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