a and two integers m and n, am × an = am+n.
Power of a Product Property
Example: This property states that the power of a product can be obtained by finding the powers of each factor and multiplying them. That is, for any two non-zero real numbers a and b and any integer m, (ab)m = am × bm.
Power of a Power Property
Example: This property states that the power of a power can be found by multiplying the exponents. That is, for a non-zero real number a and two integers m and n, (am)n = amn.

17. Quotient of Powers Property
Example: This property states that to divide powers having the same base, subtract the exponents. That is, for a non-zero real number a and two integers m and n,
Power of a Quotient Property
Example: This property states that the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them. That is, for any two non-zero real numbers a and b and any integer m,

18. Zero Power Property Negative Power Property
Example: Any number raised to the zero power is equal to “1”.
Negative Power Property
Example: Change the number to its reciprocal.

19. Zero Product Property
Example: Zero - Product Property states that if the product of two factors is zero, then at least one of the factors must be zero.
If xy = 0, then x = 0 or y = 0.

20. Product of Roots Property
For all positive real numbers a and b,
That is, the square root of the product is the same as the product of the square roots.
Quotient of Roots Property
For all positive real numbers a and b, b ≠ 0:
The square root of the quotient is the same as the quotient of the square roots.

21. Root of a Power Property
Example:
Power of a Root Property
Example:

22. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer.
1. a + b = b + a
Answer: Commutative Property (of Addition)

23. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer.
2. am × an = am+n
Answer: Product of Powers

24. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer.
3. For every real number a,
a · 0 = 0 and 0 · a = 0
Answer: Multiplicative Property of Zero

25. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer.
4. the sum of any two real numbers equals another real number.
Answer: Closure Property of Addition

26. 5. There is a unique real number 1 such that for every real number a,
Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer.
5. There is a unique real number 1 such that for every real number a,
a · 1 = a and 1 · a = a
Answer: Identity property of Multiplication

27. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer. 6.
Answer: Zero Power Property

28. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer. 7.
Answer: Quotient of Powers Property

29. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer.
8. (ab)m = am
Answer: Power of a Product

30. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer. 9.
Answer: Negative Power Property

31. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer.
10.
Answer: Prop of Reciprocals or Inverse Prop. of Multiplication

32. Now you will take a quiz! Look at the sample problem and give the name of the property illustrated.
Click when you’re ready to see the answer.
11. (ab)c=a(bc)
Answer: Associative Property of Multiplication

33. Solving Inequalities

34. Solution Set: {x: x > -4}
Solving Inequalities
Remember the Multiplication Property of Inequality! If you multiply or divide by a negative, you must reverse the inequality sign.
-2x < 8
x > -4
Solution Set: {x: x > -4}
Graph of the Solution: -4

35. Solving Inequalities {x: -5 < x ≤ 8}
Open endpoint for these symbols: > <
Closed endpoint for these symbols: ≥ or ≤
Conjunction must satisfy both conditions
Conjunction = “AND”
{x: -5 < x ≤ 8}
Click to see solution graph -5 8

36. Solving Inequalities {x: x < -6 or x ≥ 8} 8 -6
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 < -6 or x ≥ 8}
Click to see solution graph -6 8

37. Solving Inequalities – Special Cases
Watch for special cases
No solutions that work: Answer is Ø
Every number works: Answer is {reals}
When the disjunction goes the same way you use one arrow.
{x: x > -6 or x ≥ 8}
Click to see solution graph -6 8

38. Solving Inequalities – Special Cases
Watch for special cases:
No solutions that work: Answer is Ø
Every number works: Answer is {reals}
{x: -2x < -4 and -9x ≥ 18}
Click to see solution Ø

39. Solving Inequalities 2x > 6 or -16x ≤ 32 -2 < x or x ≤ 3
Now you try this problem
2x > 6 or -16x ≤ 32
Click to see solution and graph
-2 < x or x ≤ 3 -2 3

40. Solving Inequalities 4x-8 < 12 and -x < 10-4 -6 < x < 5
Now you try this problem.
4x-8 < 12 and -x < 10-4
Click to see solution and graph
-6 < x < 5 -6 5

41. Type the answer here. Set to fade-in on click
Type a sample problem here. Blah blah blah. You can duplicate this slide.
Click when ready to see the answer.
Type the answer here. Set to fade-in on click
Type any needed explanation or tips here. Set to fade-in 3 seconds after the answer appears above.

42. In the next slides you will review: Linear equations in two variables
In the next slides you will review: 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.

43. Linear Equations Slope= Point-Slope Formula= Slope-Intercept Formula=
Midpoint Formula=
Standard/ General Form= Ax+Bx=C
Distance Between Two Points Formula=

44. Slope Pt-Slope Formula (9,12) and (13, 20)
Use when you only have solution points.
(9,12) and (13, 20)
Would be negative if it had a negative sign in front of it. It would then be a falling line and not a rising line.

45. Midpoint Distance (9,12) and (13, 20)
Use to find the Distance between to points.
(9,12) and (13, 20)
Use to find the middle point on a line.

46. Equations in Two Variables
The pairs of numbers that come out for each variable can be written as an (x,y) value. (ordered pair)
You give the solutions in alphabetical order of the variables. So, it would be (a,b) and not (b,a).

47. Standard Form
ax+by=c
All linear equations can be written in this form.
A, b, and c are real numbers and a and b are non-zero. A, b, and c are integers.
To change to slope intercept:
Ax+bx=c bx=ax+c

48. How to graph
To graph the slope-intercept form: you can take the y intercept and use the slope to determine the points on the line.
To graph the standard form you have to change it to slope-intercept, explained in the last slide, and then graph it.

49. To find the y-intercept
To find the x-intercept
To find the y-intercept
F(x)=mx+b
Set f(x)=0
0=mx+b
Divide out
Example:
F(x)=4x-8
0=4x-8
8=4x
2=x
F(x)=mx+b
Set x to 0
F=b
Example:
F(x)=4x+6
F(0)=4(0)+6
F=6

50. In the next slides you will review: Linear Systems. A
In the next slides you will review: Linear Systems A. Substitution Method B. Addition/Subtraction Method (Elimination ) C. Check for understanding of the terms dependent, inconsistent and consistent

51. When two line share solution points…
Null set (if they are parallel)
This will be called an INCONSISTENT SYSTEM
One point (if they cross)
This will be called a CONSISTENT SYSTEM
Infinite Set or All Pts on the Line (if same line is used twice)
This will be called a DEPENDENT SYSTEM (It is also consistent. Dependent is the better name for it than consistent.)

52. Solution: Null set (if they are parallel)
INCONSISTENT SYSTEM
One point (if they cross)
CONSISTENT SYSTEM
Infinite Set or All Pts on the Line (if same line is used twice)
DEPENDENT SYSTEM

53. The SOLUTION of a SYSTEM is the INTERSECTION SET
Where do the two lines
Meet intersect.

54. Two Different Equation on the same graph are called a SYSTEM OF EQUATIONS.
Think about:

55. Solution to this system is:
Method 1: To estimate the solution of a system, you have to find out where they intersect.
Solution to this system is:
{(2, 1)}

56. Example: You can use Trace on a graphing calculator to help you estimate the solution of a system. It can find where they intersect.

57. Solution to this system appears to include TWO pts:
Example: You can use Trace on a graphing calculator to help you estimate the solution of a system. It can find where they intersect
Solution to this system appears to include TWO pts:
{ (-2.38, 1.79), (3, 0) }

58. Summary of Method 1: Estimate the SOLUTION of a SYSTEM on a graph
Summary of Method 1: Estimate the SOLUTION of a SYSTEM on a graph. (Goal: Find intersection pts.)
Disadvantages: Might only give an estimate.
It might not be possible to graph some equations yet. Advantages: If the graph is easy, this is a good way to check.
It is good for a quick answer.

59. Method 2: Substitution Method (Goal: Replace one variable in one equation with the set from another.)
Step 1: Look for a variable with a coefficient of one.
Step 2: Move everything else to the other side.
Equation A now becomes: y = 15-x
Step 3: SUBSTITUTE this expression into that variable in Equation B
Equation B now becomes 4x – 3( 15-x ) = 38
Step 4: Solve the equation.
Step 5: Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

60. Method 2: Substitution (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 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!)

61. Example: Substitution (Goal: replace one variable with the set of another equation.)
Step 1: Look for a variable with a coefficient of one.
Step 2: Move everything else to the other side.
Step 3: SUBSTITUTE this expression into that variable in Equation B
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!)

62. Example: Substitution (Goal: replace one variable with an equal expression.)
Step 1: Look for a variable with a coefficient of one.
Step 2: Isolate that variable
Step 3: SUBSTITUTE this expression into that variable in Equation B
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!)

63. Method 2 Summary: Substitution Method (Goal: replace one variable with an equal expression.)
Disadvantages: Avoid this method when it requires messy fractions  Avoid IF no coefficient equals one. Advantages: This is the algebra method to use when degrees of the equations are not equal.

64. Method 3: Elimination Method or Addition/Subtraction 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 the equation.
Step 5: Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.)

65. Method 3: Elimination or Addition/Subtraction 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.
A becomes: 10x – 6y = 10
B becomes: 9x + 6y = -32
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.) +

66. Method 3: Elimination or Addition/Subtraction 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.)
Step 2: Multiply each equation by the necessary factor.
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.) +

67. Example: Elimination or Addition/Subtraction 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.)
Step 2: Multiply each equation by the necessary factor.
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.) +

68. Example: Elimination or Addition/Subtraction 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.)
Step 2: Multiply each equation by the necessary factor.
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.) +

69. Method 3 Summary: Elimination Method or Addition/Subtraction Method (Goal: Combine equations to cancel out one variable.)
Disadvantages: Avoid this method if degrees and/or formats of the equations do not match. Advantages: Similar to getting an LCD, so this is intuitive, and uses only integers until the end of the problem.

70. Three Methods Method 1: Graphing Method Method 2: Substitution Method Method 3: Elimination Method or Addition/Subtraction Method

71. Factoring Factor GCF  for any # terms
Difference of Squares  binomials
Sum or Difference of Cubes  binomials
PST (Perfect Square Trinomial)  trinomials
Reverse of FOIL  trinomials
Factor by Grouping  usually for 4 or more terms

72. Take out what the two side share in common to simplify.
GCF
3x2-9x2
3x2(1-3)
Take out what the two side share in common to simplify.

73. Hint: Remember that a “glob” can be part of your GCF.
3(1-6)2+6(1-6)2
3(1-6)2(1+3)
You can take out a glob and then combine with the other globs.
Hint: Remember that a “glob” can be part of your GCF.

74. Difference of Squares 25x2-4x2 (5x-2)(5x+2)
Recall these binomials are called conjugates.

75. IMPORTANT!
Remember that the difference of squares factors into conjugates . . .
The SUM of squares is PRIME – cannot be factored.
a2 + b2  PRIME
a2 –b2  (a + b)(a – b)

76. Sum/Difference of Cubes
x3-y3
(x-y)(x2+xy+y2)

77. Sum/Difference of Cubes
x3 - y3
(x - y) ( )
Cube roots w/ original
sign in the middle

78. Sum/Difference of Cubes
x3 - y3
(x - y) (x y2)
Squares of those cube roots.
Note that squares will always be positive.

79. Sum/Difference of Cubes
x3 - y3
(x - y) (x2 + xy + y2)
The opposite of the product
of the cube roots

80. Sum/Difference of Cubes
x3 - 8
(x - 2)
Cube roots of each Squares of those cube roots &
with same sign opp of product of roots in middle
(x )
+ 2x

81. Special Case- * 1ststep: Diff of Squares * 2nd step: Sum/Diff of Cubes
x6 – 64y6
( ) ( )
( )( ) ( )( )

82. Special Case * 1ststep: Diff of Squares * 2nd step: Sum/Diff of Cubes
x6 – 64y6
(x3 – 8y3) (x3 + 8y3)
( )( ) ( )( )

83. Special Case * 1ststep: Diff of Squares * 2nd step: Sum/Diff of Cubes
x6 – 64y6
(x3 – 8y3) (x3 + 8y3)
(x–2y)(x2+2xy+4y2) (x+2y)(x2-2xy+4y2)

84. PST x2-10x+25 (x-5)(x-5) (x-5)2 Recall PST test:
If 1st & 3rd terms are squares and the middle term is twice the product of their square roots.

85. PST x
(3x-4)(3x-4)
Conjugates

86. Reverse FOIL (Trial & Error)
12x2-45x+42
(3x-6)(4x-7)
Conjugates

87. Reverse FOIL (Trial & Error)
Hint: don’t forget to read the “signs”
ax2 + bx + c  ( )( )
ax2 – bx + c  ( – )( – )
ax2 + bx – c  ( )( – )
positive product has larger value
ax2 – bx – c  ( )( – )
negative product has larger value

88. Example 11: Factor by Grouping (4 or more terms)
a(x-y)+x-y(x-y)
(x-y)3(a)

89. Example 11: Factor by Grouping (4 or more terms)
8x-2y+16x-4y
2(4x-y)+4(4x-y)
(4x-y)2(2+4)
2X2

90. Factor by Grouping 3 X 1 x2+xy+y2-4x2 (x-y)(x-y)-4x2
[(x-y)-2x][(x-y)-2x]
3X1 (PST)

91. Factor by Grouping 3 X 1 x2-10+25-4x2 (x-5)(x-5)-4x2

92. Rational Expressions

93. Rational Numbers
Thinking back to when you were dealing with whole-number fractions, one of the first things you did was simplify them: You "cancelled off" factors which were in common between the numerator and denominator. You could do this because dividing any number by itself gives you just "1", and you can ignore factors of "1".
Using the same reasoning and methods, let's simplify some rational expressions.
Simplify the following expression:
To simplify a numerical fraction, I would cancel off any common numerical factors. For this rational expression (this polynomial fraction), I can similarly cancel off any common numerical or variable factors.
The numerator factors as (2)(x); the denominator factors as (x)(x). Anything divided by itself is just "1", so I can cross out any factors common to both the numerator and the denominator. Considering the factors in this particular fraction, I get:
Then the simplified form of the expression is: 

94. Additoin and Subtraction of Rational Numbers
  3           5               8               (2)(4)            2        +              =              =                   =          20         20             20              (4)(5)            5
Notice the steps we have done to solve this problem.  We first combined the numerators since the denominators are the same.  Then we factored both the numerator and denominator and finally we cross cancelled.

95. Multiplication And Division of Rational Numbers
 x2 - y2                           is a rational expression. (x - y)2   To simplify, we just factor and cancel:
(x - y)(x + y)            x  + y                          =                     (x - y)2                  x - y

96. Quadratic Equations

97. Quadratic Equations
A quadratic equation is an equation that can be written in this form.
ax2+bx+c=0
The a,b, and c here represent real number coefficients. So this means we are talking about an equation that is a constant times the variable squared plus a constant times the variable plus a constant equals zero, where the coefficient a on the variable squared can't be zero, because if it were then it would be a linear equation.

98. Completing the Square
a² + 2ab + b²=(a + b)².The technique is valid only when 1 is the coefficient of x².
 1)  Transpose the constant term to the right:x² + 6x  =  −2
 2)  Add a square number to both sides.  Add the square of half the coefficient of x.  In this case, add the square of 3:x² + 6x + 9  =  −2 + 9.
The left-hand side is now the perfect square of  (x + 3).
(x + 3)²  =  7.
3 is half of the coefficient 6.
This equation has the form
a² = b   which implies a = ± .          Therefore,x + 3 = ±  
x = −3 ± .
That is, the solutions to
x² + 6x + 2  =  0
are the conjugate pair,
−3 + ,  −3 − .

99. Quadratic Formula

100. Quadratic Formula
on multiplying both c and a by 4a, thus making the denominators the same (Lesson 23),
This is the quadratic formula.  

101. Discriminant
The radicand  b² − 4ac  is called the discriminant.  If the discriminant is
a)   Positive:The roots are real and conjugate. 
b) Negative:  The roots are complex and conjugate. 
c)   Zero:The roots are rational and equal -- i.e. a double root.

102. Factoring Problem 2. Find the roots of each quadratic by factoring.
a)  x² − 3x + 2 b)  x² + 7x + 12
(x − 1)(x − 2)  (x + 3)(x + 4)
x = 1  or  2.  x = −3  or  −4.

103. In the next slides you will review: Functions. D
In the next slides you will review: Functions D. Quadratic functions – explain everything we know about how to graph a parabola

104. Functions
A function is an operation on numbers of some set (domain) that gives (calculates) one number for every number from the domain. For example, function 3x is defined for all numbers and its result is a number multiplied by 3. The notation is y=f(x). x is called the argument of the function, and y is the value of the function. The inverse function of f(x), is a different function of y that finds x that gives f(x) = y.

105. Functions
We have seen that a function is a special relation. In the same sense, real function is a special function. The special about real function is that its domain and range are subsets of real numbers “R”. In mathematics, we deal with functions all the time – but with a difference. We drop the formal notation, which involves its name, specifications of domain and co-domain, direction of relation etc. Rather, we work with the rule alone. For example,
f(x) =x2+2x+3
This simplification is based on the fact that domain, co-domain and range are subsets of real numbers. In case, these sets have some specific intervals other than “R” itself, then we mention the same with a semicolon (;) or a comma(,) or with a combination of them :
f(x) =\(x+1) 2−1;x<−2,x≥0
Note that the interval “ x<−2,x≥0 ” specifies a subset of real number and defines the domain of function. In general, co-domain of real function is “R”. In some cases, we specify domain, which involves exclusion of certain value(s), like :
f(x) =
11−x,x≠1
This means that domain of the function is R−{1{ . Further, we use a variety of ways to denote a subset of real numbers for domain and range. Some of the examples are :
x>1: denotes subset of real number greater than “1”.
R−{0,1{ denotes subset of real number that excludes integers “0” and “1”.
1<x<2: denotes subset of real number between “1” and “2” excluding end points.
(1,2]: denotes subset of real number between “1” and “2” excluding end point “1”, but including end point “2”.
Further, we may emphasize the meaning of following inequalities of real numbers as the same will be used frequently for denoting important segment of real number line :
Positive number means x > 0 (excludes “0”).
Negative number means x < 0 (excludes “0”).
Non - negative number means x ≥ 0 (includes “0”).
Non – positive number means x ≤ 0 (includes “0”).

106. Quadratic Functions
A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.
The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape. All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola.
You know that two points determine a line. This means that if you are given any two points in the plane, then there is one and only one line that contains both points. A similar statement can be made about points and quadratic functions.
Given three points in the plane that have different first coordinates and do not lie on a line, there is exactly one quadratic function f whose graph contains all three points. To graph simplify (see quadratic equations) and plot the points.

107. Simplifying expressions with exponents

108. Simplifying Exponents
Use the Power of a Power Property, the Product of a Power Property, the Quotient of a Power Property, the Power of a Quotient Property, the

109. Simplifying exponents
Use These

110. Simplifying Exponents
=317

111. Simplifying expressions with radicals

112. Simplifying radicals
When presented with a problem like , we don’t have too much difficulty saying that the answer 2 (since 2x2=4 ). Our trouble usually occurs when we either can’t easily see the answer or if the number under our radical sign is not a perfect square or a perfect cube. A problem like may look difficult because there are no two numbers that multiply together to give 24. However, the problem can be simplified. So even though 24 is not a perfect square, it can be broken down into smaller pieces where one of those pieces might be perfect square. So now we have
Simplifying a radical expression can also involve variables as well as numbers. Just as you were able to break down a number into its smaller pieces, you can do the same with variables. When the radical is a square root, you should try to have terms raised to an even power (2, 4, 6, 8, etc). When the radical is a cube root, you should try to have terms raised to a power of three (3, 6, 9, 12, etc.). For example,

113. Simplifying Radicals
Use the root of a power, power of a root, product of a root, and quotient of a root properties to solve.

114. 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.)

115. Suppose that it takes Janet 6 hours to paint her room if she works alone and it takes Carol 4 hours to paint the same room if she works alone. How long will it take them to paint the room if they work together?
Click to see answer.
First, we will let x be the amount of time it takes to paint the room (in hours) if the two work together.
Janet would need 6 hours if she did the entire job by herself, so her working rate is of the job in an hour. Likewise, Carol’s rate is of the job in an hour.
In x hours, Janet paints of the room and Carol paints of the room. Since the two females will be working together, we will add the two parts together. The sum equals one complete job and gives us the following equation:

116. Continued answer
Multiply each term of the equation by the common denominator 12
Simplify
Collect like terms
Solve for x

117. Suppose Kirk has taken three tests and made 88, 90, and 84
Suppose Kirk has taken three tests and made 88, 90, and 84. Kirk’s teacher tells the class that each test counts the same amount. Kirk wants to know what he needs to make on the fourth test to have an overall average of 90 so he can make an A in the class.
Steps

118. Suppose a bank is offering its customers 3% interest on savings accounts. If a customer deposits $1500 in the account, how much interest does the customer earn in 5 years?
I is the amount of interest the account earns.
P is the principle or the amount of money that is originally put into an account.
r is the interest rate and must ALWAYS be in a decimal form rather than a percent.
t is the amount of time the money is in the account earning interest.
If we want to find out the total amount in the account, we would need to add the interest to the original amount. In this case, there would be $1725 in the account

119. The Smith’s have a rectangular pool that measure 12 feet by 20 feet
The Smith’s have a rectangular pool that measure 12 feet by 20 feet. They are building a walkway around it of uniform width.
The length of the larger rectangle is , which simplifies to
Length larger rectangle =
The width of the larger rectangle is , which simplifies to
Width larger rectangle =
The area for the larger rectangle then becomes
Area larger rectangle =
The pool itself has an area of square feet

120. Answer continued
Rearrange the terms for easier multiplication and find the sum of 68 and 240.
Multiply the binomials.
Combine like terms and subtract 308 from each side.
Factor.
Solve each factor.
Since dimensions of a pool and a walkway around a pool cannot be negative our answer is that the width of the walkway is 1 foot.

121. Line of Best Fit or Regression Line

122. Line of best fit
A line of best fit is a straight line that best represents the data on a scatter plot.  This line may pass through some of the points, none of the points, or all of the points.

123. Line of Best Fit
You can find the line of best fit by estimation or by using graphing calculators.
The line of best fit is good for estimating the average of the points on the graph.

124. Line of best fit (how to solve)
1. Separate the data into three groups of equal size according to the values of the
horizontal coordinate.
2. Find the summary point for each group based on the median x-value and the
median y-value.
3. Find the equation of the line (Line L) through the summary points of the outer
groups.
4. Slide L one-third of the way to the middle summary point.
a. Find the y-coordinate of the point on L with the same x-coordinate as the
middle summary point.
b. Find the vertical distance between the middle summary point and the line by
subtracting y-values.
c. Find the coordinates of the point P one-third of the way from the line L to the
5. Find the equation of the line through the point P that is parallel to line L.

125. Line of Best Fit
Try to find the line of best fit for the points (3,9)(4,8)(6,6)(7,5)(9,7)(11,9)(13,12)(14,17)(13,19).