Final Exam Study Guide- 2011-2012 Dynamics of Algebra 2

Chapter Section Example Chapter 2

What is a function?

Properties of a function Is it a function? Explain.

Finding the Slope

Slope

2 1

2 1

y ymx x

−=−

Find the slope between two points: (-2,3) and (4,-8)

Writing the Equation of a Line

Writing the equation given a word problem Write the equation of a line given a point and a slope. y mx b= + Write the equation of a line given a point and a line parallel or perpendicular.

- Parallel – same slope - Perpendicular – opposite

reciprocals (flip it and switch it)

A businessman earns a salary of $30 per week and an additional 10% in commission of his sales, c. Write an equation to model the amount of money, s, the businessman earns per week.

2; ( 2,1)m = − Point: ( 2,1)− Line parallel to 5 5y x= − + Point: ( 2,1)− Line perpendicular to

1 53

y x= − +

Write the equation of a line given a graph.

Absolute Value Equations

Find the vertex.

: ( , ):

y a x h kVertex h kSlope a

= − +

2 5 3: ____________

y xVertex= − +

Chapter 3

Solving a system of Equations

- Substitution o Solve for the lonely variable.

- Elimination o Pick a variable, get same

number, different signs. - Graphing

o Intersection of two lines is your answer.

o Word problems like the cell phone project (which is better plan and when?)

Cases:

- One Solution: Intersecting Lines - Infinitely many solutions:

Coinciding Lines - No Solution: Parallel Lines

Word Problems

4 6 122 3 6x yx y− = −

− + =

The senior classes at High School A and High School B planned separate trips to the county fair. The senior class at High School A rented and filled 11 vans and 6 buses with 510 students. High School B rented and filled 5 vans and 2 buses with 194 students. Each van and each bus carried the same number of students. Find the number of students in each van and in each bus.

Linear Inequalities

- Slope intercept form - Standard form

Get into slope-intercept form >,< : dotted ≥, ≤ : solid

23 2

y xy x< − −≥ +

Chapter 5

Graphing

Different Forms

Vertex form: ( )2y a x h k= − +

Standard form: 2y ax bx c= + +

Intercept form: ( )( )y a x p x q= − −

Roles of a, h, k :a shaper, up/down, wide/narrow :h horizontal shift :k vertical shift

What form is ( )22 1 1y x= + − in?

What form is 22 5 1y x x= + − in? What form is ( )( )2 2 4y x x= − + in?

Describe the graph using a,h, and k.

( )22 1 1y x= − + −

Graph

1. Find the vertex.

2. Graph using the a-Table

a = – 2 1a 1(-2) -2 Right one,

down 2 3a 3(-2) -6 Right one,

down 6 5a 5(-2) -10 Right one,

down 10

Graph: ( )22 1 1y x= − + −

Graph: 22 4 1y x x= − +

Find the vertex: Vertex form ( , )h k Standard form

:2bVertex xa

plug in to equation to solve for y

−=

Find the vertex:

( )22 3 1y x= + +

22 4 1y x x= − +

Converting

Vertex standard form - Open up the present - FOIL

( )21 2y x= − +

Intercept standard form - FOIL

( )( )3 1 4y x x= − − −

Graphs of Zeros

Relating a graph to zeros (x – Intercepts) Number of solutions and intercepts

- 1 real solution – 1 x-intercept - 2 real solutions – 2 x -intercepts - No real solutions – 0 x -intercepts

Estimate the real zeros of the function graphed below. Round to the nearest whole number.

Sketch a graph with one solution. Sketch a graph with zero solutions. Sketch a graph with two solutions.

Factoring

Old School/Difference of squares

Factor: 2 12 32x x+ +

2 64x −

24 25x −

Solving quadratic equations

What are the solutions/x-intercepts to the equation?

23 11 10 0x x− + =

Radicals - Simplify

Solve for the x-intercepts using radicals.

42

18 2− ⋅

22 10 90x + =

Solving quadratic

equations for x intercepts

Quadratic Formula 2 42

b b acxa

− ± −=

Find the zeros. 23 4 2 0x x+ − =

Chapter 6

Exponents Exponents - simplifying ( )234x

23 0

43x zy

⎛ ⎞⎜ ⎟⎝ ⎠

Polynomials

Adding/subtracting Add/Subtract and write in standard form: 2 2( 4 3) ( 4 )x x x x− + + − +

2 2(2 4 3) ( 4 )x x x x+ + − − +

Dividing using synthetic division

3 2(2 4 3) ( 1)x x x x+ − + ÷ −

Graphing End Behavior

Describe the end behavior: ( ) ________( ) ________f x as xf x as x

→ → +∞→ →−∞

x→ +∞ means as you move to the right x→−∞ means as you move to the left f(x) means the graph

Sketch the graph of the equation: 3 22 4 3y x x x= − + − +

Describe the end behavior: ( ) ________( ) ________f x as xf x as x

→ → +∞→ →−∞

4 24 3y x x= + + Describe the end behavior: ( ) ________( ) ________f x as xf x as x

→ → +∞→ →−∞

Vocabulary - Leading Coefficient - Degree - Constant

Name the leading coefficient, degree, and constant of the polynomial shown below:

3 22 4 3x x x+ − +

Chapter 7

Rational Exponents

Exponential Radical Notation with coefficients Radical Notation Exponential

3/ 42x

( )43 x

Find the inverse

2 3y x= −

Composition of functions 2

( ) 1( ) 1f x xg x x

= += +

Simplify ( )( )f g x

Simplify ( )( )g f x

Solving radical equations

12x x− = 3 2 1 3x+ =

Chapter 8

Logs

Evaluate 4log 64

Converting - exponential log - log exponential

Convert to logarithmic: 52 32= Convert to exponential: 3log 9 2=

Condense Write as a single logarithmic expression:

2 2log 2logx y−

Expand Expand:

2

2log x yz

Exponential Functions

Classify as Exponential Decay/Growth and identify the growth factor.

( )1 tA a r= + ( )42 33

Growth =

( )1 tA a r= − 423

3Decay ⎛ ⎞= ⎜ ⎟⎝ ⎠

Classify as Exponential Decay/Growth. 1 22

xy = ⋅

Word Problems

Exponential Growth ( )1 tA a r= +

Exponential Decay ( )1 tA a r= −

Compounded Interest n= quarterly = monthly = weekly = daily=

1ntrA P

n⎛ ⎞= +⎜ ⎟⎝ ⎠

Continuously Compounded Interest rtA Pe=