Gifted/Honors Algebra 2 - PC\|MACimages.pcmac.org/SiSFiles/Schools/GA/GwinnettCounty... · piano for $25 and on the saxophone for $20. a. Write an inequality that represents the possible

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Integrated Algebra 2

1.1 Problem Solving Strategies and Models

Verbal Model: talk out the problem before writing it using mathematical

symbols.

Using a Formula

1. A rectangular corral has an area of 3,500 square meters. If the length of the

corral is 75 meters, what is the width of the corral?

“Verbal Model”

“Algebraic Model”

3500 = 75 * w

2. A boat travels at a speed of 32 miles per hour. How long will it take the boat to

travel 144 miles?

Using a Pattern: Look for a pattern in the table. Then write an equation

that represents the table.

3. 4.

x 0 1 2 3

y 0 -3 -6 -9

x 0 1 2 3

y 1 8 15 22

Area

(square meters)

Length

(meters)

Width

(meters) equals times

5.

x 0 2 4 6

y 3 11 19 27

Draw a Diagram

6. Rock Pond You are designing a square rock pond surrounded by a brick sidewalk

of uniform width. The pond has a side length of 32 feet. The side length of

the outside square is 40 feet. Draw a diagram to find the width of the

sidewalk.

7. Construction You want to create an open rectangular box from a square piece

of cardboard. The cardboard is 24 inches by 24 inches and you will cut 3 inch

squares from each corner. Draw a diagram to find the length of the box.

Dependent:

infinitely

many solutions

4

2

-2

-4

-5 5


1.2 Solving Linear Systems by Graphing

Systems of Equations: a collection of equations in the same variable

Solution to a System: ordered pair (or triplet) that makes all equations true.

— where the equations INTERSECT

Classifying Systems of Equations:

Consistent – at least one solution OR Inconsistent – no solution

AND

OR

Graphically:

Intersecting lines Coinciding lines Parallel lines

Classification: consistent, independent consistent, dependent inconsistent

Solution: (exactly one solution) (infinitely many solutions) (no solution)

Solve Using Graphing:

1. Change all equations to y mx b form.

2. Graph each line separately and ACCURATELY.

3. The solution is where the lines intersect, if they intersect.

4. Check the solution algebraically.

Examples: Solve each system by graphing. Then, classify the solution.

1. 5

3 3

x y

x y

Independent:

exactly one

solution

10

9

8

7

6

5

4

3

2

1

2 4 6 8

2. 2 3

5 2

x y

x y

3.

5

2 2 10

x y

x y

4

2

-2

-4

-5 5

10

8

6

4

2

5 10

4. Resort Costs Resort A charges $70 per night, plus a one-time surcharge of $10.

Resort B charges a one-time surcharge of $30 in addition to $60 per night. After

how many nights will the total cost of the two options be the same?

(Hint: Try making a scale of 20 on the y-axis)


1.3 Solving Linear Systems Algebraically

Solve Using Substitution:

1. Solve easiest equation for x or y.

2. Plug into other equation.

3. Solve and plug in to find missing variable.

4. Check your solution.

Examples: Solve the following systems by substitution. Check your solution and

then classify each system.

1. 2 3

3 2 8

x y

x y

2.

2 3

5 2

x y

x y

3. 3 1

7 1 3

x y

y x

4. Determine if 3,1 is a solution to

5 14

4 3 10

y x

x y

Solve using elimination: adding equations together to eliminate a variable

1. Line up variables

2. Multiply one or both equations so that a variable disappears when adding

equations

3. Check your solution

Examples: Solve each system by elimination. Check your solution and then classify.

5. 7 15

3 15

x y

x y

CHECK:

6. 2 5 3

10 4 6

y x

y x

CHECK: 7.

4 3 15

5 2 10

x y

x y

CHECK:

8. 2 5 15

4 7 13

x y

x y CHECK: 9.

2 5 4

2 8

x y

y x

CHECK:

10. School T-Shirts Fairview High School sells short sleeve T-shirts that cost the school

$5 each and are sold for $8 each. Long sleeve T-shirts cost the school $7 each and

are sold for $13 each. The school spends a total of $2,450 on T-shirts and sells all of

them for $4,325. How many of each type of T-shirt are sold?


1.4 Linear Inequalities in Two Variables

Graphing Linear Inequalities:

1. Put inequality in slope intercept form.

2. Plot y-intercept and slope.

a) , dashed line

b) , solid line

3. Shade solution region: less than: below/under

greater than: above/over

Examples: Graph each linear inequality.

1. 2 y x 2. 2 3 y x 3. 2 3 3 x y

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

Check: Check: Check:

Vertical and Horizontal Lines:

Vertical: 2x Horizontal: 3y

Examples: Graph each linear inequality.

4. 1 y 5. 2 x 6. 3 12

07

y

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

Check: Check: Check:

4

2

-2

2

-2

-4

Examples: Write the inequality for each graph.

7. 8.

Examples: Tell whether the given ordered pairs are solutions of the inequality.

9. 2,4 , 1, 34 2; x y 10. 3

2,4 , 0, 32

5 15; x y

Examples: Graph each absolute value inequality.

11. 1 xy 12. 1 2 xy

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

10

8

6

4

2

5

Examples: Solve the multi-step problem

13. Perimeter The perimeter of a rectangle is to be more than 16 inches. Write an

inequality describing the possible dimensions of the rectangle. Then graph the

inequality and identify three solutions.

14. Music Lessons Your parents have budgeted $550 for you to take music lesson on the

piano for $25 and on the saxophone for $20.

a. Write an inequality that represents the possible number of piano lessons, x, and

saxophone lessons, y, you can take this summer.

b. Is it possible to take 12 piano lessons and 15 saxophone lesson this summer?

c. If you take 14 piano lessons, what is the maximum number of saxophone lessons

you can take?


1.5 Systems of Linear Inequalities

Graphing systems of linear inequalities:

1. Graph each inequality separately.

2. Solution region: where all shadings overlap

Examples: Graph each system and find the solution (feasible) region.

1. 3 4

6

y x

y x

2.

1

2 5

1

y x

y x

y

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

3.

1

2 1 2

1

y x

y x

x

4.

0

0

2 1 5

3 1

x

y

y x

y x

4

2

-2

-4

-6

-5 5

4

2

-2

-4

-6

-5 5

5. 3 2 x 6. 1 3y

4

2

-2

-4

-6

-5 5

4

2

-2

-4

-6

-5 5

Write the system of inequalities whose solution is graphed below.

7. 8.

4

2

-2


1.6 Linear Programming

Linear programming: used to find optimal solutions such as maximum revenue or

profit for a real-world situation

a) constraints – the inequalities contained in the problem

b) feasible region – the solution to the set of constraints

c) objective function – the function to be maximized or minimized

Corner-Point Principle: The maximum and minimum values of the objective function each

occur at one of the vertices of the feasible region.

Examples:

1. Find the minimum value and the maximum value of the objective function 3 2C x y

subject to the following constraints.

0

0

3 15

4 16

x

y

x y

x y

14

12

10

8

6

4

2

5 10 15

2. Crafts Toy wagons are made to sell at a craft fair. It takes 4 hours to make a

small wagon and 6 hours to make a large wagon. The owner of the craft booth will

make a profit of $12 for a small wagon and $20 for a large wagon and has no more

than 60 hours available to make wagons. The owner wants to have at least 6 small

wagons to sell.

a) Write a system of inequalities to represent the constraints.

b) Graph the feasible region.

14

12

10

8

6

4

2

5 10 15

c) How many of each size should be made to maximize profit?

3. A farmer wants to plant corn and wheat. He wants to plant no more than 120

acres of both crops. At least 30 acres of corn will be planted. He also wants to

plant at least 20 acres of wheat, but no more than 60 acres. The farmer makes

a profit of $357.53 per acre for planting corn and $159.31 per acre for planting

wheat.

a) Write a system of inequalities to represent the constraints.

b) Graph the feasible region.

120

110

100

90

80

70

60

50

40

30

20

10

20 40 60 80 100 120

c) Write an objective function for the farmer’s total profit and find his

overall maximum profit.


1.7 Solve Systems of Linear Equations in Three Variables

A solution of a system with three variables is an ordered triplet , ,x y z

whose coordinates make each equation true.

Using the Elimination Method: Rewrite the system as a linear system in two

variables and then solve and check.

Examples: Solve the system.

1.

5

2 3 2

4 8

x y z

x y z

z

2.

2 3 13

3 3 11

3

x y z

x y z

x y z

3.

2 2 2 9

5

3 2 4

x y z

x y z

x y z

4.

3 1

2 2 2 0

3 3 2

x y z

x y z

x y z

5.

3

3

3 3 9

x y z

x y z

x y z

6. Harvest Yields A farmer makes three deliveries to the feed mill during one

harvest. The harvest produced 2,885 bushels of corn, 1,335 bushels of wheat,

and 1,230 bushels of soybeans. Use the table to write a system of equations to

find the total number of bushels in each delivery.

Crop First Delivery Second Delivery Third Delivery

Corn 50% 75% 40%

Wheat 30% 10% 30%

Soybeans 20% 15% 30%

Entry/Element


1.8 Perform Basic Matrix Operations

Matrix: a rectangular array of numbers enclosed in a single set of brackets

Dimensions: the number of horizontal rows and vertical columns in a matrix

example: 2 0 7

3 4 12A

2 3

21 0a

32 12a

row column

Examples: Give the dimensions of A, B, and C, then give the entry 21a ,

32b , and 23c

1.

2 4

1 0

5 3

A

2.

1 4 7

2 5 8

3 6 9

B

3. 1 5 3

2 6 4C

Examples: Addition and Scalar Multiplication…

(You can only add or subtract matrices if they have the same dimensions.)

2 0 1 5 7 1Let and .

5 7 8 0 2 8A B

4. A B 5. A B

0 0 10 5

Let 4 1 and 0 4 .

3 5 7 3

A B

6. A B 7. A B

6

4

2

-2

-4

-5 5

A (-3,3)

B (3,2)

C (3,-2)

D (-2,-1)

0 0 10 5

Let 4 1 and 0 4 .

3 5 7 3

A B

8. 4A 9. 1

2A B

Properties of Matrix Addition: For matrices A, B, and C, each with dimensions of m n …

Commutative A B B A

Associative A B C A B C

Additive Identity The m n matrix having 0 as all of its entries is the m n

identity matrix for addition.

Additive Inverse For every m n matrix A, the matrix whose entries

are the opposite of those in A is the additive inverse of A.

Note: Two matrices are equal if they have the same dimensions AND

corresponding entries are equal

Examples: Solve for x and y.

10. 3 4 3 2

5 8 2 1 8

y

x

11.

2.5 3 5 10 2

4

x y

y x y

Example: Given the coordinates of the vertices of polygon P, graph the polygon

that is represented by each of the following matrices. Then, graph the polygons

along with the original figure. -3 3 3 -2

= 3 2 -2 -1

P

13. 1

2P 12. 2P


1.9 Matrix Multiplication

Matrix Multiplication:

- Matrices can be multiplied only if inner dimensions are the same.

- Outer dimensions become the dimensions of the product matrix.

3 31 1 1 1

Always multiply each row by each column…

1st row 1st column 1st row 2nd column

5 _____

2 1 3 6 1

4 0 2 _____ _____1

3 6 1

2 1 5 10 2

24 0 1 _____ _____

2nd row 1st column 2nd row 2nd column

2 1 5 6 1 10 2

3

4 0 1 122 0 _____

5

4 0 2 2

2 1 3 6 1 10 2

1 12 0 0 0

Finally, add to find the final entries…

2 1 3 5 6 1 10 2

4 0 1 2 12 0 2

7 8

12 200 0

inner

outer

Dimension of product matrix

Examples: Find each product.

2 3 7 1 322 3 6 0 5 0

0 5 Q 4 5 041 5 4 7 4 7

2 0 9 2 13

H G R W B

1. Find .HG 2. Find .GH

3. Find .RW 4. Find .WR

5. Find QB

6. School Supplies You and a friend are purchasing school supplies. You buy 4 binders and

1 notepad. Your friend buys 2 binders and 3 notepads. Each binder costs $3.50 and each

notepad costs $2. Write a supplies matrix and a cost per item matrix. Then use matrix

multiplication to write a total cost matrix.

-

+

-

- - -

+

-

+

-


1.10 Evaluating Determinants

Square Matrix: a matrix with the same number of rows and columns

Determinant of a Matrix: used to tell if a matrix has an inverse…

+

2 2 Matrix: det( )a b a b

A A A add

bcc c d

Examples: Find the determinant of each matrix.

1. 7 8

6 7A

2. 1 1

2 2B

3.

1 2

2 3

37

4

C 4. 6 2

4 3

3 3 Matrix:

1 2 3

4 5 7

0 3 2

C

1 2 3 1 2

4 5 7 4 5

0 3 2 0 3

C

1 2 3 1 2

4 5 7 4 5

0 3 2 0 3

C

10 0 3de 0 21 16t( ) 9 6 C C

+

-

+

-

+

-

- - -

a b c a b

C d e f d e

g h i g h

gecae hfi abfg cdh idb

Examples: Find the determinant of each matrix.

5.

2 1 3

3 2 1

0 1 2

D

6.

2 1 1

1 3 4

2 1 0

Q

Linear Systems and Matrices: matrices can help us solve linear systems

Example: Create a matrix equation (3 matrices) given the linear system.

7. 7 5 16

4 2 14

x y

x y 8.

2 5 2 2

3 3

10

x y z

x y

y z

9. Pricing Two bottles of water and 1 bottle of juice sell for $2.55. Two bottles

of juice and 1 bottle of milk sell for $2.45. One bottle of water, 1 bottle of

juice, and 1 bottle of milk sell for $2.30. Create a linear system for the prices

of the water, juice, and milk.


1.11 Inverses of a Matrices and Solving Linear Systems

Square Matrix: a matrix with the same number of rows and columns

Identity Matrix: a square matrix with ones down the main diagonal and zeros

everywhere else

10

1 0I

1

1

1

0 0

0 0

0 0

I

Any matrix multiplied by I has a product of the original matrix.

Any matrix multiplied by its inverse has a product of I.

Determinant of a Matrix: used to tell if a matrix has an inverse…

If det( ) 0,A then matrix A has an inverse.

Finding the Inverse of a Matrix: 1 1If , then .

det( )

a bA A

c d A

d b

c a

1. Find the determinant (det).

2. Switch the main diagonal (a & d) and change the sign of the other diagonal (c & b).

3. Multiply this “new” matrix by the fraction 1

det.

* Remember, you can only find the inverse of a SQUARE matrix and when det 0 *

Examples: Find the inverse of each matrix.

1. 1 2

3 5A

2. 6 8

5 7F

3. 2 1

6 3P

4.

1 2

2 3

6 5

C

main diagonal

Using Matrices to Solve Systems:

2 10

3 9

x y

x y

Write a Matrix Equation: AX B

(A) Coefficient Matrix (X) Variable Matrix (B) Constant Matrix

2 1

1 3

x

y

10

9

**To solve the matrix equation, you want to solve the variable matrix. Since we

cannot divide by matrix A, we will have to multiply both sides by the inverse of matrix A.

1A 2 1

1 3

x

y

=10

9

1A

1 3

4

X A B

,x y

Steps to Solving:

1. Create a matrix equation: coefficient matrix (A), variable matrix (X), constant

matrix (B).

2. Find the inverse of A.

3. Multiply A inverse times B ( 1X A B ).

4. Create a coordinate to show the intersection point.

Examples: Solve each system by first setting up a matrix equation.

5. 3 2 1

4 2

x y

x y 6.

3 4 3

6 8 18

x y

x y

1A AX = 1A B

I X = 1A B

X = 1A B

x

y

1A A = I the Identity Matrix

7. 2 4

13

x y

x y

8.

5 2 7

2 2 0

3 17

x y z

x y z

y z

9.

6 4 3 8

4 2 3

8

y z x

x y z

y z

10.

12 7

11 2 2

9 9 0

x y z

x y

x y

Examples: Write the matrix equation that represents each system.

11.

5 7 3 5

2 3 4 2 9

0

x y z

z y x

x y z

Examples: Write the system of equations represented by each matrix equation.

12.

2 3 5 2

0 2 1 6

7 0 9 8

x

y

z

Edge

Vertex

F ED

C

B

A


1.12 Vertex-Edge Graphs

Vertex-Edge Graph: a collection of points and line segments connecting some (possibly

empty) subset of the points.

Example 1: Transportation An airline serves four cities: Bedford, Columbia, Dunwich,

and Exton. There are flights between Bedford and Columbia, Bedford and Dunwich, and

Columbia and Exton. Draw a vertex-edge graph to represent this situation.

Example 2: Transportation An airline serves fives cities: Lowell, Montour, Newman,

Peoria, and Orlando. There are flights between Lowell and Montour, Lowell and Orlando,

Montour and Orlando, Newman and Orlando, and Newman and Peoria. Draw a vertex-edge

graph to represent this situation.

Using Matrices to Represent Vertex-Edge Graphs: zeros in a matrix represent no

connection, ones in a matrix represent a connection between the two initial elements.

Example 3: Draw a vertex-edge graph based on the following matrix.

Flight patterns at airport A

A =

B C D E

B

C

D

E

1 1 1 0

1 0 0 1

1 0 0 0

0 1 0 0

PZ

T

F

B

Example 4: Use matrix A from example 3 to findA2 . What does the matrix represent?

Example 5: Write a matrix to represent the vertex-edge graph.


2.1a Evaluating and Graphing Polynomial Functions

Definitions:

- monomial: one term (number, variable, etc.)

- constant: monomial with no variables (number)

- coefficient: number in front of a variable

- binomial: two terms

- trinomial: three terms

- polynomial: sum or difference of monomials, binomials, and trinomials

- degree of a polynomial: greatest exponent

example: 4 32x +3x -x+1 degree : 4

Classifying polynomials by degree:

Examples: Classify each polynomial by degree and by number of terms.

1. 3 52x -3x+4x 2. 3 4 3-2x +3x +2x +5 3. 2 3x +4-8x-2x

Evaluating polynomials:

4. Evaluate 3 2x -3x +4x for x =-2 5. Evaluate 4 3 212 x +5x -x for x =2

Degree Name Example

0 Constant 3

1 Linear 5x+4

2 Quadratic 2-x +11x-5

3 Cubic 3 24x -x +2x-3

4 Quartic 4 29x -3x +x-1

5 Quintic 5 4 3-2x +3x -x +6

6

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-5 5

Adding and Subtracting Polynomials: Steps…

1. Get rid of parentheses and combine like terms.

2. Write in standard form: biggest exponent smaller exponent

Examples: Find the sum or difference.

6. 2 3 3(-2x -3x +5x+4)+(-2x +7x-6) 7. 4 3 4 2(2x +4x +5x-2)+(-2x -7x +8x-10)

8. 3 2 3 2(-6x -6x +7x-1)-(3x -5x -2x+8) 9. 3 2 2(3x -12x -5x+1)-(-x +5x+8)

Graphing Polynomial Functions: Graph and describe the shape of each function.

10.

f x x

g x x

2

2

2

2

11.

f x x

g x x

3

3

4

4

1

4

2

-2

-4

-5 5


2.1b Evaluating and Graphing Polynomial Functions

Relative Minimums and Maximums:

Increasing and Decreasing Functions:

2f(x) =x -2 increasing:

decreasing:

Examples: Look at the graph of each function. State the domain and range and

approximate the local maxima or minima to the nearest tenth (if necessary). Find the

intervals over which the function is increasing and decreasing.

1. 3 2Q(x) =x +3x -x-3 2. 4 2T(x) =-2x +4x -2

4

3

2

1

-1

-2

-3

-4

-4 -2 2 4

f x = x3+3x2 -x-3

2

1

-1

-2

-3

-4

-5

-6

-4 -2 2 4

f x = -2x4+4x2 -2

2

Even, Odd, or Neither?

Test for EVEN and ODD Functions:

A function valuef is EVEN if, for eachx in the domain off , f x f x .

A function valuef is ODD if, for eachx in the domain off , f x f x .

End Behavior of Polynomial Functions:

The behavior of the graph as the X-VALUE at the “end” of the function rises or falls.

Describing the end behavior of a function can be written as (use rises or falls)…

Left End: Right End:

Examples: Describe the end behavior of each function by graphing and state whether

each function is even, odd, or neither. Use a t-chart as necessary.

3. 3f x = x -x 4. 2g x = x +1

Symmetric to y-axis

EVEN

Symmetric to origin

ODD

Symmetric to x = 2

NEITHER

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5


2.2a Translating Graphs of Polynomials

Transformation: an alteration of a function (translation, stretch/compression, reflection)

Translations: Given ( )y f x . add or subtract

Vertical: OUTSIDE

Horizontal: INSIDE (think “opposite” or “what makes inside zero”)

Vertical (up/down): ( )y f x k outside Horizontal (left/right): ( )y f x h inside

a) Add: up a) Add: left

b) Subtract: down b) Subtract: right

Examples: Graph each pair of functions and describe the transformation.

1. 2 2( ) and ( ) 3 f x x g x x 2. 332 and 2 2 h t t k t t

3. 22( ) and ( ) 1 3 f x x g x x 4.

333 3( ) and ( ) 2 1

2 2 f x x g x x

turning points?

Examples: Write an equation for the function that is described by the given

characteristics.

5. An s-shape, but translated four units to the right and two units down.

6. A u-shape, but upside down and translated five units to the left.

7. A w-shape, but translated three units up and three units to the right.

8. Geometry The volume of a cube with side length x feet is given by 3

1V x . The

volume of a cube with side length 2x feet is given by 3

2 2V x .

a. Explain how the graphs are related. b. Find the volume of each cube when 6x


2.3a Factor and Solve Polynomials

Multiplying Polynomials: Always write in standard form.

1. 2

3 4x x 2. 2x+2 x +4x+1

Examples: Factor each of the following polynomials.

GCF: 3. 3 2x -5x -6x 4. 3 22 32 128x x x

Factoring

By 5. 3 2x +4x +2x+8 6. 3 25x +6x -20x-24 Grouping:

Difference of Two Squares

- -

7. - 8. -

Factoring the Sum and Difference of Two Cubes:

3 3 2 2a +b = a+b a -ab+b

3 3 2 2a -b = a-b a +ab+b

Examples: Factor each completely.

9. 3x 27 10. 3x 125

11. 38x -343 12. 364x 216

Factor Theorem:

Given the polynomial P(x) =..., x-r is a factor if and only if P(r) =0.

Examples: Use the Factor Theorem to determine whether each binomial is a factor

of the given polynomial.

13. 3 2x+2 ; x -2x -5x+6

14. 3 2x-1 ; x -x -5x-3


2.3b Factor and Solve Polynomials

Solving Polynomials by factoring: set each factor equal to zero and solve.

(*All solutions may not be real solutions)

1. - 2. -

Examples: Factor each of the following polynomials to solve for x.

3. 4. - -

5. 6.

7. - 8. -

6

4

2

-2

-4

-6

-5 5


2.4 Solving Polynomial Inequalities

Solving Polynomial Inequalities Algebraically: Steps…

1. Set equal to zero

2. Solve using the zero product property

3. Use test values from number line to fill in solutions

Examples: Solve each polynomial algebraically.

1. 3 2x 3x 10x 2. 3 23x 10x 8x

Solving Polynomial Inequalities Graphically: Steps…

1. Graph polynomial

2. Find zeros of polynomial

3. State intervals for the inequality (less than or greater than)

Examples: Solve each inequality graphically.

3. 3 22x x 6x 0 4. - -

-5 5

30

20

10

-10

-20

-30


2.5a Dividing Polynomials

LONG DIVISION:

1. 3 23 4 12 2x x x x

2. 3 2 26 3 24 2 3 4 x x x x x

3. 3 2 23 13 15 2 3x x x x x

SYNTHETIC DIVISION:

Steps…

1. Find r value and list coefficients of polynomial.

2. Bring down first number.

3. Multiply number by r and place under next coefficient.

4. Add.

5. Repeat process until you find a remainder.

* Answer starts with a degree one less than original polynomial*

Examples: Divide by using synthetic division.

4. 22 7 3 3 x x x

5. 3 8 4x x

Use synthetic division to factor completely.

6. 3 28 9 18 3x x x x

Remainder Theorem: If a polynomial is divided by x – a, then the remainder is

the number ( )P a .

7. Use substitution and synthetic division to find 3( 3) if 10 P P x x x .

a. substitution: b. synthetic division:


2.5b Dividing Polynomials

Divide using long division.

1. 4 3 22 2 1 1x x x x x

2. 3 2 23 3 1x x x x

Divide using synthetic division.

3. 4 24 5 5 2x x x x

Factor the polynomial completely given that 3x is a factor.

4. 3 22 5 6x x x

A polynomial f and one zero of f are given. Find all the zeros of f.

5. 3 23 4; 2f x x x

6. 3 23 8 24; 3 f x x x x


2.6a Finding Rational Roots

Finding ALL zeros: Use any method to find the roots/zeros of the polynomial.

Examples: Find all zeros of each polynomial.

1. 22 12 14f x x x 2. 4 2( ) 4 5 Q x x x

3. 3 2( ) 3 12 36 P x x x x

What happens when we cannot solve by factoring?

Rational Root Theorem: used to find all possible roots of a polynomial.

p is a factor of the constant term

q is a factor of the leading coefficient

p

q is a possible root

Finding all zeros/roots of a polynomial:

1. Use theorem to list all possible roots.

2. Use a graph to find all test values.

3. Check all test values using synthetic division.

Remember…some of these solutions may be imaginary solutions.

15

10

5

-5 5

2

-2

2

-2

Examples: Find all possible and actual rational roots.

4. 3 28 10 11 2 0x x x

5. 3 24 13 11 2 0x x x

6. 3 23 10 10 4g x x x x

2

-2

-4

5

15

10

5

-5

-10

5


2.6b Finding Rational Roots

Examples: List the possible roots and then find all the real zeros.

1. 3 2( ) 7 14 8f x x x x

2. 3 2( ) 3 17 18 8f x x x x

2

-2

Examples: List the possible roots and then find ALL of the zeros (real and imaginary).

3. 3 23 10 10 4g x x x x

Use any method to find ALL the zeros of the function.

4. 4 2( ) 2 13 15g x x x

60

50

40

30

20

10

-10

-2 2


2.7a Fundamental Theorem of Algebra

Fundamental Theorem of Algebra:

Every polynomial of degree 1n has at least one complex zero.

Corollary: Every polynomial function of degree 1n has exactly n complex zeros,

counting multiplicities.

In our own words: The degree of a polynomial function tells us how many factors there

are…including factors that are listed more than once. We need this information to write

polynomial equations below.

Writing Polynomial Functions:

1. Write binomials with zeros: (x – r).

2. Write polynomial in factored form.

3. Write in standard form if indicated.

Examples: Write each polynomial in factored form.

1. P(x) is a cubic polynomial that has zeros of -2, 1, 2.

What would you do to put this in standard form?

*the multiplicity determines how many times

a particular zero appears…. what’s the

difference between even and odd multiplicities?

2. P(x) is a degree 5 polynomial with zeros at 1 (multiplicity of 3), 2 (multiplicity of 2).

20

10

-10

-20

-30

-40

2

-2

Complex Conjugate Root Theorem:

If a bi is a root of ( ) 0P x , then a bi is also a root.

3. P is of degree 4 and has zeros at 2, 5, 3i

What would you do to put this in standard form?

4. P is of degree 3 and has zeros at 1, 2 i

Examples: Find all the zeros of the polynomial function.

5. 4 3 22 1h x x x x x

6. 5 4 3 22 3 20 30 18 27q x x x x x x


2.8 Analyzing Graphs of Polynomials

Example 1: Use x- and y-intercepts to help graph 2 2

( ) 0.2 3 1f x x x .

Example 2: Use a graphing calculator to graph 4 3 2( ) 2 3 4 4g x x x x x . Identify

the x- and y-intercepts and where the local maximums and minimums occur.

6

4

2

-2

5

2

-2

-4

-6

-8

Example 3: Graph the function 31( ) 6 8

2P x x x . Identify the x- and y-intercepts and

the points where the local maximums and minimums occur.

Example 4: Graph the function 2 22( ) 1 6

3T x x x x . Identify the x- and y-

intercepts and the points where the local maximums and minimums occur.

5

8

6

4

2

-2

-4

-6

-8

-10

-12

-14

-16

-18

-5 5

10

8

6

4

2

-2

-4

-6

-8

-10


3.1a Evaluate nth Roots and Use Rational Exponents

Index Root Symbol

2 square

3 cube 3

4 fourth 4

5 fifth 5

n nth n

Example 1: Using Radical Notation for nth Roots

a. 2

3 b. 3

3 6

c. 5

5 4 d. 6

6 7

http://www.google.com/imgres?q=radical+expressions+pictures&hl=en&sa=X&biw=1366&bih=540&tbm=isch&tbnid=tJ-9zE9mdC8zGM:&imgrefurl=http://www.jolenemorris.com/mathematics/lectures/MTH209_Wk5_lecture.htm&docid=J70xVtj0boGhnM&w=500&h=256&ei=wXUsTtS-JpPqgQekqaGsCw&zoom=1

Example 2: Evaluate radical expressions.

a. 5

3 8 b. 2

4 625

c. 32 216 3 d. 2

310 1000 5

3 512e.

4

23f. 343

Example 3: Rewrite the expression using radical notation.

a. 134 b.

153

c. 176 d.

2512

Example 4: Rewrite the expression using rational exponents.

a. 5 12 b. 6 16

c. 9 20 d. 4

3 18


3.1b Evaluate nth Roots and Solve Equations Using nth Roots

Example 1: Evaluate the expression using a calculator. Be sure to round your answer to

the nearest hundredth.

a. 6 185 b. 1

4245

c. 2

5342 d. 3 96

Example 2: Evaluate the expression without using a calculator.

a. 2

3125 b. 3

216

c. 3

2100 d. 4

327

Example 3: Solve the equations by using nth roots. Give the exact answer and an

approximated answer to the nearest hundredth when appropriate.

a. 62 1458x b. 5

2 35x

c. 3 9 31x d. 4

8 2x

Example 4: Use nth roots in problem solving.

Beach Ball The volume of a beach ball can be approximated using the model 34

3V r

where r is the radius of the ball in inches. Estimate the radius of the ball if the volume is

382 cubic inches.


3.2a Apply Properties of Rational Exponents

Recall the Properties of Exponents from previous math courses and apply these to rational

exponents.

Example 1: Simplify the expression involving rational exponents.

a. 41

3 34 4 b.

14

34

5

5 c.

14 358

d. 136 6 e.

1364

54

f. 2

6 3 38p q

Product and Quotient Properties for nth Root Radicals

Example 2: Simplify each radical expression by using the Properties of nth Roots.

a. 2 45 b. 3 4 12

Product: m n m na a a Quotient:

mm n

n

aa

a

Power of a Power: n

m mna a Power of a Product: n n nab a b

Power of a Quotient:

n n

n

a ab b

Definitions: 0 1

1 and n

na a

a

Product: n n nab a b Quotient: n

nn

a ab b

c. 3

3

108

4 d.

3 37 49

Example 3: Simplify radical expressions involving variables.

a. 4 372a b c b. 3 3 6 281w u v

c. 4

48

81

16

m

n d.

126 4

2 2

24

12

x z

x z


3.2B Apply Properties of Rational Expressions

Example 1: Simplify each sum or difference.

a. 6 12 7 75 b. 12 18 15 4 2

c. 4 27 10 48 d. 5 18 6 50

Example 2: Simplify each product.

a. 3 5 2 4 2 2 b. 4 3 2 3 5

c. 4 2 3 4 2 3 d. 2

3 5

Rationalizing: The process of rewriting a quotient without a radical in the denominator.

Example 3: Write each expression with a rational denominator.

a. 2

10 b.

7 3

6

c. 3

4

3

d.

4

3 7

4

2

-2

5

8

6

4

2

5 10


3.3A Graph Square Root and Cube Root Functions

Square Root Function:

( )f x x

Domain: 0x

Range: 0y

Square Root Functions can be transformed based on the following information below.

Note: The domain and range will always be limited.

Example 1: State the transformations. Then sketch the graph and identify the domain

and the range of the function.

a. 2 4 3 y x

b. 1

2 12

y x

c. 2 1y x

Example 2: Write an equation for the function described below.

a. A square root function with a horizontal translation of 4 units right, a vertical

translation of 5 units down, and a vertical stretch by a factor of 2.

b. A square root function with a reflection across the x-axis, a horizontal translation

of 8 units left, a vertical translation of 3 units down, and a vertical compression by

a factor of ½ .

-2

5

2

-2

5 10

-10 -5 5 10

8

6

4

2


3.3b Notes Graph Square Root and Cube Root Functions

Example 1: Identify the transformations of the cube root functions. Then sketch the

graph and identify the domain and the range of the function.

a. 3 2y x

-10 -5 5 10

8

6

4

2

-10 -5 5 10

8

6

4

2

b. 32 1 2y x

c. 3 2 1y x

d. 311

2y x

2

-2

-4

-6

-5 5

2

-2

-5 5

2

-2

5


3.4 Solving Radical Roots

SQUARE ROOT GRAPHS

Four basic steps to solve radical equations or equations raised to a rational exponent

1. Isolate the radical or rational exponent

2. “Undo” the radical

3. Solve for what is left over

4. Check your solution

Walk-through Examples:

ONE RADICAL/EXPONENT

Given 32 5 11 6x

3 5 11 3x Divide by two to isolate the radical (Step 1)

33

35 11 3x Cube each side to eliminate the cube root (Step 2)

5 11 27x Solve for what is left over (Step 3) 11 22x 2x

Given 3

2( 1) 2 6x

3

2( 1) 8x Add two to both sides to get the rational exponent by itself (Step 1)

3

2

2233( 1) 8x

Raise each side to the reciprocal to get rid of the rational exponent

(Step 2)

1

231 (8 )x Solve for what is left over (Step 3)

1 4x

5x

TWO RADICALS

Given 1 3 1x x

1 3 1x x Get one radical on either side (Step 1)

2 2

1 3 1x x Undo the radical *Careful on the right side* (Step 2)

1 3 2 3 1x x x This creates just one radical!

2 2 3x x Isolate the radical

3x x

2

2 3x x Undo the radical

2 3x x Old stuff… set equal to zero, factor and solve

2 3 0x x

( 3) 0x x

0, 3x x Substitute and check

3x

Examples: Solve for x. Be sure to check your solution.

a. 3

42 16x b. 1 2 1 x x

c. 4 5 3x d. 32 3 9 15x

e. 4 4x x f. 1

34 2 6 x


4.1a Exponential Growth Functions

Exponential Function

The function xy ab is an exponential function.

a: initial amount b: base (b > 0, b 1)

Example 1: Evaluating exponential expressions

Evaluate each expression when 5x and 4y

a. 22 1x b. 1

100 3y

Example 2: Bacterial Growth

Predict the population of bacteria for the given situation and time period.

The initial population of bacteria in a lab test is 200. Find the bacteria population for the given

time period.

a. doubles every hour after 4 hours b. triples every hour after 4 hours

c. doubles every 30 minutes after 6 hours

Growth Factor

Growth is: 100% %r

OR

1 r (r is a decimal)

(where 1 r b in the equation xy ab )

Example 3: Find the growth factor for each rate of exponential growth.

a. 9% growth b. 0.75% growth

Exponential Growth ( )xY a b (where b > 1)

Population Growth 1t

Y P r t: time (in years)

Y: new population r: rate (as a decimal)

P: initial population (1 + r or b : growth factor)

Example 4: The population of the United States was 248,718,301 in 1990 and was projected to

grow at a rate of about 8% per decade.

Predict the population, to the nearest hundred thousand, for the given years.

a. year: 2020 b. year: 2025

Compound Interest

t

y = P 1+r

n

n

Example 5: Find the final amount for each investment with the given information.

a. $500 initial amount at 6% interest compounded annually for 20 years.

b. $1800 initial amount at 8.5% interest compounded quarterly for 15 years.

c. $10,000 initial amount at 4.75% interest compounded daily for 30 years.

Y: Total

investme

nt

(amount

over

time)

n: # of times

compounded per year

t: time in years P: Principal

(initial

amount)

r: annual interest rate

(as a decimal)

8

6

4

2

-2

-5 5


4.1b Graphing Exponential Growth Functions

Two types of Exponential Growth Functions:

Increasing Function Decreasing Function

Key pieces to recognize and help graph:

1. “Starting Point” (without transformations): (0,1)

2. Asymptote (without transformations): y = 0

3. Domain: All Real Numbers; Range (without transformations): y > 0

4. End Behavior: Left falls to zero; Right rises to infinity

5. Then identify and implement transformations to adjust points (use a t-chart as

necessary), domain, range, etc.

6. Don’t forget to include at least 3 or 4 points!

With each of the following functions:

a) Graph each function (including asymptote) b) Find the y-intercept c) Find the equation for

the asymptote d) Describe the end behavior (write “rises to __” or “falls to __”)

e) State the Domain and Range

1. xy 2

y-intercept: _______ Asymptote: ________

Left:_____________ Right: ____________

Domain: ___________ Range: ____________

12

10

8

6

4

2

-2

-5 5 10

2. xf(x) 4 2

3. x 1

f(x) 3

4.

x3

y 32

5. x 1f(x) 5 3

6

4

2

-2

-5 5

2

-2

-4

-6

-8

-5

10

8

6

4

2

-2


Left:_____________ Right: ____________

Domain: ___________ Range: ____________


Left:_____________ Right: ____________

Domain: ___________ Range: ____________


Left:_____________ Right: ____________

Domain: ___________ Range: ____________


Left:_____________ Right: ____________

Domain: ___________ Range: ____________

14

12

10

8

6

4

2

-5 5

4

2

-2

-4

-6

-8

-10

-12

-14

-16

-18

-20

-5

6. (x 2)f(x) 3 2 3

7. x 2

f(x) 2 3 1

(watch for different scales on the x- and y-axis)


Left:_____________ Right: ____________

Domain: ___________ Range: ____________


Left:_____________ Right: ____________

Domain: ___________ Range: ____________


4.2 Exponential Decay Functions

Example 1: Evaluating exponential expressions

Evaluate each expression when 2x and 4y .

a.

31

2

x

b. 9 3y

Decay Factor

Decay is: 100% %r

OR

1 r (r is a decimal), (where 1 r b )*

*remember exponential function equation ( )xy a b from 4.1a notes.

Example 2: Find the decay factor for each rate of exponential decay.

a. 8.2% decay b. 0.05% decay

Exponential Decay x

y a b (where 0 < b < 1)

Decreased Value 1t

Y P r t: time

Y: current value r: rate (as a decimal)

P: initial value 1 - r or b : decay factor

Example 3: Decreased Value

Suppose that you bought a car 6 years ago for $15,000. If the car’s value decreases at a rate of

25% per year,

a. determine the value of the car today b. predict its value 3 years from now.

2

-2

-4

-6

-8

-5 5

Example 4: Exponential Decay Graphs

Sketch the graph of each exponential decay function using a table of values as necessary.

Determine the domain, range, y-intercept, asymptote, and end behavior.

a. 1

2

x

y

b.

11

2

x

y

c. 1

9 23

x

y

Example 5: Identify Functions

Determine if the functions are linear, quadratic, or exponential

a. 10 3y x b. 24 6 12y x x c. 5 3x

y

d. 77y x x e. 24 xy f. 23 4y x x x

8

6

4

2

-2

-5 5

12

10

8

6

4

2

-5 5

Domain: ______ Range: ______

y-int: ______ asymptote: ____

Left: ________ Right: _________

Domain: ______ Range: ______


Left: ________ Right: _________

Domain: ______ Range: ______


Left: ________ Right: _________


4.3a Use Functions Involving e

The letter e is called the Natural Base

The value of e is approximated by the decimal: 2.718281828e

A function of the form rty ae is called a natural base exponential function.

Example 1: Evaluate the natural base expressions to the nearest thousandth.

a. 8e b. 5e c. 2e d. 1

2e

Recall the properties of exponents below:

Example 2: Simplify expressions involving the natural base e.

a. 8e e b. 5 34 7e e c. 9

4

21

7

e

e

d. 10

4

15

45

e

e e.

532 xe f.

23

7

8

2

e

e

Product: m n m na a a Quotient:

mm n

n

aa

a

Power of a Power: n

m mna a Power of a Product: n n nab a b

Power of a Quotient:

n n

n

a a

b b

Remember… 0 1a and 1 1

aa

Example 3: Sketch the graph of the Natural Base Functions (including their asymptotes).

Determine if the function is a growth or decay (and whether it’s increasing or

decreasing). Be sure to fill in the box below each graph.

a. 2 3xy e b. 23 xy e

c. 4xy e d. 2 1xy e

8

6

4

2

-2

-4

-5 5

2

-2

-4

-6

-8

-5

12

10

8

6

4

2

-2

5

2

-2

-4

-6

-8

-10

-5

Growth/Decay _________ Inc/Dec __________


Left:_____________ Right: ____________

Domain: ___________ Range: ____________



Left:_____________ Right: ____________

Domain: ___________ Range: ____________



Left:_____________ Right: ____________

Domain: ___________ Range: ____________



Left:_____________ Right: ____________

Domain: ___________ Range: ____________


4.3b Use Functions Involving e

Recall from the 4.3a notes the following information:

The letter e is called the Natural Base

The value of e is approximated by the decimal: 2.718281828e

A function of the form rty ae is called a natural base exponential function.

Continuously Compounded Interest

t: time in years

rty Pe

Y: Total r: annual interest rate (decimal)

P: Principal

Example 1: Solve each continuously compounded interest problem

a. You deposit $500 in an account that pays 3% annual interest compounded continuously. Determine the amount in the account after 2 years.

b. You deposit $12,500 in an account that pays 5.6% annual interest compounded continuously. Determine the amount in the account after 30 years.

c. You take out a loan to buy a house for $200,000 in an account that charges 4% annual

interest that is compounded continuously. Determine how much interest you paid on the

initial $200,000 loan after 30 years.

Example 2: Solve a Multi-Step Problem

A population of bacteria can be modeled by the function 0.2270 tP e where t is the time

(in hours). Graph the model and use the graph to estimate the population after 4 hours.

Use a graphing calculator as an aid.

Example 3: Determine if the functions below represent exponential growth or exponential

decay.

a. y 3( ) 4 xf x e b.

41

( )x

g xe

c. ( ) xh x e

d. 4( ) 5 3xf x e e. 1

2( )x

g x e

500

450

400

350

300

250

200

150

100

50

-50

5 10


4.4a Logarithmic Functions

Logarithmic Functions are the inverse of Exponential Functions.

Logarithms are represented by log and its equation is written as logby x .

Natural logarithmic equations have a base of e and are written as log lney x x .

Logarithmic graphs are “mirror images” of each other when reflected about the line y x .

The domain and range roles are reversed.

The graph to the left displays the exponential function

10xy and the logarithmic function 10

logy x . What is

the domain and range for each of these functions?

Logarithms are used to find unknown exponents in exponential

models.

Logarithmic functions define many measurement scales in the

sciences, including the pH, decibel, and Richter scales.

Converting from Exponential form to Logarithmic Form… 25

5 25 log 25 2

Example 1: Write each equation in logarithmic form.

a. 42 16 b. 43 81 c. 04 1 d. 19 9

e.

21

255

f.

31

273

g. 3 0.05e

Example 2: Write each equation in exponential form.

a. 3log 81 4 b. 5

log 1 0 c. 27log 27 1 d.

2

1log 5

32

e. 1

9

log 81 2 f. ln4 1.39 g. log 0.135 2e

8

6

4

2

5

One-to-One Property of Exponents

If yxb b , then x y

Example 3: Evaluate the following logarithms by converting into exponential form.

a. 3log 81 x b. 4

log 64 x

c. 4

1log

4x d.

5

1log

5x

e. 1

6

log 36 x f. 1

2

log 16x

g. 27log 3x h. 81

log 3 x


4.4b Logarithmic Functions

Example 1: Evaluate each logarithmic expression by using a calculator. Round answers to

the nearest thousandth.

a. 10

log 110 b. log55 c. ln2 d. ln 5

e. ln0 f. log0 g. log( 1) h. ln3

Exponential-Logarithmic Inverse Properties

For 0b and 1b :

log xb b x and logb xb x for 0x

Example 2: Simplify using inverse properties.

a. 7

log 49x b. 23

log 81 x c. 5log 125 x d. 8log 18

x

Guidelines for Finding the Inverse of Exponential and Logarithmic Functions

1. Switch x and y (and isolate exp/log as necessary).

2. Convert from exponential form into logarithmic or vice versa.

3. Solve for y.

Example 3: Find the inverse of each function.

a. 8xy b. 5

logy x

c. ln 3y x d. 4xy e

Example 4: Sketch the graph of each logarithmic function. Determine the domain and

range, end behavior, and the asymptote.

a. 3

logy x b. 2log 1y x

4

2

-2

-4

5 10

6

4

2

-2

-4

5 10

c. 3

log 1y x d. 2log 3 1y x

6

4

2

-2

-4

5 10

4

2

-2

-4

-6

-5 5 10

Domain: __________ Range: ________

Left:_____________ Right: ________

Asymptote: ___________

Domain: __________ Range: ________

Left:_____________ Right: ________


Domain: __________ Range: ________

Left:_____________ Right: ________


Domain: __________ Range: ________

Left:_____________ Right: ________



4.5 Interpret Graphs of Exponential and Logarithmic Functions

Average Rate of Change

Finds the slope of a line that passes through a curve through two given points

2 1

2 1

y yr

x x

Example 1: Graph and analyze each exponential function.

a. 2xy b. 1

32

x

y

c. 32 xy e d. 4xy e

8

6

4

2

-2

-5 5

8

6

4

2

-2

-4

-5 5

2

-2

-4

-6

-8

-5

6

4

2

-2

-4

-6

-5 5

Domain: _____, Range: ______, y-intercept: ____

Asymptote: _____, Inc/Dec Interval: __________

Average Rate of Change: from (0, 1) to (3, 8)_____

Domain: _____, Range: ______, y-intercept: _____

Asymptote: _____, Inc/Dec Interval: __________

Average Rate of Change: from (-3, 5) to (0, -2)______

Domain: _____, Range: ______, y-intercept: _____

Asymptote: _____, Inc/Dec Interval: ____________

Average Rate of Change: from (-3, -2) to (-2, -5.4):_____

Domain: _____, Range: ______, y-intercept: _______

Asymptote: _____, Inc/Dec Interval: ______________

Average Rate of Change: from (0, -3) to (-2, 3.4): _____

Example 2: Graph and analyze each logarithmic function.

a. 2log 1y x b.

2log 3y x

c. lny x d. ln 2 3y x

4

2

-2

-4

5 10

6

4

2

-2

-5 5 10

4

2

-2

-4

5 10

6

4

2

-2

5 10



Average Rate of Change: from (2, 0) to (5, 2)________



Average Rate of Change: from (2, 2) to (8, 0)________



Average Rate of Change: from (1, 0) to (10, 2.3)________



Average Rate of Change: from (-1, 3) to (6, 5.1)________


4.6a Applying Properties of Logarithms

Properties

Product: log log logb b bmn m n

Quotient: log log logb b b

mm n

n

Power: log ( ) logpb bm p m

Example 1: Expand a logarithm: Write each expression as a sum, difference, or power of

logarithms. Simplify if possible.

A. 2

log (3 4) B. 13

log5

C. 2

ln5

c

D. 32

log 9 E. 2

ln4

x y

z

Example 2: Condense a logarithm: Write each expression as a single logarithm. Then

simplify if possible.

A. 8 8

1log 12 log 5

3 B. log 2 log 3log log 7z z z za b cf

C. 12 12

2log 6 log 4 D. log 18 log 2

Example 3: Use Properties of Logarithms: Given that 2

log 5 2.3219 , approximate the

value for each expression below by using the product and quotient

properties (log 1b b )

A. 2

log 10 B. 2

5log

2

C. 2

25log

8

Use Properties of Logarithms in Real Life:

Example 4: Average student scores on a memory exam are modeled by the function

where t is the time in months.

a. Use properties of logarithms to write the model in condensed form.

b. Find the average score after 3 months.


4.6b Applying Properties of Logarithms

Reviewing Properties of Logarithms

Product: log log logb b bmn m n

Quotient: log log logb b b

mm n

n

Power: log ( ) logpb bm p m

Example 1: Write each expression as a sum or difference of logarithms. Then simplify, if

possible.

a. 8 2

64log

y b.

9

3log

7

a

Example 2: Write each expression as a single logarithm. Then simplify, if possible.

a. 3log log 4 logb b bx x b. 12 12

2log 6 log 2

c. 3 3 3

1log 81 log 6 log 2

2x x

Example 3: Use ln2 0.693 and ln3 1.099 to evaluate the following logarithms.

a. 4

ln3

b. ln18

Change-of-Base Formula

Right now we only know how to evaluate 10

ln and log in our calculator…the following

will allow us to evaluate ANY BASE in our calculator!

For any positive real numbers 1, 1, and 0 :a b x

10

10

log lnlog

log lnb

x xx

b b 10

310

log 15log 15

log 3 OR

ln 15

ln 3

Example 4: Use the change of base formula to evaluate. Round to the nearest hundredth.

a. 8

log 97 b. 83

log 4 c. 1

3

3 log 15

Example 5: Use the change of base formula to solve each equation for x. Round your

answers to the nearest hundredth.

a. 3 22x b. 1

172

x

c. 272 35x

d. 5

log 10 x e. 3 17x (You have to think about this one!!)


4.7a Solve Exponential and Logarithmic Equations

How do you solve…

a. log15 b. 2

log 8 c. 725 125x d. 10 1.76x

*What about…3 20x ?

Solving Exponential Expressions by Using Logarithms…

Common log: 10

log logx x Natural Log: e

ln logx x

Properties:

Inverse: xblog b = x & blog xb = x

One-to-One Property: yxIf b =b , then x = y.

One-to-One Property: b b

If log x =log y, then x = y.

Change-of-Base Formula: b

logx lnlog x = or

logb ln

xb

Helpful hints when solving (in no specific order):

* Create the same base (exponential form)

* Convert from exp. to log. or from log. to exp…don’t forget to ISOLATE!

* Use a log or ln on both sides

* Don’t forget your logarithmic properties!

Examples: Solve for x. Round to the nearest thousandth and CHECK your solution!

1. 5 62x 2. 1

826

x

3. 18.2 3 52x 4. 272 35x

5. 47 7x 6. 35 25x

7. 2 58 56xe 8. 3 5 7 25xe

No decimals!

9. 4

1log

16x 10.

1

3

log 27 x

11. 5

log 625 2x 12. 2

log 16 3 2x


4.7b Solving Exponential and Logarithmic Equations

Common log: 10

log logx x Natural Log: e

log lnx x

Properties: * b > 0 (Remember domain restriction of logs)

Inverse: xblog b = x & blog xb = x

One-to-One Property: yxIf b =b , then x = y.

One-to-One Property: b b

If log x =log y, then x = y.

Change-of-Base Formula: b

logxlog x =

logb

Examples: Use the properties of logarithms to solve each expression for x. Round to the

nearest thousandth and be sure to check your solution!

1. ln 3 2ln4x 2. ln2 ln 2 ln3x x

3. 3 3

1log 2 log 49

2x 4. 7 7 7

log 2 log 8 log 30x

5. 3ln 9 1x 6. log log 3 1x x

7. log 48 log 2x x


4.8a Solving Exponential and Logarithmic Inequalities

Interval Notation:

( ) “not included” or “open” [ ] “included” or “closed”

(1, 5]

Steps:

1. Change inequality to an = sign

2. Solve for x (just like in 4.7 & 4.8)

3. Test one point to either side of your x to determine direction of solution

4. Use interval notation for your final answer.

Example 1: Solve the exponential inequality algebraically and write your final answer using

interval notation.

45 125x

Example 2: Solve the exponential inequality algebraically.

14 32x


2

5 3 1 11x


2

81 3 193

x

Example 5: Your family purchases a new car for $25,000. Its value depreciates by 12% each

year. During what interval of time does the car’s value exceed $16,000?

Example 6: You deposit $700 in an account that pays 2.75% annual interest. How long does it

take the balance to reach

a. $2000 when interest is compounded continuously

b. $1000 when interest in compounded quarterly


4.8b Solve Exponential and Logarithmic Inequalities

Steps:

1. Change inequality to an = sign

2. Solve for x (just like in 4.7 & 4.8)

3. Test one point to either side of your x to determine direction of solution

4. Use interval notation for your final answer (be sure to consider domain

restrictions for logarithms!)

Solve each logarithm inequality algebraically.

Example 1: 5

log 2x

Example 2: 3

log 2x

Example 3: 4

log 8 11x

Example 4: 4log 3 7 8x

Example 5: 4ln 2 3x

35

30

25

20

15

10

5

-5

5


4.9a Write & Apply Exponential Functions

Example 1: Write an exponential function xy ab whose graph passes through (1,6) and (2,18).

Step 1: Substitute the coordinates of the two given points into xy ab (2 different equations).

Step 2: Solve for a in the first equation and substitute into the

second equation.

Step 3: Because 3b , it follows that 6

23

a . So, 2 3xy .

Example 2: Write an exponential function xy ab whose graph passes through (2,8) and (3,32)

Step 1: Substitute the coordinates of the two given points into xy ab .


second equation.

Step 3: Because _________b , it follows that __________a . So, ___________y .

18

16

14

12

10

8

6

4

2

-2

5



8

6

4

2

5

80

70

60

50

40

30

20

10

-10

-5

60

50

40

30

20

10

-10

5

10

-10

-20

-30

-40

-50

-60

-70

5


4.9b Write and Apply Power Functions

Example 1: Write a power function, by ax , whose graph passes through 2,16 and 3,54 .

Step 1: Substitute the coordinates of the two given points into by ax .


second equation.

Step 3: Because b = ________, it follows that a = _________. So, y = ___________.

Example 2: Write a power function, by ax , whose graph passes through 1, 4 and 4, 64 .

Step 1: Substitute the coordinates of the two given points

into by ax .

Step 2: Solve for a in the first equation and substitute into

the second equation.

Step 3: Because b = _________, it follows that a = __________. So, y = __________.



10

8

6

4

2

5 10

20

18

16

14

12

10

8

6

4

2

-2

-5 5


5.1a Graph and Write Equations of Parabolas

Parabola: The set of all points in a plane

whose distance to a fixed point, the focus,

equals its distance to a fixed line,

the directrix.

Standard Equation of a Parabola: Vertex at (0,0)

Horizontal directrix Vertical directrix

Steps to graphing parabolas:

1. State whether directrix is vertical or horizontal.

2. Find p.

3. Plot vertex, focus, and directrix.

4. Create an x-y chart to find reference points.

P is the

distance from

vertex to focus!

P2 Focus

P1

D1 D2 Directrix

21

4

0 : opens up

0 : opens down

focus : 0,

directrix :

y xp

p

p

p

y p

21

4

0 : opens right

0 : opens left

focus : ,0

directrix :

x yp

p

p

p

x p

4

2

-2

-4

-6

-5

8

6

4

2

-2

-5 5

Examples: State whether each parabola has a horizontal or vertical directrix. Graph each

parabola and label the vertex, focus, and directrix.

1. 21

4y x

Directrix (Horizontal/Vertical): _____________

Vertex: ________________________________

P: _______ Focus: ______________________

Directrix Equation: _______________________

2. 21

8x y


Vertex: ________________________________

P: _______ Focus: ______________________


6

4

2

-2

-4

-5 5

4

2

-2

-4

-6

5

3. 212y x


Vertex: ________________________________

P: _______ Focus: ______________________


4. 24x y


Vertex: ________________________________

P: _______ Focus: ______________________


2

-2

-4

5. Write the standard form of the equation of the parabola with the given focus and vertex at (0, 0).

a. ( 0, 1) b. ( 3, 0)

6. Write the standard form of the equation of the parabola with the given directrix and vertex at (0, 0).

a. directrix: y = 4 b. directrix: x = 4

7. Write an equation of the parabola shown.

Directrix (Horizontal/Vertical): _________

P: _______

Vertex: ________________

Equation: __________________________


5.1b Graph and Write Equations of Parabolas

Standard Equation of a Translated Parabola: Vertex at (h,k)

Horizontal directrix Vertical directrix

Examples: Graph each parabola. Label the vertex, focus, and directrix.

1. 211 ( 2)

12y x

Directrix (Horizontal/Vertical): ________

Vertex: _______________

Focus: ________________

Directrix: _____________

6

4

2

-2

-4

-6

-5 5

21

4

0 : opens up

0 : opens down

focus : ,

directrix :

y k x hp

p

p

h k p

y k p

21

4

0 : opens right

0 : opens left

focus : ,

directrix :

x h y kp

p

p

h p k

x h p

2. 213 ( 1)

8x y


Vertex: _______________

Focus: ________________

Directrix: _____________

3. 2 8 8 8 0y y x


Vertex: _______________

Focus: ________________

Directrix: _____________

4. Write an equation of a parabola in standard form with vertex at (-8, 1) and focus at (-8, 3).

5. Find the vertex and value of p for…

a. 2 6 10 1x x y b. 2

3 2 4y x

6

4

2

-2

-4

-6

-5 5

10

8

6

4

2

-2

-5 5


5.2a Graph and Write Equations of Circles

Circle: The set of all points in a plane that are a constant distance, the radius, from a

fixed point, the center.

Standard Equation of a Circle: Center at 0,0

2 2 2x y r

Example 1: Identify the radius of the circle and then graph.

a. 2 225x y b. 2 29y x

Radius: _______ Radius: _______

Example 2: Write the standard equation of a circle whose center is at the origin. Then,

sketch the graph.

a. A circle with a radius of 3 b. A circle with a radius of 4

Equation: Equation:

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

r: radius

Example 3: The point (6, 2) lies on a circle whose center is the origin. Write the

standard form of the equation of the circle.

Example 4: State whether the given point is inside, outside, or on the circle whose

equation is given.

a. 2 2 10x y ; (-2, 1) b. 2 24x y ; 4, 5

c. 29x y ; (0, 3)

Example 5: Find an equation of the line tangent to the circle 2 2 10x y at (-1, 3).

* Hint: A tangent line is perpendicular to the radius.

Example 6: A furniture store advertises free delivery up to a 50 mile radius from the

store. If a customer lives 28 miles east and 41 miles north of the store, does

the customer qualify for free delivery? (Draw a picture to help with the

situation.)

6

4

2

-2

-4

-5 5


5.2b Graph and Write Equations of Circles

Standard Equation of a Translated Circle: Center at ,h k

2 2 2x h y k r

Example 1: Write the equation for the translated circle graphed.

a. b.

Example 2: State whether the given point is inside, outside, or on the circle whose

equation is given.

a. 2 212,3 ; 12 2 12P x x y y b. 2 22, 3 ; 8 2 3P x x y y

Completing the Square:

1. Isolate the x’s and y’s

2. Draw blanks on both sides of = sign

3. Half the middle term, square it, and ADD to BOTH sides

4. Factor

8

6

4

2

-2

-5 5

2

-2

-4

-6

-8

-5 5

r: radius

Example 3: Write each equation in standard form. Then, state the coordinates of the

center and give the radius.

a. 2 2 4 12x y y b 2 2 4 6 3 0x y x y

c. 2 22 2 2x x y y d. 2 2 10 12 7x y x y

Example 4: Graph the given equation. Label the center and identify the radius.

a. 2 23 9x y b.

2 22 5 20x y

4

2

-2

-4

-5

-2

-4

-6

-8

-10

5


5.3a Graph and Write Equations of Ellipses

Ellipse: The set of all points in a plane such that the sum of the distances from p to two

fixed points, foci, is a constant.

Standard Equation of an Ellipse: Center at (0, 0)

Horizontal major axis Vertical major axis

2 2

2 21

x y

a b

2 2

2 21

x y

b a

Steps to graphing translated ellipses:

1. State whether the ellipse has a horizontal or vertical major axis.

2. Find a, b, and c.

3. Plot center: 0,0 or ,h k .

4. Count from center to plot vertices, co-vertices, and foci.

5. List or label each set of coordinate points.

Minor axis (shortest)

Major axis (longest)

vertex vertex

co-vertex

co-vertex

focus focus

a: vertices

b: co-vertices

c: foci

2 2

2 2 2

length of major axis: 2

length of minor axis: 2

a b

a b c

a

b

Example 1: Sketch and label the graph of each ellipse. List the coordinates of the center, foci,

vertices, and co-vertices.

a. 2 2

14 9

x y

a: _____, b: _____, c: _____

center: _________________

vertices: ________________

co-vertices: ______________

foci: ____________________

b. 2 2

19 1

x y

a: _____, b: _____, c: _____

center: _________________

vertices: ________________

co-vertices: ______________

foci: ____________________

Example 2: Write the standard equation for an ellipse with foci at ( 8,0) and (8,0) and

with major axis of 20.. Then, sketch the graph.

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

10

8

6

4

2

-2

-4

-6

-8

-10

-10 -5 5 10

6

4

2

-2

-4

-6

-5

6

4

2

-2

-4

-6

-5 5

Example 3: Graph the equation 2 29 36x y . Identify the center, vertices, co-vertices,

and foci of the ellipse.

a: _____, b: _____, c: _____

center: _________________

vertices: ________________

co-vertices: ______________

foci: ___________________

Example 4: Write an equation of the ellipse that has a vertex at (5, 0), a co-vertex at

(0, 4), and a center at (0, 0).

Example 5: Write an equation for given ellipse.


5.3b Graph and Write Equations of Ellipses

Standard Equation of a Translated Ellipse: Center at ( , )h k

Horizontal major axis Vertical major axis

Example 1: Sketch and label the graph of each ellipse. List the coordinates for the

center, foci, vertices, and co-vertices.

a. 2 2( 2) ( 1)

14 9

x y b.

2 2( 2) ( 2)1

16 1

x y

4

2

-2

-4

-5

6

4

2

-2

5

2 2

2 2 2

length of major axis: 2

length of minor axis: 2

a b

a b c

a

b

a: vertices

b: co-vertices

c: foci 2 2

2 21

x h y k

a b

2 2

2 21

x h y k

b a

a = ____ b = ____ c = ___

Center: ______________

Vertices: ____________________

Co-vertices: __________________

Foci: ________________________

COUNT from center to identify

and plot each coordinate!

a = ____ b = ____ c = ___

Center: ______________

Vertices: ____________________

Co-vertices: __________________

Foci: ________________________

Example 2: Write the standard equation for an ellipse.

An ellipse with its center at (2, 4) and with a horizontal major axis of 10 and a

minor axis of 6. Sketch the graph.

Example 3: Write each ellipse in standard form.

a. 2 24 10 24 45 0x y x y

b. 2 216 4 96 8 84 0x y x y

-2

-4

-6

-8

5 10


5.4a Graph and Write Equations of Hyperbolas

Hyperbola: The set of all points in a plane such that the absolute value of the difference between

the distances from P to foci, is equal.

Parts of a hyperbola:

Standard Equation of a Hyperbola: Center at (0,0)

Horizontal transverse axis Vertical transverse axis

22

2 21

yx

a b

2 2

2 21

y x

a b

F1 F2

P

Q

2 2 2

transverse axis : 2

conjugate axis : 2

a b c

a

b

a: vertices

b: co-vertices

c: foci

Steps to Graphing Hyperbolas:

1. State whether the hyperbola has a vertical or horizontal transverse axis.

2. Find a, b, and c.

3. Plot center and count a, b, and c to plot vertices, co-vertices, and foci.

4. Draw boundary box and asymptotes.

5. Draw both u-shapes at vertices.

Example 1: Write the standard equation for each hyperbola and sketch the graph. Label the

vertices, co-vertices, and foci.

a. 2 2

116 9

y x b.

22

149 16

yx

a: b: c: a: b: c:

vertices: __________________ vertices: ________________________

co-vertices: ________________ co-vertices: ______________________

foci: _____________________ foci: ____________________________

Example 2: Write an equation of the hyperbola with foci at (0, 7) and (0, -7) and vertices at (0, 6)

and (0, -6).

6

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-10 -5 5 10

Asymptotes of a Hyperbola: If the transverse axis is…

Horizontal: b

y xa

Vertical: a

y xb

Example 3: Find the equations of the asymptotes and the coordinates of the vertices for each.

a. 2 2

116 36

y x b.

22

116 25

yx

Example 4: Graph 2 29 4 36x y . Identify the vertices, foci, and asymptotes of the hyperbola.

a: b: c:

Vertices: _________________ co-vertices: ____________________

Foci: ____________________ asymptotes: ___________________

6

4

2

-2

-4

-6

-8

-5 5


5.4b Graph and Write Equations of Hyperbolas

Standard Equation of a Translated Hyperbola: Center at (h,k)

Horizontal transverse axis Vertical transverse axis

2 2

2 21

x h y k

a b

2 2

2 21

y k x h

a b

Example 1: Write the standard equation for each translated hyperbola.

a. Horizontal transverse axis b. Vertical transverse axis

center: ( 1,2) center: (3,4)

4 and 6a b 5 and c 7a

2 2 2

transverse axis : 2

conjugate axis : 2

a b c

a

b

a: vertices

b: co-vertices

c: foci

4

2

-2

-10 -5 5

Example 2: Graph each translated hyperbola. Label the center, vertices, co-vertices, and foci.

a.

2 22 3

125 16

y x b.

22 ( 1)( 2)1

25 4

yx

center: ________________ center: ________________

vertices: _______________ vertices: _______________

co-vertices: _____________ co-vertices: _____________

foci: ___________________ foci:____________________

Example 3: Write the standard equation for each hyperbola.

a. 2 24 24 10 5 0x y x y b. 2 22 4 6 3 0x y x y

10

8

6

4

2

-2

-4

5


5.5 Classify Conic Sections

Classifying Conic Sections: 2 2 0Ax Cy Dx Ey F

Type Coefficients

Ellipse 0AC

Circle , 0, 0A C A C

Parabola 0AC

Hyperbola 0AC

Example 1: Classify each conic section.

a. 2 24 8 6 13x x y y b. 26 12 3 9x x y

c. 2 29 18 4 8 23x x y y d. 2 218 4 111x y y x

Examples 2: Graph each translated conic section.

a.

2 24 3

19 16

x y b.

213 2

12x y

2

-2

-4

-6

5

8

6

4

2

-2

-4

-6

5

Transverse Axis (H/V)______________

Center________, a____, b____, c____

Vertices________________________

Co-vertices_____________________

Foci___________________________

Directrix (H/V): ______________ Vertex: _________, P = ________ Focus: ________ Directrix Equation: ___________

c.

2 22 1

19 1

x y

d.

2 23 1 16x y

e. 2 29 4 18 40 55 0x y x y

-2

-5

6

4

2

-2

-4

-6

5

12

10

8

6

4

2

-2

-5

Major Axis (H/V)_________________

Center________, a____, b____, c____

Vertices________________________

Co-vertices_____________________

Foci___________________________

Center: _______________ Radius: ________________

Transverse Axis (H/V)______________

Center________, a____, b____, c____

Vertices________________________

Co-vertices_____________________

Foci___________________________


5.6 Solve Quadratic Systems

Example 1: Solve the system using the substitution method.

2 5

0

x x y

y x

Example 2: Solve the system using substitution.

2 2 25

1

x y

y x

6

4

2

-2

5

4

2

-2

-4

-5 5

Example 3: Determine the points of intersection of a parabola and line using the

substitution method.

2

12 6

4

y x

x y

Example 4: Determine the points of intersection of an ellipse and a circle using the

elimination method.

2 2

2 2

4 13 100

9

x y

x y

Example 5: Solve the system by elimination.

2 2

2 2

25 0

10 10 25 0

x y

x x y y


5.7 Use Figures in Three-Dimensional Space

Review Distance Formula

Distance Formula: 2 2

2 1 2 1d x x y y

Example: Find the distance between P and Q . Give exact answers and approximate

answers to the nearest hundredth when appropriate.

7, 2 and 5, 1P Q

Three-Dimensional Space

Three-dimensional space or 3-space: formed by the intersection of an x-axis, y-axis,

and z-axis. The axes determine three coordinate planes: xy-plane, xz-plane, yz-plane

which divide the 3-space into eight octants.

Each point in a 3-space is represented by an ordered triple: x,y,z

Steps to drawing a rectangular prism:

1. Plot endpoints of the diagonal

2. Find differences in x, y, and z values:

2 1x x length of prism

2 1y width of prismy

2 1z height of prismz

3. Decide which endpoint is in foreground/background.

4. Decide which endpoint is on right/left.

5. Draw bases and connect height.

Example 1:

A. Draw a rectangular prism having a diagonal with endpoints 1,0, 2 and 2, 1,3 .

Distance Formula in 3-Space: distance between 1 1 1 2 2 2x , y ,z and x , y ,z

22 2

2 1 2 1 2 1d x x y y z z

B. Find the length of the diagonal using the distance formula in 3-Space.

Standard Form of a Plane: Ax By Cz D

The graph of a linear equation in three variables is a plane. In order to sketch a

plane, we must find the intercepts of each axis.

Example 2: Find the x-intercept, y-intercept, and z-intercept. Then, graph the plane.

a. 2x y 3z 6 b. 2x 3y 4z 12

Standard Equation of Sphere: Center at 0 0 0x ,y ,z

22 2 2

0 0 0x x y y z z r

Example 3: Write equation of a sphere with the given center and radius.

a. C 1,5, 3 r 5 b. C 4, 2, 6 r 7

Multiple-Use Classroom Resources 18

Isometric Dot Paper

Teaching Aid Master 18©

Prentice-H

all, Inc. All rights reserved.

Multiple-Use Classroom Resources 18

Isometric Dot Paper

Teaching Aid Master 18©

Prentice-H

all, Inc. All rights reserved.


6.1a Binomial Distribution

Vocabulary:

RANDOM VARIABLE:

a variable whose value is determined by the outcomes of a random event

examples: picking a card from a deck, or guessing a question right on a test, or average number of points scored per game in a baseball season

DISCRETE RANDOM VARIABLE:

a variable that has a countable number of distinct values

example: number of correct answers on a student’s quiz or the number of days with a low temperature below freezing the month of January

CONTINUOUS RANDOM VARIABLE:

a variable that has an uncountable, infinite number of possible values, often

over a specified interval

example: the height of the players on the basketball team or amount of time it takes to brush your teeth

PROBABILITY DISTRIBUTION:

shows the probability of each possible value of a random variable

BINOMIAL DISTRIBUTION: shows the probability of the outcomes of a binomial experiment

BINOMIAL EXPERIMENT:

A probability experiment is a binomial experiment if both of the following

conditions are met:

1. The experiment consists of n trials whose outcomes are either successes or

failures.

2. The trials are identical and independent with a constant probability of success,

p, and failure, 1-p .

BINOMIAL PROBABILITY:

n-kk

n k 1-P pC p=

Examples: Find the probability.

1. Suppose that the probability a seed will germinate is 80%. What is the probability

that 7 of these seeds will germinate when 10 are planted?

p: probability of success

1-p: probability of failure

2. A landscaping plan specifies that 10 trees of a certain type are to be planted in

front of a building. When this type of tree is planted in the autumn, the probability

that it will survive the winter is 85%. What is the probability that no fewer than 8

of the 10 trees will survive the winter if planted in the autumn?

3. Surgery Success: A surgical technique is performed on seven patients. You are

told there is a 70% at least chance of success. Find the probability that the

surgery is successful

a) exactly five patients

b) at least five patients

c) less than five patients

4. Favorite Cookie: Ten percent of adults say oatmeal raisin is their favorite

cookie. You randomly select 12 adults and ask each to name his or her favorite

cookie. Find the probability that the number who say oatmeal raisin is their

favorite Cookie is;

a) exactly four

b) at least four

c) less than four


6.1b Binomial Distribution

Example 1:

Let X be a random variable that represents the number of questions that students

guessed correctly on a quiz with three true-false questions.

n = p = 1-p =

a) Complete the table and a histogram showing the probability distribution for X.

b) Find the probability that a student guesses at least two questions correctly.

Example 2:

Let X be a random variable that represents the number of girls that a family has with 4

children.

n = p = 1-p =

a) Complete the table and a histogram showing the probability distribution for X.

b) Find the probability that a family had at most 3 girls.

X (number correct) 0 1 2 3

Outcomes 1 3 3 1

P(X)

X (number girls) 0 1 2 3 4

Outcomes 1 4 6 4 1

P(X)

Example 3:

In a standard deck of cards, 25% are hearts. Suppose you choose a card at random,

note whether it is a heart, then replace it. You conduct the experiment 5 times.

Draw a probability histogram of the binomial distribution for your experiment

showing the probability for 0, 1, 2, 3, 4, or 5 hearts.

Example 4:

In a standard deck of cards, 1/13 of the cards are Aces. Suppose you choose a card at

random, note whether it is an ace, and then replace it. You conduct the experiment 4

times.

Draw a probability histogram of the binomial distribution for your experiment showing the

probability for 0, 1, 2, 3, 4 ACES.


6.2a Using Normal Distributions

The NORMAL DISTRIBUTION is symmetric, bell shaped, and characterized by its mean, X , and standard

deviation, . The probability within any particular number of standard deviations of X is the same for all normal

distributions.

EMPIRICAL RULE: This probability equals 0.68 within 1 standard deviation, 0.95 within 2 standard deviations, and

0.997 within 3 standard deviations.

EXAMPLE #1… The mean value of land and buildings per acre from a sample of farms is $1500, with a

standard deviation of $200.

A. What percent of farm land is valued at or below $1300?

B. What percent of farm land is valued at or above $1100?

C. What percent of farms whose land and building values per acre are between $1300 and $1700?

2

D. What percent of farms whose land and building values per acre are below $1300 Or above $1700?

EXAMPLE #2… A normal distribution has mean X and standard deviation of . Find the indicated

probability for a randomly selected x-value from the distribution.

A. ( 2 )P x X

B. ( 2 )P x X

C. ( 2 )P X x X

D. ( 3 )P x X Or ( 3 )P x X

3

EXAMPLE #3…Referring back to Example 1 and using the normal curve, what ranges

of dollar amounts would contain…

a) …about 95% of the data?

b) …about 32% of the data?

c) …about 99.7% of the data?

d) …about 5% of the data?

e) … about 4.7% of the data?


6.2B Z-SCORE FOR A VALUE OF A RANDOM VARIABLE

When in doubt, sketch your curve out!

The Z-SCORE for a value x of a random variable is the number of standard deviations that x

falls from the mean . It is calculated as

x xz

EXAMPLE #1… The heights of 300 women at a particular college are normally distributed with a mean of 65

inches and a standard deviation of 2.5 inches.

A. Find the probability that a randomly selected college woman has a height of at most 62.5 inches.

i. using the normal curve ii. using the z-score table

B. Find the probability that a randomly selected college woman has a height of at most 68 inches.

i. using the normal curve ii. using the z-score table

2

Example 2… Assuming The heights of 300 women at a particular college are normally distributed with a mean of 65

inches and a standard deviation of 2.5 inches. Redraw the curve for each step.

A. About what percent of college women have heights below 70 inches?

B. About how many of the college women have heights above 60 inches?

C. About how many of the college women have heights between 60 and 65 inches?

D. Find the probability that a randomly selected college woman has a height of at most 61 inches.

About how many women is this?


6.3a Approximate Binomial Distributions

Vocabulary:

o Consider the binomial distribution of n trials with probability p of success on each

trial. The binomial distribution can be approximated by a normal distribution with

the following mean and standard deviation.

Mean: X np Standard Deviation: (1 )np p

Find the mean and standard deviation of a normal distribution that approximates the binomial distribution with

n trials and probability p of success on each trial.

a. n = 130, p = 0.79 b. n = 120, p = 0.08

BINOMIAL AND THE EMPIRICAL RULE

1. According to a survey conducted by the Harris Poll, 23% of adults in the United States

favor abolishing the penny and making the nickel the lowest denomination coin. You are

conducting a random survey of 500 adults. What is the probability you will find at most 106

adults who favor abolishing the penny?

a. Find the mean: b. Find the Standard Deviation:

c. Now apply to the Normal Curve using the empirical rule.

BINOMIAL AND Z-SCORE

1. Use the fact that hyperopia, or farsightedness, is a condition that affects

approximately 25% of the adult population in the United States. Consider a random

sample of 280 U.S. adults.

What is the probability that 63 or more people are farsighted? 63P x

a. Find the mean: b. Find the Standard Deviation:

c. Find the z-score. d. Now use the Normal table

2. What is the probability that 80 or fewer people are farsighted? 80P x

b. Find the mean: b. Find the Standard Deviation:

d. Find the z-score. d. Now use the Normal Table


6.3b Approximate Binomial Distributions and Test Hypotheses

To test a hypothesis about a statistical measure for a population, use the following steps:

1. State the hypothesis you are testing.

2. Collect data from a random sample of the population and compute the statistical

measure of the sample.

3. Assume the hypothesis is true and calculate the resulting probability P of obtaining the

sample statistical measure. If the probability is small, such as P(x)<.05, you should reject

the hypothesis. In other words, if the probability of an event is less than 5%, you should

reject the hypothesis.

1. A recent Harris Poll claimed that 44% of adults cut back on their spending in order to pay the

increased price of gasoline. To test this finding, you survey 60 adults and find that 21 of them

cut back on their spending in order to pay the increased price of gasoline. Should you reject the

Harris Poll’s findings?

a. State the hypothesis-H:

b. State what really happened-

c. n = p = x =

d. X

e. z

f. Find P(x )

g. Compare your results to .05. If it is smaller than .05, then reject the hypothesis. If it is

not smaller, then do not reject the hypothesis.

2. You read an article that claims 35% of seniors will buy a class ring. To test this claim, you survey

55 randomly selected seniors in your school and find that 11 are planning to buy a class ring.

Should you reject the claim?

h. State the hypothesis-H:

i. State what really happened-

j. n = p = x =

k. X

l. z

m. Find P(x )

n. Compare your results from step to .05. If it is smaller than .05, then reject the

hypothesis. If it is not smaller, then do not reject the hypothesis.


6.4 Select and Draw Conclusions from Samples

Vocabulary

Census- Count or measure of entire population

Sampling- Measures a part of the population

Survey- a study of one or more characteristics of a group

Population- the entire group you want information about

Type of Sample Definition Example

Simple Random

Sample

every member of the population

has an equal chance at being

selected

Chick-fil-A wants to gather information on

customer satisfaction. The manager asks 45

customers to complete a survey before they leave.

Stratified Random

Sample

the population is divided into

distinct groups. Members are

selected at random from each

group.

CGHS want to know where students want to go on a

trip at the end of the year. The administrators

randomly survey 30 freshman, 30 sophomores, 30

juniors, and 30 seniors.

Systematic

Sample

a rule is used to select members

of the population

Every person in the phonebook whose last name

starts with a Z is selected by a company to

complete a survey

Convenience

Sample

only members of the population

who are easily accessible are

selected

I ask 25 people at my softball game if they order

more ice cream in the winter than in the summer.

Self-selected

Sample

members of the population select

themselves by volunteering

Chili’s is conducting a survey to determine the

quality of their food service. The phone number to

participate in the survey is located at the bottom of

the receipt for customers to call in and participate.

Representative

Sample

a sample that accurately reflects

the characteristics of a

population

Research is being conducted to determine the

amount of interest in the Georgia Bulldogs at CGHS.

Students of all grade levels were used in the study.

Biased Sample a sample that is not

representative of the population

A group of students at your school wants to gather

information about the need for additional funding

for the baseball team. They survey members on the

baseball team.

Biased Question A question that encourages a

particular response

Don’t you think that this wedding invitation is cuter

than the other one since it has a bow?

Example 1: Identify the type of sample described. Then tell if the sample is biased.

Explain your reasoning.

A. A baseball league wants to know who the fans think was the league’s best pitcher. Fans

are asked to vote on the league’s website.

B. The owner of a miniature golf course wants to find the number of times people in the

community play a round of miniature golf each year. The owner has the front desk

employee survey customers when they pay for a round.

C. An online vendor wants to know how people that purchased a new product feel about

changes from the older models. A computer is used to randomly generate a list of 100

customers to survey from a list of all customers that bought that product.

Margin of Error: 1

En

The difference from a sample to a population.

Example 2: Calculate the margin of error for a survey that has the given sample size (n).

Round your answer to the nearest tenth of a percent.

a. n = 482 b. n = 664 c. n = 801

Example 3: Find the sample size required to achieve the given margin of error. Round

your answer DOWN to the nearest whole number. 2

1n

E

a. 1.8%E b. 2.3%E c. 0.6%E


6.4b Margin of Error Intervals

Example 1: Computers A survey reported that 2048 respondents out of n = 3200 had detected a virus

on their computer at least once during the last two years.

a. What is margin of error for the survey? Round your answer to the nearest tenth of a

percent. 1

En

b. Give an interval that is likely to contain the exact percent of people that have detected a

virus on their computer at least once during the last two years. (Probability E )

Example 2: Environment A survey claims that the percent of the population that makes purchasing

decisions based on the effect it will have on the environment is between 88.75% and

91.25%. The remainder of the people in the survey do not let effects on the environment

influence their purchasing decisions.

a. How many people were surveyed?

Hint: To find E, subtract your percents and divide by 2

Hint: To find n, use 2

1n

E

b. Give an interval that is likely to contain the exact number of people in the population

that do not let effects on the environment influence their purchasing decisions.

Step 1: To find the probability for people that DO, add your percents together and divide by 2

Step 2: To find the percent of people that do not let effects on the environment, subtract

from 100%.

Step 3: Now your new interval for people that do not, Probability E


6.5 Experimental and Observational Studies

Vocabulary

Experimental Group- a group that undergoes some procedure or treatment

Control Group- a group that does not undergo the procedure or treatment

Experimental Study- The investigator is in charge of assigning the individuals to the experimental group or the

control group

Observational Study – The investigator has no control of assigning individuals to the experimental group or the

control group

For each example below, tell whether the study is an experimental study or an observational study.

Explain your reasoning. Then identify any flaws/bias in the research.

1. You want to study the effects of regular exercise on a person’s heart rate. You measure the heart

rate of a person at rest and again after jogging in place for 5 minutes. The control group is students

who are not on a school athletic team. The experimental group is students who are on a school

athletic team.

2. You want to study the effects that music has on the ability to recall knowledge. Each individual in

your study is given material to read on the same unfamiliar topic and then asked to take a factual

quiz on the material. The control group is individuals who read the material and take the quiz in a

quiet room. The experimental group is individuals who read the material and take the quiz in a room

that has classical music playing.

3. A scientist wants to study the effects that a nutritional supplement has on the growth of mice. The

weight of each mouse is recorded daily. The control group consists of mice that do not receive the

supplement. The experimental group consists of mice that receive a safe amount of the supplement.

4. You want to study the effects that using a calculator has on the time it takes to complete a math

test. You record how long it takes a student to complete the test. The control group is students

that choose not to use calculators. The experimental group is students that choose to use

calculators.

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Gifted/Honors Algebra 2 - PC\|MACimages.pcmac.org/SiSFiles/Schools/GA/GwinnettCounty... · piano for $25 and on the saxophone for $20. a. Write an inequality that represents the possible