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Algebra 2
Fall Semester Exam Review
Test Format
• Final Exam is all calculator• 35 Questions• All Multiple Choice
Key Concepts on Test• Graphing Parent Functions and their
characteristics• Domain/Range/Functions• Interval Notation and Inequality Notation• Transformations
–Order of transformations–Graphing using transformations
• Graphing Absolute Value Functions • Solving Absolute Value Equations and
Inequalities
Key Concepts on Test
• Horizontal and Vertical Parabolas– Graphing them given an equation– Finding Key Info (Vertex, Focus, Directrix)– Writing Equations given 2 pieces of info– Complete the Square to convert formats
Key Concepts on Test
• Linear/Quadratic Regressions (STAT)• Data Analysis (Zoom 9)• Quadratic Equations
– Simplify positive and negative radicals– Simplify Complex Numbers– Factoring Methods– Square Roots Method– Complete the Square– Quadratic Formula
Calculator
• Can be used to solve 60% of your test• Know the following:
– How to graph– 2nd trace (zeros and maximums)– Linear & quadratic regressions– Plug in numbers (watch out for negatives)
Testing Hints• If you can graph it in the calculator, then do so• Double graphing to compare• Be careful of negatives when solving equations• Questions with graphs! Look carefully at each
graph so you select the one you really want• Plug in solutions to calculator to check
In Class Review: Today
• Relations/Functions• Domain/Range• Transformations• Calculator Regression/Data Analysis• Quadratic Word Problems
RelationsOrdered Pairs
(2, 3)
(-3, 1)
(1, -2)
X Y
2 3
-3 1
1 -2
Tables
GraphsMapping
2
-3
1
3
1
-2
X Y
Example :
• Given the following ordered pairs, find the domain and range. Is it a function
• {(4,5), (-2,3), (5,6)}
• Domain is {-2, 4, 5}• Range is {3, 5, 6}• YES, no duplicated x-values
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
Domain( , )
Range[2, )
Domain( , )
Range[0, )
( )y af bx c d
xR
VS or VC
R y
(+) Left
(-) Right
(+) Up
(-) Down
HS or HC
Example 1
( ) ( 5) 3f x g x
5 , 3Right Up
Example 2
( ) ( 2) 1f x g x
2 , , 1yLeft R Down
Example 3
( ) 2 | 3 | 7f x x
3 , 2, , 7xR VS R U
Data AnalysisHeight(meters)
15 30 45 60 75 90 105
DistanceKm
13.833 19.562 23.959 27.665 30.931 33.883 36.598
Zoom 9
What Parent Function??
STAT Plotter “ON”
Weeks Experience
4 7 8 1 6 3 5 2 9 6 7
Speed (wpm)
33
45
49
20
40
30
38 22 52 44 42
1 2 63 4 5 7 8 9 10
20
15
10
5
35
2530
40
45
x-axis
y-axis
0
4.064 16.300y x
.986r
Application Problems
• Need to change the viewing WINDOW
• x-min, x-max• y-min, y-max
2.0035 2 5y x x Put in Calculator
Window
Max Height
Max Distance
(Vertex Pt)
(Zero)
290.7
573.9
Inverse Concept
• The main concept of an inverse is the x and y coordinates have switched places
( , )x y ( , )y x
Inverses
• The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation.
• Example:• { (1, 2), (3, -1), (5, 4)} is a relation• { (2, 1), (-1, 3), (4, 5) is the inverse.
Graphing an Inverse
• Pick some Critical Points off Original Graph (x, y)
• SWITCH the x and y values• Re-plot the newly formed
ordered pairs.
GRAPH the inverse
Inverse Concept
• The main concept of an inverse is the x and y coordinates have switched places
( , )x y ( , )y x
28
NOTATION FOR THE INVERSE FUNCTION
f x 1 ( )
f x 1 ( )We use the notation
for the inverse function of f(x).
Find Inverse of f(x)= 3x + 2
• y = 3x + 2 (Replace f(x) with “y”)• x = 3y + 2 (Swap variables)• 3y = x - 4
1 4
3 3y x
-1Inverse is a function so replace y with f (x)
1 1 4( )
3 3f x x
Function Composition
Notation
( )( )f g x x( )( )g f x x
Absolute Value Equations
There are ALWAYS 2 cases:
- Positive case
- Negative case
So for this Ex: |x-25|=17
• Case 1 (+ case)• (x –25) = 17• x=42• Check:• |42-25|=17• |17|=17• 17=17
• Case 2 (- Case)• -(x - 25) =17• -x + 25 = 17• - x = - 8• x = 8• Check:• |8-25| =17• |-17|=17• 17=17
BIG DIFFERENCEInequalities
If you multiply or divide by a negative number then the order of the inequality must be switched.
3 9x 3 9
3 3
x
3x
Solve: |2x+4| > 12
• |2x+4| > 12• (2x + 4) > 12 or -(2x + 4) > 12• 2x > 8 or -2x - 4 > 12• x > 4 or -2x > 16
or x < -8• Solution set: x > 4 or x < -8
-8 4
Parabola• A parabola is a set of points in a plane that are all
the same distance from a fixed line called the directrix and a fixed point not on the line called the focus .
2( )y a x h k
Vertex Point: (h, k)Opens Up
Opens Down
a
a
Vertical Parabola
2( )x a y k h
Vertex Point: (h, k)
Opens Right
Opens Left
a
a
Horizontal Parabola
Key Concept
ap
4
1
Distance from Vertex to
1Focus is
4a
Distance from Vertex to
1Dirextrix is also
4a
Vocabulary
• The perpendicular WIDTH of parabola at the focus point is the LR.
LR
1LR
a
Example 1: 21( 3) 6
12y x
Opening Direction?
Vertex Point?
Distance Calculation?
Width Calculation?
Down
(3,6)
1
4a
11
412
3
1
a 1
1
12
12
Opening Direction? Down
Vertex Point? (3,6)
Distance Calculation? 3
Width Calculation? 12
Focus Point?
Directrix Line?
(3,3)
Axis of Symmetry?
9y
3x
Example 2: 21( 2) 4
8x y
Opening Direction?
Vertex Point?
Distance Calculation?
Width Calculation?
Right
( 4, 2)
1
4a
11
48
2
1
a 1
1
8
8
Opening Direction? Right
Vertex Point?( 4, 2)
Distance Calculation? 2
Width Calculation? 8
Focus Point?
Directrix Line?
( 2, 2)
Axis of Symmetry?
6x
2y
:
( 2,9)
( 2,6)
Given
Vertex
Focus
Opens Down
2( )y a x h k 2( 2) 9y a x
1 3
4 1a 12 1a
1
12a
21( 2) 9
12y x
Distance Calculation
:
(1, 3)
: 5
Given
Vertex
Directrix x
Opens Left
2( )x a y k h 2( 3) 1x a y
1 4
4 1a 16 1a
1
16a
21( 3) 1
16x y
Distance Calculation
Converting to Vertex• y = x2 - 12x + 27• y = (x2 - 12x + ____) + 27• y = (x2 - 12x + _36_) +27 - 36• y = (x - 6)2 - 9• Vertex Point (6, - 9)
Converting to Vertex23 12 18x y y 2(3 12 ) 18x y y 23( 4 ) 18x y y
23( 4 _____) 18x y y
23( 2) 6x y
412