1. Algebra 1
2nd Quarter - Review

2. Topics from Semester II
Functions and Linear Equations
Systems of Linear Equations
Quadratics and Polynomials
Exponential Functions and Sequences
Scatter Plots, Trends and Statistics

3. Functions and Linear Equations
f(x) = mx + b

4. Functions and Linear Equations

5. Functions

6. An Ordered Pair (x, y) is a pair of numbers that describes a location in the coordinate plane.
A Relation is a set of ordered pairs.
{ (3, 5), (4, 7), (5, 9) (6, 11) }
A Function is a relation where each input (x) has only one output (y).

7. Mapping
Functions (and relations) are sometimes presented as mapping one set of numbers (input) to another (output).
The arrows connect each input (x)
with the output (y)
it produces.
input
output

8. Function: Each x has only one y
Not a Function
Function

9. Vertical Line Test A Vertical Line
hits the graph of a function only once.
This IS the graph of a Function

10. Vertical Line Test
This
IS NOT the graph of a Function

11. Domain and Range Domain
The Domain of a Relation is the set of all Input values (x).
This is also called the
Independent Variable.
Notice:
Domain
Input
Independent
all contain the word “in”
Domain

12. Domain and Range Range The Range of a Relation is the set of all
Output values (y).
This is called the Dependent Variable because its value depends on the input.
Domain Range
Input Output
Independent Dependent
Range

13. Evaluating a Function
Many Functions are defined by algebraic rules (or equations).
To evaluate the function for a given input value, substitute the value into the equation and simplify.
f(x) = 3x – 5
{f is the name of the function and x is the input }
If x=4, then f(4) = 3(4) – 5 = 7
So f(4) = 7 and the point (4,7) is on the graph of the function.

14. Linear Equations

15. Linear Equations Graphing Linear Equations
Graphing Linear Inequalities
Writing Linear Equations

16. Graphing Linear Equations
Graph: y = 2x -5
y = 2(0) – 5
y = 2(1) – 5
From a table: X Y
– 5 1
– 3
Pick values for x
That will be easy to use.
And evaluate for y.
x = zero
is usually easiest
Pick another x
Close by.

17. Graphing Linear Equations
Graph: y = 2x -5
From a table: X Y
– 5 1
– 3
Plot each Point
And draw the
Line through them.

18. Graphing Linear Equations
Graph: y = -⅖x +5
y = -⅖(5) + 5
y = -⅖(0) + 5
From a table:
-2 + 5 X Y 5 5 3
Pick values for x
That will be easy to use.
And evaluate for y.
x = zero
is usually easiest
Pick denominator
As second x

19. Graphing Linear Equations
Graph: y = -⅖x +5
From a table: X Y 5 5 3
Plot each Point
And draw the
Line through them.

20. Graphing Linear Equations
Graph: y = ⅔x -3
Using y-intercept
and slope.
Plot the y-intercept
on the y-axis.
run = 3
Use slope to find
A second point.
rise = 2
Draw line through
the 2 points.

21. Graphing Linear Equations
Graph: 4x - 3y = 24
-3y = 24
4x = 24
From standard form: X Y -8 6
Because 24 is a multiple
of both 4 and 3, it’s easy
to use x and y intercepts.
x = zero
for y-intercept
y = zero
For x-intercept

22. Graphing Linear Equations
Graph: 4x - 3y = 24
From standard form: X Y -8
x-intercept on x-axis 6
Plot each Point
And draw the
Line through them.
y-intercept on y-axis

23. Graphing Linear Equations
Graph: 5x - 3y = 24
From standard form:
Because 24 is not
a multiple of 5.
Convert to
y = mx + b

24. Convert from Standard form to Slope-Intercept
Standard Slope-Intercept
ax + by = c y = mx + b
5x - 3y = 24
-5x
-5x
Move x-term to other side
by adding the opposite
-3y = -5x + 24
Solve for y -3 -3 -3
y = 5/3x - 8

25. Graphing Linear Equations
Graph: 5x - 3y = 24
y = 5/3x - 8
y = 5/3(0) - 8
y = 5/3(3) - 8 X Y -8 3 -3

26. Graphing Linear Inequalities
Graph the Line
If < or > use a dashed line (not part of solution).
If ≤ or ≥ use a solid line.
Identify a point in the solution set:
Usually, check the origin (0,0)
If point is a solution, shade that side of the line
Otherwise, shade the other side of the line.

27. Graphing Linear Inequalities
Graph: y <⅔x -3
Using y-intercept
and slope.
Plot the y-intercept
on the y-axis.
Use slope to find
A second point.
Draw line through
the 2 points.

28. Graphing Linear Inequalities
Graph: y < ⅔x -3
Shade below the line?
Check a point.
Is (3, -4) a solution?
Shade ?
y < ⅔x -3
-4 < ⅔(3) -3 ?
-4 < (2 -3) ? √

29. Graphing Linear Inequalities
Graph: y < ⅔x -3
Shade below the line!

30. Writing Linear Equations
Linear Equations in Three Forms
Point-Slope Form
y - y1 = m( x – x1)
Slope-Intercept Form
y = mx + b
Standard Form
ax + by = c

31. Writing Linear Equations
Point-Slope Form
y - y1 = m( x – x1)
Point (x1, y1)
Slope
Slope-Intercept Form
y = mx + b
y-intercept

32. Slope Intercept ⅔ Slope-Intercept Form y = mx + b
Write the equation of a line with a slope of ⅔
And a y-intercept of 9
Slope-Intercept Form
y = mx + b ⅔ 9
Slope = ⅔
y-intercept = 9

33. Slope Intercept Slope-Intercept Form y = m x + b
Write the equation of a line
Parallel to y = -3x +5
with a y-intercept of -6
Parallel lines have
the same slope.
Slope-Intercept Form
y = m x + b -3
- 6
Slope = -3
y-intercept = -6

34. Slope Intercept Write the equation of a line
Perpendicular to y = ½x - 13
Through the point (0, 5)
The slopes of
Perpendicular lines
are negative inverses.
Negative inverse
of ½ is -2
Slope-Intercept Form
y = mx + b -2
The y-intercept…
is where x = 0.
Slope = -2
the point (0, 5) 5
y = -2x + b

35. Slope Intercept Write the equation of a line with a slope of -3
Through the point (3, 8)
Slope-Intercept Form
y = mx + b -3
We don’t know the y-intercept…
Slope = -3 8
y = -3 x + b
(3)
We do know one point on the graph.
The point (3, 8)

36. Slope Intercept Write the equation of a line with a slope of -3
Through the point (3, 8)
Slope-Intercept Form
y = mx + b
y = -3x + b
1) Substitute in slope
8 = -3(3) + b
2) Substitute x and y
8 = b
3) Solve for b (y-intercept)
17 = b
y = -3x + b 17
4) Substitute for b

37. Slope Intercept Write the equation of a line with a slope of ⅖
Through the point (10, 3)
Slope-Intercept Form
y = mx + b
y = ⅖x + b
1) Substitute in slope
3 = ⅖(10) + b
2) Substitute x and y
3 = b
3) Solve for b (y-intercept)
4) Substitute for b
-1 = b
y = ⅖ x + b
- 1

38. Point-Slope Write the equation of a line with a slope of ⅔
through the point (2, 5)
Point-Slope Form
y - y1 = m( x – x1) 5 ⅔ 2
Point (2, 5)
Slope = ⅔

39. Point-Slope Write the equation of a line with a slope of 4
through the point (-2, -3)
Point-Slope Form
y - y1 = m( x – x1 )
+ 3 4
+ 2
Point (-2, -3)
Slope = 4

40. Point-Slope Write the equation of a line parallel to y= 2x - 9
through the point (-5, 7)
Point-Slope Form
y - y1 = m( x – x1) 7
+ 5 2
Point (-5, 7)
Slope = 2
Parallel Lines have the Same Slope

41. Convert from Point-Slope to Slope-Intercept
Point-Slope Slope-Intercept
y - y1 = m( x – x1) y = mx + b
y - 5 = 3( x – 2)
Distribute
y - 5 = 3x – 6 +5 +5
Solve for y
y = 3x – 1
skip

42. Convert from Point-Slope to Slope-Intercept
Point-Slope Slope-Intercept
y + 3 = ⅔( x – 6)
Distribute
y + 3 = ⅔x – 4
Solve for y
- 3
- 3
y = ⅔ x – 7

43. Convert from Slope-Intercept to Standard form
Slope-Intercept Standard
y = mx + b ax + by = c
y = 3x + 7
Move x-term to other side
by adding the opposite
- 3x
- 3x
-3x + y = 7

44. Convert from Slope-Intercept to Standard form
Slope-Intercept Standard
y = mx + b ax + by = c
y = ⅓x + 5
Move x-term to other side
by adding the opposite
- ⅓x
- ⅓x 3(
- ⅓x + y = 5 )
Clear Fractions
-x + 3y = 15

45. Convert from Standard form to Slope-Intercept
Standard Slope-Intercept
ax + by = c y = mx + b
3x + 2y = 18
Move x-term to other side
by adding the opposite
-3x
-3x
2y = -3x + 18 2 2 2
Solve for y
y = - x + 9 3 2

46. Systems of Linear Equations1

47. Linear Systems Solving by Graphing Solving Algebraically Substitution
Linear Combination

48. Linear Systems √ y = 2x – 5 1 = 2(3) – 5 y = -x + 4 1 = -(3) + 4
Solving by Graphing
Graph Each Equation.
Solution is the point where the graphs
Intersect.
Check Algebraically
(3, 1) √

49. Linear Systems 2x + 3y = 6 y = -⅔x + 4 Solving by Graphing
Graph Each Equation.
Solution is the point where the graphs
Intersect.
Parallel Lines
Don’t Intersect.
There is no solution.

50. Linear Systems 2x + 5y = 15 y = -⅖x + 3 Solving by Graphing
Graph Each Equation.
Both Equations are
For the same line.
There are infinitely
many solutions.

51. Linear Systems By Substitution: (5, -1) 4x +2y = 18 2x -5y = 15
Solve for one variable in terms of the other
Substitute in other equation
4x +2y = 18
2x -5y = 15
2y = -4x +18
y =
-2x +9
-2x +9
y = -2( )+9
2x -5( ) = 15
y = = -1
2x +10x -45 = 15
(5, -1)
12x = 60
x = 5 5

52. Substitution (5, 7) = y = 3x – 8 y=3x - 8 + 2x -2x +17 + 2x y=3(5) - 8
Set the two equations equal (since they both equal y).
y =
3x – 8 =
y=3x - 8
+ 2x
-2x +17
+ 2x
y=3(5) - 8
5x – 8 = 17
5x = 25
x = 5
y= 7
(5, 7)
Substitute in either of the equations.

53. Linear Combination + (7, 5) 3x – y = 16 2x + y = 19 5x = 35 x = 7 7
Also called Elimination because the goal is to eliminate one of the variables.
3x – y = 16
2x + y = 19
Add the two equations. +
Opposites in the two equations
will cancel out, eliminating the variable.
5x = 35
x = 7 7
Solve for the remaining variable.
2( ) + y = 19
Substitute in any of the equations.
14 + y =19
y =5
(7, 5)

54. } Linear Combination (8, 3) 2( 2x – y = 13 3x +2y = 30 ) 4x –2y = 26
Need Opposites in the two equations 2(
2x – y = 13
3x +2y = 30 )
So multiply one (or both) equation(s) }
Add these two equations.
4x –2y = 26
7x = 56
x = 8
3x +2y = 30
3( 8) +2y = 30
24 +2y = 30
(8, 3)
2y = 6
y = 3

55. } Linear Combination (3, 4) 3( 2x –3y = -6 3x +5y = 29 ) ) -2(
Need Opposites in the two equations 3(
2x –3y = -6
3x +5y = 29 )
So multiply one (or both) equation(s) )
-2( }
6x –9y = -18
Add these two equations.
-6x –10y = -58
3x +5y = 29
-19y = -76
3x +5(4) = 29
y = 4
(3, 4)
3x = 29
3x = 9
x = 3

56. } Linear Combination 2( 2x – y = 13 -4x +2y = 30 ) 4x –2y = 26 0 = 56
Need Opposites in the two equations 2(
2x – y = 13
-4x +2y = 30 )
So multiply one (or both) equation(s) }
Add these two equations.
4x –2y = 26
0 = 56
Since this is false, there is no solution.
These are parallel lines.
If both variables are eliminated
and the resulting statement is true
then there are infinitely many solutions.

57. Quadratics and Polynomials

58. Quadratics and Polynomials
Laws of Exponents
Operations with Polynomials
Special Products
Factoring
Completing the Square
Quadratic Formula
Graphing

59. ( ) Laws of Exponents = x2+3 = x5 = x7-3 = x4 = x 4∙3 = x12 = 27a3b6 =
Product property x2(x3)
Quotient Property
Power of a Power (x4)3
Power of a Product (3ab2)3
Power of a Quotient
= x2+3 = x5 x7 x3
= x7-3 = x4
= x 4∙3 = x12
= 27a3b6
( ) x7 y3 2
x7∙2
y3∙2
x14 y6 = =

60. ( ) Laws of Exponents = x5-5 = x0 = 1 = x5-7 = x-2 = x5 Zero Exponents
Negative Exponents
= x5-5 = x0 = 1
( )
2x5y3
7x3z-3
= 1 1 x2 x5 x7
= x5-7 = x-2 =
-2x-5
y7z-3 -2
In fractions x5
Terms with negative exponent move

61. Laws of Exponents = x5-5 = x0 = 1 = x5-7 = x-2 = x5 Zero Exponents
Negative Exponents
= x5-5 = x0 = 1 1 x2 x5 x7
= x5-7 = x-2 = -2 y7 -2
y7z-3 z3
In fractions x5
Terms with negative exponent move

62. Simplifying Expressions
Each variable base appears only once
There are no powers of powers
All fractions are in simplest form
(2gh4)3[(-2g4h)3]2
[-8g12h3]2
(8g3h12)
(64g24h6)
= 512g27h18

63. Simplifying Expressions
Each variable base appears only once
There are no powers of powers
All fractions are in simplest form
a7b4c5
a2b7c a5 1 1 b3 c4 1
a5c4 b3
a7b4c5
b-3c5 a7 b7 c0
= a7b7

64. Simplifying Expressions
( ) 3 3
( )
2x5y6z3
3x7y2
2y4z3
3x2
8y12z9
27x6
Parentheses first
Then exponent
Negative Exponent means the “inverse”
( )
( ) -2 2
( )
a3b4
a8b2c3 2
a8b2c3
a3b4
a10c6 b4
a5c3 b2
So you can flip the fraction first

65. Adding Polynomials (3x3 – 5x2 +x -11) + (4x3 + 7x2 -6x +23) (7x3 +2x2
Group and Combine Like Terms
Like Terms have the same variable
raised to the same power
(3x3 – 5x2 +x -11) + (4x3 + 7x2 -6x +23)
(7x3
+2x2
- 5x
+ 12)
(x3 – 5x2 + 4x) + (4x3 -6x +23)
(5x3
-5x2
- 2x
+ 23)
Include any terms that are only in one polynomial

66. Subtracting Polynomials
Add the Opposite
(3x3 – 5x2 +x -11) - (4x3 + 7x2 -6x +23)
(3x3 – 5x2 +x -11) + (-4x3 -7x2 +6x -23)
(-x3
-12x2
+ 7x
- 34)
(x3 – 5x2 + 4x) - (4x3 -6x +23)
(x3 – 5x2 + 4x) + (-4x3 +6x -23)
(-3x3
-5x2
+ 10x
- 23)

67. Multiplying Polynomials
Use the distributive property and Laws of Exponents
-5x2 (3x2 + 4x - 11)
= -15x x3 + 55x2
2x (3x - 11) +3(x -5)
= 6x x + 3x -15
= 6x x -15
(x+ 3)(x - 8)
= x2 - 8x
+ 3x - 24
= x2 - 5x - 24

68. Special Products Square of a Binomial (x+ 3)2 = (x+ 3)(x + 3)
Rewrite and distribute
(x+ 3)2
= (x+ 3)(x + 3)
= x2 + 3x +3x + 9
= x2 + 6x + 9
Or use pattern
(a+ b)2 = a2 + 2ab + b2
(x+ 8)2
= x2 + 2x(8) + 82
= x2 + 16x + 64
(a - b)2 = a2 - 2ab + b2
(x- 5)2
= x2 - 2x(5) + 52
= x2 - 10x + 25

69. Special Products Product of a sum and difference (x+ 3)(x - 3)
= x2 - 3x +3x - 9
= x2 - 9
(4x+ 5)(4x - 5)
= 16x2 - 20x +20x - 25
= 16x2 - 25
(a+ b)(a - b)
= a2 – b2
Pattern:
(x+ 7)(x - 7)
= x2 – 72
= x2 – 49

70. Factoring

71. Factoring: x2 + bx + c x2 + bx + c = (x + m)(x + p )
So, we’re looking for
two factors of c that add to b.
x2 + 7x + 12 = (x +__ )(x +__ ) 3 4
Factors of 12 1 12 2 6 3 4
= 7

72. Factoring: x2 + bx + c x2 + 9x + 18 = (x __ )(x __ ) + +3 + 6
Factors of 18 1 18 2 9 3 6
Both
Positive
Factors have
the same sign
x2 – 8x = (x __ )(x __ ) – – 4 4
Factors of 16 1 16 2 8 4
Both
Negative
Factors have
the same sign

73. Factoring: x2 + bx – c x2 + 7x – 8 = (x __ )(x __ ) – +1
Factors of 8 1 8 2 4
Larger
Positive
Factors have
different signs
x2 – 4x – 21 = (x __ )(x __ ) + – 3 7
Factors of 21 1 21 3 7
Larger
Negative
Factors have
different signs

74. Factoring: x2 – c This is a “difference of squares”
Pattern: (a2 – b2) = (a + b)(a – b)
(m – n) = (√m + √ n)(√m– √n)
x2 – 16 = (x + __ )(x – __ )
√ 16 = 4 4 4
√ 121 = 11
x2 – 121 = (x + __ )(x –__ ) 11 11
√ 4x2 = 2x
4x2 – 169 = ( x + __ )( x –__ ) 2 13 2 13
√ 169 = 13

75. Factoring: ax2 – ac This “difference of squares”
Is hidden by a common factor.
(ax2 – ab2) = a(x2 – b2) = a(x + b)(x – b)
3x2 – 75 = 3(x2 – 25) =3(x + __ )(x – __ )
√ 25 = 5 5 5
2x3 – 98x =
2x(x2 – 49) =
√ 49 = 7
2x(x + __ )(x –__ ) 7 7
4x2 – 324 =
4(x2 – 81 )
√ 81 = 9
4(x + __ )(x –__ ) 9 9

76. Factoring by Grouping = ac + ad + bc + bd a(c + d) +b (c + d)
(a + b)(c + d)
= ac + ad + bc + bd
a(c + d) +b
(c + d)
(a + b)(c + d)
To factor by “grouping”
Split the polynomial into two “groups”
Pull out a common factor on Left
See if the resulting binomial is a factor on right.
Find common factor on Right
Regroup into binomial factor s.

77. Factoring by Grouping x3 + 3x2 + 5x + 15 x2(x + 3) +5 (x + 3)
To factor by “grouping”
Split the polynomial into two “groups”
Pull out a common factor on Left
See if the resulting binomial is a factor on right.
Find common factor on Right
Regroup into binomial factors.

78. Factoring: ax2 + bx + c 3x2 + 10x + 8 3(8) = 24 3x2 + 4x+ 6x + 8
Find the product ac
3(8) = 24
3x2 + 4x+ 6x + 8
We’re looking for factors of 24
that add to 10
Split middle term.
x(3x + 4) +2
(3x + 4) 1 24 2 12 3 8 4 6
Factor by grouping.
(x +2 )(3x +4 )

79. Solving Quadratic Equations by Factoring
Zero Product Rule:
If the product of two numbers is zero,
then at least one of the numbers is zero.
(x – 5)(x +7) = 0
(x – 5)= 0 or (x +7) = 0
x = or x = -7

80. Solving Quadratic Equations by Factoring
Factors of 12
x2 - 8x + 12 = 0 1 12 2 6 3 4
(x – 6)(x - 2) = 0
(x – 6)= 0 or (x - 2) = 0
x = or x = 2

81. Solving Quadratic Equations by Factoring
Factors of 24
x2 - 5x - 24 = 0 1 24 2 12 3 8 4 6
(x – 8)(x +3) = 0
(x – 8)= 0 or (x + 3) = 0
x = or x = -3

82. Solving: x2 – c = 0 This is a “difference of squares”
Pattern: (a2 – b2) = (a + b)(a – b)
x2 – 25 = 0
√ 25 = 5
(x + __ )(x – __ ) = 0 5 5
(x + 5)= 0 or (x - 5) = 0
x = -5 or x = 5

83. Solving: x2 – c = 0 Or take the square root x2 – 49 = 0 x2 = 49
x = 7 or -7

84. Solving: take the square root
2x2 – 47 = 25
First solve for x2
2x2 = 72
Then take square root
On both sides
x2 = 36
√x2 = ±√ 36 = ± 6
x = 6 or -6

85. Solving: take the square root
2(x - 2)2 – 4 = 14
First solve for (x-2)2
2(x - 2)2 = 18
Then take square root
On both sides
(x – 2)2 = 9
√(x – 2)2 = ±√ 9
x – 2 = ± 3
x = 2+3 or 2-3
x = or -1

86. Solving: Perfect Square
Cannot take the square root
if there is an x term,
Unless it’s a perfect square.
x2 -10x +25 = 12
(x – 5)2 = 12
Rewrite as (x-5)2
√(x – 5)2 = ±√ 12
Then take square root
On both sides
x – 5 = ± √12
x = 5 ± √12

87. Completing the Square
(x + b)2
= x2 + 2bx + b2

88. Completing The Square (x +n)2= x2 + 2nx + n2 x2 + bx + c 2n = b n = ½b
Square of a binomial
x2 + bx + c
Standard form
2n = b
n = ½b
c = n2

89. Completing The Square (x +n)2= x2 + 2nx + n2 x2 + 12x + 36 2n =12
Square of a binomial
x2 + 12x + 36
Standard form
2n =12
n = ½(12) = 6
c = 62 = 36

90. Completing The Square (x +n)2= x2 + 2nx + n2 x2 + 14x + c 2n =14
Square of a binomial
x2 + 14x + c
Standard form
2n =14
n = ½(14) = 7
c = 72 = 49 49
What value must c have to represent a binomial square?

91. Completing The Square x2 + 16x + 24 = 9 x2 + 16x + = -15 x2 + 16x + 64
Subtract 24 to create a space for c
x2 + 16x = -15
n = ½b
= ½(16) = 8 8
+64
c = n2
= 82 = 64 64
x2 + 16x + 64
= 49
= 49
Add c to both sides of equation
(x + n )2

92. Completing The Square √ x2 + 16x + 24 = 9 x2 + 16x + = -1564
+64
x2 + 16x + 64
= 49
= 49 √
(x + 8 )2 = 49
Take square root
on both sides of equation
x = 7
+ and - square roots
- 8
- 8
Solve for x
x = = -1
x = = -15

93. The Quadratic Formula and Discriminant

94. Quadratic Formula ax2 + bx + c = 0 x2 + 16x + 15 = 0
Standard form
x2 + 16x + 15 = 0
a=1, b = 16, c = 15

95. Quadratic Formula x2 + 16x + 15 = 0 x = -16 + 14 = -1 2
a=1, b = 16, c = 15
x = = -1 2
x = -16 – 14 = -15 2

96. Quadratic Formula x2 + 8x + 16 = 0 x = -8 = -4 2 a=1, b = 8, c = 16
discriminant
x = -8 = -4 2
When b2-4ac = 0
there is only 1 solution

97. Quadratic Formula x2 + 6x + 10 = 0 x = -6+2i = -3 + i 2
a=1, b = 6, c = 10
discriminant
When b2-4ac < 0
2 imaginary solutions
x = -6+2i = -3 + i 2
x = i = -3 - i 2

98. Graphing Quadratic Equations

99. Graphing Quadratic Equations
Determine Coordinates of the Vertex
Vertex is the point where the graph turns around.
It represents either the Minimum or Maximum
Plot two points on the same side of the vertex.
Copy those points to the other side of vertex
The graph is a “u”-shaped parabola, so is symmetric
Draw a smooth curve to match the plotted points

100. Graphing Quadratic Equations
Determine Coordinates of the Vertex
Vertex is the point where the graph turns around.
It represents either the Minimum or Maximum
In Vertex Form: y = a(x –h)2 +k Vertex=(h, k)
y = (x – 4) Vertex=(4, 7)
y = (x + 5) Vertex=(-5, -9)
y = -2(x + 1) Vertex=(-1, 6)
Note: the value of a doesn’t effect the vertex.

101. Graphing Quadratic Equations
Determine Coordinates of the Vertex
Vertex is the point where the graph turns around.
So, the vertex occurs where the amount being squared equals zero
since squaring any other amount produces a positive value.
y = (2x – 5)2 +7
2x – 5 = 0
2x = 5
y = (2(2.5) – 5)2 +7
x = 2.5
y = (5 – 5)2 +7
Vertex=(2.5, 7)
y = 0 +7

102. Graphing from Vertex Form
Determine Coordinates of the Vertex
In Vertex Form: y = a(x –h)2 +k Vertex=(h, k)
y = (x – 4) Vertex=(4, 5)
y = (6 – 4)2 +5
y = (5 – 4)2 +5
= (2)2 +5 = 9
= (1)2 +5 = 6 x y 4 5 5 6 6 9
Plot two points on the same side of the vertex.

103. Graphing from Vertex Form
Determine Coordinates of the Vertex
In Vertex Form: y = a(x –h)2 +k Vertex=(h, k)
y = (x – 4)2 +5 x y 4 5 5 6 6 9
Copy points to the other side of the vertex.

104. Graphing from Vertex Form
Determine Coordinates of the Vertex
In Vertex Form: y = a(x –h)2 +k Vertex=(h, k)
y = (x – 4)2 +5 x y 4 5 5 6 6 9
Draw a smooth curve to match the plotted points .

105. Graphing from Vertex Form
y = (x – 4)2 +5
Vertex=(4, 5)
The graph of this Quadratic Equation is a translation of the equation y = x2.
The vertex (4,5) and every other point
have been shifted 4 units to the right
And 5 units up.
y = (x – 4)2 +5
y = x2

106. Graphing from Vertex Form
In Vertex Form: y = a(x –h)2 +k
If 0 < a < 1, graph has a vertical shrink
If a > 1, graph has a vertical stretch
y=x2
More narrow
than y=x2
Wider than
y=x2

107. Graphing from Vertex Form
y= -x2
If a < 0, graph is
Reflected over the
x-axis.
y=x2
y= -x2

108. Graphing from Standard Form
Standard Form: y = ax2 +bx + c
Quadratic Formula -b 2a
1) Axis of Symmetry: x=
This is the x value of the Vertex

109. Graphing from Standard Form
Standard Form: y = ax2 +bx + c
y = x2 +6x +4
a = 1, b = 6, c = 4
Determine Coordinates of the Vertex
This is the
x value of
the Vertex -b 2a
- 6
2(1)
1) Axis of Symmetry: x= x=
= -3
2) Substitute and calculate y
y = x2 +6x +4
y = (-3)2 +6(-3) +4
y = ­
y = -5
Vertex=(-3, -5)

110. Graphing from Standard Form
y = x2 +6x +4
Vertex=(-3, -5)
y = (-2)2 +6(-2) +4
= 4 ­ 12 +4
y = (0)2 +6(0) +4
= 4 x y -3 -5 -2 -4 4
Plot two points on the same side of the vertex.

111. Graphing from Standard Form
y = x2 +6x +4 x y -3 -5 -2 -4 4
Copy points to the other side of the vertex.

112. Graphing from Standard Form
y = x2 +6x +4 x y -3 -5 -2 -4 4
Draw a smooth curve to match the plotted points

113. Exponential Functions and Sequences

114. Exponential Functions
y = bx
If the base, 0 < b < 1
Exponential Decay
If the base, b > 1
Exponential Growth
y = 2x
y = ½x

115. Exponential Functions
y = bx
If the base, 0 < b < 1
Exponential Decay
If the base, b > 1
Exponential Growth
y = 2x
y = ½x
In either Growth or Decay there is an Asymptote
where the curve approaches a horizontal line.

116. Exponential Functions
y = bx +k
If the base, 0 < b < 1
Exponential Decay
If the base, b > 1
Exponential Growth
y = 2x+3
y = 2x
y = ½x
Asymptote: y = 3
Asymptote: y = -2
y = ½x -2
The Asymptote is always the x-axis unless the graph is
Translated up or down.

117. Exponential Functions
y = bx
Any number raised to the zero power equals one.
If the base, b > 1
Exponential Growth
So a key point on the graph is where the base is raised to the zero power.
y = 2x
(0,1)
y = 3x

118. Exponential Functions
y = b(x-h)
A key point on the graph is where the base is raised to the zero power.
If the base, b > 1
Exponential Growth
With a horizontal shift
this isn’t the y-intercept.
y = 2x
y = 2(x-2)
Shift 2 units to right.
(2,1)
(0,1)

119. Exponential Functions
y = bx
Any number raised to the first power equals that number.
If the base, b > 1
Exponential Growth
So that is another key point on the graph.
y = 2x x 2x 1 2 4 x 3x 1 3 2 9
(1,3)
(1,2)
y = 3x

120. Exponential Functions
y = bx
If the base, 0 < b < 1
Exponential Decay
b0 = 1 is still a key point
Other key points may use
negative exponents
y = ½x x ½x 1 -1 2 -2 4

121. Scatter Plots, Trends and Statistics

122. Scatter Plots, Trends & Correlation
The scatter plot shows the number of CDs
sold from 1999 to 2005.
There were about
650 million sold.
If the trend continued,
how many CDs
were sold in 2006?
Draw a trend line
(line of best fit)
So about ½ the points
Are on either side.

123. Scatter Plots, Trends & Correlation
The table shows
Predicted cost for
a middle income
family to raise a
child.
Draw a scatter plot
And describe the
relationship.
There is a strong
positive correlation

124. Line of Best Fit Write equation of the line. Slope of the line = -1
Draw a scatter plot of the data and a line of best fit
Write equation of the line.
Slope of the line = -1
y-intercept = 1
Equation of the line
y = -x + 1

125. Education
The table gives hours studying and grade. Draw the scatter plot and line of best fit.
Hours
Grade 3 84 2 77 5 92 1 70 60 4 90 75 95 90 85 80 75 70 65 60

126. Education Write the Equation of the Line of Best Fit. Slope = 95 90 85 80 75 70 65 60
Run = 4
Intercept = 60
Rise = 30 15 2
y = x + 60

127. Education No. You’ll never get more than 100%
Predict the grade for a Student who studied 6 hrs. 15 2
y = x + 60 95 90 85 80 75 70 65 60 15 2
y = (6) + 60
Could this line
go on forever?
y =
y = 105
No. You’ll never get
more than 100%
Or maybe 100?

128. Baseball Slope = 18.75 – 17.60 = 1.15 y = 1.15x + b
Use 2000 as starting point.
Then 2001 would be x=1
Slope = – = 1.15
y = 1.15x + b
17.60 = 1.15(1) + b
16.45 = b
y = 1.15x
Where x is the number
of years since 2000

129. Baseball y = 1.15x + 16.45 Use Equation to tell the
Price of a ticket in 2009.
Where x is the number
of years since 2000
y = 1.15(9)
y =
y = 26.80
This is extrapolation.
It uses a point outside
The data.

130. Statistics
Mean: Average = Sum/number of instances Median: Middle Number (when sorted) Mode: Most Common Number Range: Highest - Lowest