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
Integrated Algebra 2
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)
Integrated Algebra 2
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?
Integrated Algebra 2
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?
Integrated Algebra 2
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
Integrated Algebra 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.
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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.
-
+
-
- - -
+
-
+
-
Integrated Algebra 2
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.
Integrated Algebra 2
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
Integrated Algebra 2
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.
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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:
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 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
Integrated Algebra 2
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
Integrated Algebra 2
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
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
Integrated Algebra 2
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.
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
y-intercept: _______ Asymptote: ________
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)
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
Integrated Algebra 2
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: ______
y-int: ______ asymptote: ____
Left: ________ Right: _________
Domain: ______ Range: ______
y-int: ______ asymptote: ____
Left: ________ Right: _________
Integrated Algebra 2
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 __________
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
Growth/Decay _________ Inc/Dec __________
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
Growth/Decay _________ Inc/Dec __________
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
Growth/Decay _________ Inc/Dec __________
y-intercept: _______ Asymptote: ________
Left:_____________ Right: ____________
Domain: ___________ Range: ____________
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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: ________
Asymptote: ___________
Domain: __________ Range: ________
Left:_____________ Right: ________
Asymptote: ___________
Domain: __________ Range: ________
Left:_____________ Right: ________
Asymptote: ___________
Integrated Algebra 2
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
Domain: _____, Range: ______, y-intercept: _______
Asymptote: _____, Inc/Dec Interval: ______________
Average Rate of Change: from (2, 0) to (5, 2)________
Domain: _____, Range: ______, y-intercept: _______
Asymptote: _____, Inc/Dec Interval: ______________
Average Rate of Change: from (2, 2) to (8, 0)________
Domain: _____, Range: ______, y-intercept: _______
Asymptote: _____, Inc/Dec Interval: ______________
Average Rate of Change: from (1, 0) to (10, 2.3)________
Domain: _____, Range: ______, y-intercept: _______
Asymptote: _____, Inc/Dec Interval: ______________
Average Rate of Change: from (-1, 3) to (6, 5.1)________
Integrated Algebra 2
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.
Integrated Algebra 2
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!!)
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Example 3: Solve the exponential inequality algebraically.
2
5 3 1 11x
Example 4: Solve the exponential inequality algebraically.
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
Integrated Algebra 2
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
Integrated Algebra 2
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 .
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 .
18
16
14
12
10
8
6
4
2
-2
5
Example 3: Write an exponential function xy ab whose graph passes through (0,4) and (2,1)
Example 4: Write an exponential function xy ab whose graph passes through (1,10) and (4,80)
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
Integrated Algebra 2
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 .
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 = ___________.
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 = __________.
Example 3: Write a power function, by ax , whose graph passes through 2,1 and 4,3 .
Example 4: Write a power function, by ax , whose graph passes through 4,8 and 6,18 .
10
8
6
4
2
5 10
20
18
16
14
12
10
8
6
4
2
-2
-5 5
Integrated Algebra 2
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
Directrix (Horizontal/Vertical): _____________
Vertex: ________________________________
P: _______ Focus: ______________________
Directrix Equation: _______________________
6
4
2
-2
-4
-5 5
4
2
-2
-4
-6
5
3. 212y x
Directrix (Horizontal/Vertical): _____________
Vertex: ________________________________
P: _______ Focus: ______________________
Directrix Equation: _______________________
4. 24x y
Directrix (Horizontal/Vertical): _____________
Vertex: ________________________________
P: _______ Focus: ______________________
Directrix Equation: _______________________
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: __________________________
Integrated Algebra 2
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
Directrix (Horizontal/Vertical): ________
Vertex: _______________
Focus: ________________
Directrix: _____________
3. 2 8 8 8 0y y x
Directrix (Horizontal/Vertical): ________
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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.
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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___________________________
Integrated Algebra 2
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
Integrated Algebra 2
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.
Integrated Algebra 2
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
Integrated Algebra 2
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.
Integrated Algebra 2
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?
Integrated Algebra 2
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?
Integrated Algebra 2
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
Integrated Algebra 2
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.
Integrated Algebra 2
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
Integrated Algebra 2
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
Integrated Algebra 2
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.