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Intermediate Algebra

Algebra II Notebook

Wasatch High School

2011-2012

Student Name _______________________

Teacher Name _______________________

x

y

=

Algebra 2

Standard I: Students will use the language and operations of algebra to evaluate, analyze and solve problems.

Objective 1: Evaluate, analyze, and solve mathematical situations using algebraic properties and symbols.

a. Solve and graph first-degree absolute value equations of a single variable.

b. Solve radical equations of a single variable, including those with extraneous roots.

c. Solve absolute value and compound inequalities of a single variable.

d. Add, subtract, multiply, and divide rational expressions and solve rational equations.

e. Simplify algebraic expressions involving negative and rational exponents.

Objective 2: Solve systems of equations and inequalities.

a. Solve systems of linear, absolute value, and quadratic equations algebraically and graphically.

b. Graph the solutions of systems of linear, absolute value, and quadratic inequalities on the coordinate plane.

c. Solve application problems involving systems of equations and inequalities.

Objective 3: Represent and compute fluently with complex numbers.

a. Simplify numerical expressions, including those with rational exponents.

b. Simplify expressions involving complex numbers and express them in standard form, a + bi.

Objective 4: Model and solve quadratic equations and inequalities.

a. Model real-world situations using quadratic equations.

b. Approximate the real solutions of quadratic equations graphically.

c. Solve quadratic equations of a single variable over the set of complex numbers by factoring, completing the square, and using the quadratic formula.

d. Solve quadratic inequalities of a single variable.

e. Write a quadratic equation when given the solutions of the equation.

Standard II: Students will understand and represent functions and analyze function behavior.

Objective 1: Represent mathematical situations using relations.

a. Model real-world relationships with functions.

b. Describe a pattern using function notation.

c. Determine when a relation is a function.

d. Determine the domain and range of relations.

Objective 2: Evaluate and analyze functions.

a. Find the value of a function at a given point.

b. Compose functions when possible.

c. Add, subtract, multiply, and divide functions.

d. Determine whether or not a function has an inverse, and find the inverse when it exists.

e. Identify the domain and range of a function resulting from the combination or composition of functions.

Objective 3: Define and graph exponential functions and use them to model problems in mathematical and real-world contexts.

a. Define exponential functions as functions of the form

x

yab

=

, b > 0, b ≠ 1.

b. Model problems of growth and decay using exponential functions.

c. Graph exponential functions.

Objective 4: Define and graph logarithmic functions and use them to solve problems in mathematics and real-world contexts.

a. Relate logarithmic and exponential functions.

b. Simplify logarithmic expressions.

c. Convert logarithms between bases.

d. Solve exponential and logarithmic equations.

e. Graph logarithmic functions.

f. Solve problems involving growth and decay.

Standard III: Students will use algebraic, spatial, and logical reasoning to solve geometry and measurement problems.

Objective 1: Examine the behavior of functions using coordinate geometry.

a. Identify the domain and range of the absolute value, quadratic, radical, sine, and cosine functions.

b. Graph the absolute value, quadratic, radical, sine, and cosine functions.

c. Graph functions using transformations of parent functions.

d. Write an equation of a parabola in the form y = a(x − h)2 + k when given a graph or an equation.

Objective 2: Determine radian and degree measures for angles.

a. Convert angle measurements between radians and degrees.

b. Find angle measures in degrees and radians using inverse trigonometric functions, including exact values for special triangles.

Objective 3: Determine trigonometric measurements using appropriate techniques, tools, and formulas.

a. Define the sine, cosine, and tangent functions using the unit circle.

b. Determine the exact values of the sine, cosine, and tangent functions for the special angles of the unit circle using reference angles.

c. Find the length of an arc using radian measure.

d. Find the area of a sector in a circle using radian measure.

Standard IV: Students will understand concepts from probability and statistics and apply statistical methods to solve problems.

Objective 1: Apply basic concepts of probability.

a. Distinguish between permutations and combinations and identify situations in which each is appropriate.

b. Calculate probabilities using permutations and combinations to count events.

c. Compute conditional and unconditional probabilities in various ways, including by definitions, the general multiplication rule, and probability trees.

d. Define simple discrete random variables.

Objective 2: Use percentiles and measures of variability to analyze data.

a. Compute different measures of spread, including the range, standard deviation, and interquartile range.

b. Compare the effectiveness of different measures of spread, including the range, standard deviation, and interquartile range in specific situations.

c. Use percentiles to summarize the distribution of a numerical variable.

d. Use histograms to obtain percentiles.

Utah State Standards

4.1.a Distinguish between permutations and combinations and identify situations in which each is appropriate

4.1.b Calculate probabilities using permutations and combinations to count events

ACT Warm up:

**Exploration – Permutations and Probability

Outcome-

Ex. Possible outcomes to flipping a coin.

Sample space-

Event-

Independent events-

Ex.

Dependent events-

Ex.

Fundamental Counting Principle

Ex for Independent Events. A sandwich menu offers customers a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayonnaise. How many different combinations of bread and spread are possible?

Ex for more than 2 Independent Events. Many voice mail systems allow owners to call from another phone and get their messages by entering a 3-digit code. How many codes are possible?

Ex. for Dependent Events. Carla wants to take 6 different classes next year. Assuming that each class if offered each period, how many different schedules could she have?

Factorials-

Notation

On a calculator, look for the ! in the MATH button menu.

By definition, 0! = 1.

Permutation-

Ex.

Notation-

Formula for Permutations

Ex. Eight people entered the Best Pizza Contest. How many ways can blue, red, and green ribbons be awarded?

Formula for Permutations with Repetitions.

Ex. How many different ways can the letters of the word BANANA be rearranged?

Ex. How many different ways can the letters of the word MISSISSIPPI be rearranged?

Combination-

Ex.

Notation-

Formula for Combination

Ex. Five cousins at a family reunion decide that three of them will go to pick up a pizza. How many ways can they choose the three people to go?

Ex. Six cards are drawn from a standard deck of cards. How many hands consist of two hearts and four spades?

**Exploration – Winning the Lottery

Homework Problems. p. 635 #11-19 odds

Homework Problems. p. 641 #13-19 odd, 23-29 odd


4.1.d Define simple discrete random variables

4.2.c Use percentiles to summarize the distribution of a numerical variable

4.2.d Use histograms to obtain percentiles

ACT Warm up:

Probability-

Notation

(different than permutation notation)

*you need to calculate the possible number of outcomes!

(sometimes this involves the Fundamental Counting Principle)

Practice Problems.

1. When two coins are tossed, what is the probability that both are tails?

2. When three coins are tossed, what is the probability that all three are heads?

3. Roman has a collection of 26 books- 16 are fiction and 10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability that he chooses 4 fiction and 4 nonfiction?

Random variable-

Ex. D represents the number showing when rolling a die.

Possible values:

Notation

P(X=n)

Probability distribution-

Ex. Probability distribution for rolling a die.

Histogram-

Ex. Probability distribution of the sum of the numbers of two dice rolled.

S=Sum

2

3

4

5

6

7

8

9

10

11

12

Probability

1/36

1/18

1/12

1/9

5/36

1/6

5/36

1/9

1/12

1/18

1/36

Draw a histogram.

Using the histogram, which outcome is most likely? What is the probability?

Use the table and histogram to find P(S=9). What other sum has the same probability?

Homework Problems. p. 648 #19-29 all, 57-60.


4.1.c Compute conditional and unconditional probabilities in various ways, including by definitions and probability trees

ACT Warm up:

Probability of Independent Events

If two events, A and B are independent, then the probability of both events occurring P(A and B) is ________________________.

*This formula can be applied to any number of independent events.

Ex. Gerardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and replaces it. He then randomly selects another coin. What is the probability that both of the coins he selects are dimes?

Ex. When three dice are rolled, what is the probability that two dice show a 5 and the third die shows an even number?

Probability of Dependent Events

If two events, A and B, are dependent, then the probability of both events occurring P(A and B) is ___________________________.

*This formula can be applied to any number of dependent events.

Ex. The host of a game show is drawing chips from a bag to determine the prizes for which contestants will play. The host draws from a bag of 20 chips, of which 11 say computer, 8 say trip, and 1 says truck. Drawing at random and without replacement, find each of the following probabilities.

a. What is the probability that the host draws a trip, then a computer.

b. What is the probability that the host draws a truck, then two trips.

Homework Problems. p.655 #15-23 odd, 31, 33

CDAS: Probability


4.2.a Compute and compare different measures of spread, including the range, standard deviation, and interquartile range

4.2.b Compare the effectiveness of different measures of spread, including the range, standard deviation, and interquartile range in specific situations

ACT Warm up:

Measures of Central Tendency

Mean-

Use when the data are spread out and you want an average of the values.

Median-

Use when the data contain outliers or points that are extremely high or low.

Mode-

Use when the data are tightly clustered around one or two values.

Ex. A new Internet company has 3 employees who are paid $300,000, 10 who are paid $100,000, and 60 who are paid $50,000.

1. Find the mean, median and mode.

2.Which measure of central tendency best represents the pay at this company?

3. Which measure of central tendency would recruiters for this company be most likely to use to attract job applicants?

Measures of Spread, Variation, or Dispersion

Range-

*Only gives information of the high and

the low values including outliers!

*Does not give information on the

majority of values

Variance-

Notation

Steps to Calculate Variance

1. Find the mean

2. Find the difference between each value in the set of data and the mean.

3. Square each difference.

4. Find the mean of the squares.

Standard Deviation-

*The typical variation for the data items from the mean.

Ex. The following table shows the length in thousands of miles of some of the longest rivers in the world. Find the standard deviation for these data.

River

Length

(thousands of miles)

Nile

4.16

Amazon

4.08

Missouri

2.35

Rio Grande

1.90

Danube

1.78

Interquartile Range-

*Should be used when data set contains outliers

Steps to Calculate

1. Find the median of the data set.

2. Find the median of the 1st half of the data Q1 (lower quartile)

3. Find the median of the 2nd half of the data Q3 (upper quartile)

4. Find the difference between Q3 and Q1

Ex.A year ago, Angela began working at a computer store. Her supervisor asked her to keep a record of the number of sales she made each month.

The following data set is a list of her sales for the last 12 months:

34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37

Use Angela's sales records to find:

a) the median

b) the range

c) the upper and lower quartiles

d) the interquartile range

Box Plots:

Draw a box plot for Angela’s sales records above.

**Exploration – Pulse Rates

Homework Problems. p. 667 #11, 13, 17,18, 24-26, worksheet on interquartile range

CDAS: Percentiles and Variability

Utah State Standard I: Students will use the language and operations of algebra to evaluate, analyze, and solve problems

ACT Warm up:

Properties of Real Numbers

Identity Properties:

Identity Property of Addition:

a

a

=

+

0

Ex:

4823

+

can be written as

48023

++

. (I know, it seems silly, but wait!)

(

)

(

)

(

)

48023

482223

482232

5021

71

++=

++-+=

++-=

+=

Ex: Why is

2714

+

the same as

3011

+

?

Identity Property of Multiplication:

a

a

=

×

1

Ex. Simplify:

3

4

5

6

(Show two ways)

Ex. Simplify:

9

4

3

2

Inverse Properties:

Inverse Property of Addition:

(

)

0

bb

+-=

Ex. What are the additive inverses of the following numbers?

a. 4

b.

2

1

c. – 5

Inverse Property of Multiplication:

1

1

b

b

×=

1

and

b

b

are called multiplicative inverses or reciprocals of each other.

Ex. What are the multiplicative inverses of the following numbers?

a. 10

b.

3

1

c.

5

4

-

Properties of Equality

Commutative Property of Addition: For any real numbers

and ,

ab

abba

+=+

Ex. 4 + 3 =

Commutative Property of Multiplication: For any real numbers

and ,

ab

abba

×=×

Ex. The array

shows a representation of the product

ab

×

.

How might this array also represent the product

ba

×

?

Explain why is there no Commutative Property of Subtraction or Commutative Property of

Division? Illustrate with specific examples.

Associative Property of Addition: For any real numbers a, b, and c,

(

)

(

)

abcabc

++=++

Draw a diagram that illustrates this property.

Associative Property of Multiplication: For any real numbers a, b, and c,

)

(

)

(

c

b

a

c

b

a

×

×

=

×

×

Distributive Property: For any real numbers

,, and

abc

c

a

b

a

c

b

a

×

+

×

=

+

×

)

(

Ex: a. 5(3 + 4) =

b. x(2 – y) =

Using what we know about Properties of real numbers, answer the following questions:

1. How can we write

c

a

¸

as a multiplication problem?

2. Explain why

(

)

/

ab

abc

cc

+=+

.

Homework

Match each equation with the property it illustrates?

1.

(

)

(

)

537537

++=++

________A. Commutative Property of

Multiplication

2. 2 + 5 + (-5) = 2 + 0

________B. Identity Property of

Multiplication

3.

(

)

(

)

(

)

(

)

50363650

+×+=+×+

________C. Associative Property of Addition

4.

5315

7321

æöæö

=

ç÷ç÷

èøèø

________D. Commutative Property of

Addition

5. 3 + (2 + 8) = 3 + (8 + 2)

________E. Identity Property of Addition

6. 7 + 4 + 0 = 11

________F. Associative Property of

Multiplication

7. 5(2) + 7(2) = ( 12)(2)

________G. Inverse Property of

Multiplication

8.

25

551

52

××=×

________H. Inverse Property of Addition

Use the Properties of the real numbers to simplify each expression. Please NO calculators.

9.

2

1

3

57

69

+

+

10.

7827

+

11.

(

)

742

12.

45722

¸

13.

1

1

1

1

1

1

1

2

+

+

+

14.

57

1218

+

Utah State Standard I: Students will use the language and operations of algebra to evaluate, analyze, and solve problems

ACT Warm up:

Order of Operations

Step 1

Step 2

Step 3

Step 4

Ex. Simplifying an expression

Find the value of [18 - 3(6-5)3] / 3

Variables-

Algebraic expressions-

Ex. Evaluating an expression

Evaluate g – t(g2 – t) if g = 2 and t = 6

Ex. Evaluating an expression

Evaluate

5

8

2

3

+

+

y

z

xy

if x = 5, y = -2, and z = -1

*Be careful using a calculator. Sometimes parentheses are necessary to get the correct answer.

Ex. To evaluate

)

4

(

5

)

12

(

4

, you must enter 4*12/(5*4).

Number Types and an example of each

Reals

Wholes

Rationals

Naturals

Integers

Complex

Irrationals

Map of relationship of number types

Ex. Classify Numbers

Name the sets of numbers to which each number belongs

1.

3

2

-

2. 9.99999

3.

6

4.

100

5. -23.3

6. 7 + 3i

Turning words into algebraic expressions

Key phrases: Sum of

Difference between

Product of

Quotient of

More than

Fewer than

Is

Opposite

Ex. Write an expression for each phrase

1. a number

2. 5 more than a number

3. six times the cube of a number

4. a number decreased by 7

5. the square of a number decreased by 11

Equation-

*Neither true nor false until all the variables have been replaced by numbers.

Solution-

Solving equations: think of the order of operations in reverse

Isolate the chosen variable.

Ex. Solve the equation and check your solution.

1. a – 5.48 = 0.02

2.

t

2

1

18

=

3. 53 = 3(y – 2) – 2(3y – 1)

Ex. Solving an equation or formula for a chosen variable.

1. Solve for

l

2

r

rl

S

p

p

+

=

(this is the formula for the surface area of a cone)

Ex. Writing an equation from a word problem and then solving for a chosen variable.

1. in-class ex. #8 p.23

2. p.26 #66

Cartesian Coordinate System (created by ________________________)

· horizontal axis _____________

· vertical axis ______________

· The point where the axes meet is called ________________ and has coordinates ( , )

· Broken into four regions called ___________________. Label these on the coordinate system below.

y

x

y

x

f

x

y

x

y

+

=

+

£

£

³

)

,

(

1

2

6

1

Homework problems. P.9 #19, 29, 39, 49, 53, 57

p.15 #19-25 odds, 51,53,65

p.25 #45,49,59, 67


Standard II: Students will understand and represent functions and analyze function behavior.

ACT Warm up:

Graphing

We can graph any equation using a __________________

x

y

=

Example:

2

yx

=

+ 3

X

2x

Y = 2x + 3

Some rules that will help when graphing:

Always follow order of operations when solving for y.

Exponent rules:

0

x

=

Examples:

0

0

0

2

55

(3)

=

=

-=

Negative exponents:

1

x

-

=

Examples:

1

2

4

2

4

3

-

-

-

=

=

=

Absolute Value:

Examples:

|6|

=

|17|

-=

Practice graphing:

Graph each of the following equations by making a T-chart.

x

y

±

=

1.

51

yx

=-

x

5x

y = 5x – 1

a

b

2.

|2|

yx

=+

x

x + 2

y = |x + 2|

3.

2

(1)4

yx

=+-

x

x + 1

2

(1)

x

+

2

(1)4

yx

=+-

4.

6

yx

=+

x

y =

5.

2

x

y

=

x

y =

6.

2

(3)1

yx

=-+

x

y =

7.

23

yx

=-+

x

y =

8.

32

x

y

=+

x

y =

9.

||3

yx

=-

x

y =

10.

1

yx

=-

x

y =

Group the 10 graphs above into categories based on similarities or differences. Explain the reasoning for your categories.

Domain -

Range -

Do the equations give us any clues as to what the graph might look like? Explain.

Types of equations:

Homework Problems: Worksheet


2.1.b Describe a pattern using function notation

2.1.c Determine when a relation is a function

2.1.d Determine the domain and range of relations

2.2.a Find the value of a function at a given point

ACT Warm up:

Functions –

A collection of data, or a set of ordered pairs is called a _____________________.

The _______________ is the set of all the first coordinates.

The ______________ is the set of all the second coordinates.

** Supplemental Reading 4.2 Function Notation p.178 & 181

A ________________ is a special kind of relation in which each element of the domain is paired with exactly one element of the range.

*What does this mean?

If you have a number that repeats in the ____________ (x-coordinates), it has to have the same range value (y-coordinate).

If it does not have the same range value, it is not a function.

If there is a repeat in a range value:

Mapping:

How to spot a function in a crowd.

1. Take a look at the two relations below.

Domain

Range

-3

1

0

2

2

4

6

7

Map:

Are there any repeats in the domain?

Is it a function?

Domain

Range

-3

0

1

1

5

0

-3

6

Map:

Are there any repeats in the domain?

Do the repeats have the same range value?

Is it a function?

Ex. The chart below gives the average and maximum lifetime for some animals. Determine if the data in this chart represents a function. Explain your reasoning.

Animal

Average Lifetime (years)

Maximum Lifetime (years)

Cat

12

28

Cow

15

30

Deer

8

20

Dog

12

20

Horse

20

50

Source: The World Almanac

2. Take a look at the graphs of the two relations shown below.

Ex.1

Ex.2

In the second relation, we had a repeat domain value of -3. Do you notice anything about the two points that have the same domain value of -3?

*We can use something called the ___________________________ test on a graph, to see whether a set of points is a function.

Use your pencil as a vertical line.

If no vertical line intersects a graph in more than one point, the graph represents a __________________.

If some vertical line intersects a graph in two or more points, the graph does _________ represent a function.

Ex. Problems using the vertical line test

Use the vertical line test to determine whether a graph represents a function or simply a relation. Hint: Remember graphs are made up of a gazillion ____________!

1.

2.

3.

When we have a lot of data, we can sometimes come up with a general way to describe the relationship of the information. The way we describe a relationship in math is with an ____________________.

The solution(s) of an equation: The set(s) of ___________________ that make the equation true.

*A graph is the scatter plot of all the points that make the equation true.

Finding a graph without a calculator

Make a T-Chart.

Ex. Make a t-chart with at least 5 points, graph the relation, and decide whether it is a function or not.

1. Many of the equations we work with will have a domain (x-values) that is made up of all real numbers. So when we make a t-chart, we can usually pick any numbers we want and put them in the x-column.

y = 3x-1

t-chart:

2. Sometimes it’s easier to start with y-values.

1

2

+

=

y

x

t-chart:

For functions only:

The domain values (usually x-values) are represented in an equation by the _________________ variable.

The range values (usually y-values) are represented in an equation by the _________________ variable. (Because its values depend on the x).

Function notation-

Naming a function

The function depends on ______

Sometimes we say ______ is a function of _______.

We shorten it to __________. How do we say it out loud?

* This does not mean multiplying.

This new symbol takes the place of _____ in the equation of a function.

The equation y = 5x + 7, can be written as _____________________.

We can use this notation when making our t-charts, or finding out range (y-values) for a given x-value.

Ex. 1 f(x) = 5x+7

Find f(2) means that the function depends on x = 2. Plug in 2 for x and solve.

f(2)=

Try

f(-1)=

Ex. 2 g(x)=x2+1

g(1)=

g(0)=

Ex. 3

5

)

(

2

-

=

x

x

h

=

)

(

t

h

=

-

)

2

(

a

h

Homework Problems: p. 61 #17-33 odd, 46-53 all, 57, 58


2.2.b Compose functions when possible

2.2.c Add, subtract, multiply, and divide functions

2.2.e Identify the domain and range of a function resulting from the combination or composition of functions

ACT Warm up:

**Reading – p. 383 in text – Why is it important to combine functions in business?

If we have two functions f(x) and g(x), we are allowed to…

Add

(f + g)(x)=

Subtract(f – g)(x)=

Multiply

)

(

x

g

f

×

=

Divide

=

)

(

x

g

f

*denominator cannot be =0!!!

The domain and range of the new functions could be different than the originals..

Practice Problems

If f(x) = 3x2+7x and g(x) = 2x2-x-1

What are the domain and range for f(x) and g(x) (approximate if necessary)?

1. find (f+g)(x). Find the domain and range.

2. find (f-g)(x). Find the domain and range.

If f(x) = 3x2-2x+1 and g(x)=x-4

What are the domain and range for f(x) and g(x)?

3. find

)

)(

(

x

g

f

×

. Find the domain and range.

4. find

)

)(

(

x

g

f


**Reading – Compositions of Functions

Composite Functions

*Plugging a function into another function.

Notation:

How do you say this out loud?

*Order is important

Practice problems.

For f(x) = 3x2-x+4 and g(x)=2x-1

What are the domain and range for f(x) and g(x)?

1. Find

)

](

[

x

g

f

o


2. Find

)

](

[

x

f

g

o


3. Find

)

2

](

[

-

g

f

o

4. Find

)

2

](

[

-

f

g

o

Homework Problems. p. 387 #17-21 odd (determine domain and range for each), 29-41 odd

CDAS: Operations with Functions


2.2.d Determine whether or not a function has an inverse, and find the inverse when it exists

ACT Warm up:

Inverse functions

Notation

Domains and Ranges are switched.

Ex. f(x)

f-1(x)

X

Y

3

7

2

9

1

0

5

8

-3

3

X

Y

Finding the inverse of a function algebraically

*remember: the domain (x) and the range (y) are switched for inverse functions.

Ex.

()46

fxx

=+

Find the inverse.

Look at the top row of the t-chart:

x

4x

y = f(x) = 4x + 6

We want to undo the order of operations.

Write the answer in inverse notation:

Ex.

1

2

1

)

(

+

-

=

x

x

f

Find the inverse.

Ex.

6

()

3

x

fx

-

=

How to Verify if two functions are inverses.

*Use the composite function.

Two functions are inverses if and only if

Ex Determine whether the two functions are inverses.

6

4

3

)

(

-

=

x

x

f

8

3

4

)

(

+

=

x

x

g

**Exploration – Application of Inverse Functions

Homework Problems. p. 393 #14-36 even, 53, 55

Utah State Standards.

2.1.a Model real-world relationships with functions

ACT Warm up:

Linear Regression

**Supplemental Reading – Fitting a Line to Data

*All the equations come from sets of data (relations).

Ex. Finding an equation from data:

The median income of U.S. familes for the period 1970-1998.

Year

Income

1970

9867

1980

21,023

1985

27,735

1990

35,353

1995

40,611

1998

46,737

Rewrite this information as ordered pairs:

Why do we have these ordered pairs? They represent ____________ that we can study and use to make predictions.

This collection of data, or set of ordered pairs is called a ___________________ .

Plotting the points forms a __________________ plot.

How to put a scatter plot in your calculator.

· We will put our data into lists. So first, we need to clear the lists of anything that might already be in there!

Use the buttons

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INCLUDEPICTURE "../../../../AppData/Documents%20and%20Settings/CB1580/Local%20Settings/Temp/2nd.png" \* MERGEFORMAT


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Your screen should look like

Then hit .

Your screen will say “done”.

· To enter the data into columns, use the button

and select the Edit option1.

You should now have lists ready for entering data.

We will use L1 as our x-coordinates or first coordinate, and L2 as our y-coordinates.

Press a number you’d like to input and press enter. The cursor moves down the column. Use the arrow keys to get to move between columns.

Enter the data from the table above. Your screen should look as follows.

· To graph the data you have entered, hit the buttons

INCLUDEPICTURE "../../../../AppData/Documents%20and%20Settings/CB1580/Local%20Settings/Temp/Y=.png" \* MERGEFORMAT .

Your screen should look like this.

Hit the Enter button to enter Plot1. And hit enter when your cursor is sitting on the the word “On”.

We also want to double check that our list of x-coordinates (Xlist) comes from L1 and our y-coordinates (Ylist) comes from L2. *If it doesn’t, we can select those! Ask me!

If we hit the the button, our data would not show up in our calculator window. We would see a blank coordinate system.

We need to set the window size so we can see our information! Hit .

And now we need to think about our data.

Our x-values vary. The smallest number in L1 is ______

The greatest number in L1 is ______.

Our y-values vary. The smallest number in L2 is ______.

The greatest number in L2 is ______.

We’ll change our min and max x-values to correspond to our L1.

We’ll change our min and max y-values to correspond to L2.

Make your window look like the one at the left.

*Don’t touch the Xscl, Yscl, or Res.

Then hit the button. You should see a scatter plot of your data.

If we want to start again and create another plot, we need to clear out the values from out lists. Hit the buttons

INCLUDEPICTURE "../../../../AppData/Documents%20and%20Settings/CB1580/Local%20Settings/Temp/4.png" \* MERGEFORMAT . We now need to specify which lists we want to clear. Hit the buttons




INCLUDEPICTURE "../../../../AppData/Documents%20and%20Settings/CB1580/Local%20Settings/Temp/2.png" \* MERGEFORMAT to specify L1 and L2. Then hit enter. Your screen will say “done”.

Before we start an example, we need to tell our calculator how to round decimals.

Press . Next to the word “float”, select the #2. We are telling the calculator to round to 2 decimal

places. Then press and we are ready to roll.

Make sure your StatPlot is on. The graph of our data.

To let our calculator work for us, we use something called________________.

We’re going to tell our calculator to draw a line that best fits our data points.

This process is called ____________________________.

*We have to tell our calculator where to find the x-values, the y-values, and where it should store the equation it comes up with.

13

12

5

A

Use the

INCLUDEPICTURE "../../../../AppData/Documents%20and%20Settings/CB1580/Local%20Settings/Temp/RightArrow.png" \* MERGEFORMAT buttons

to get to this menu.

We will select #4 LinReg (ax+b).

LinReg stands for ________________________.

The following buttons will get you to the complete command for the calculator.






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INCLUDEPICTURE "../../../../AppData/Documents%20and%20Settings/CB1580/Local%20Settings/Temp/Enter.png" \* MERGEFORMAT

INCLUDEPICTURE "../../../../AppData/Documents%20and%20Settings/CB1580/Local%20Settings/Temp/Enter.png" \* MERGEFORMAT

1

Your screen should look like this.

Hit Enter twice.

The calculator will spit out the equation of a line that best fits the data.

y

x

y

x

f

x

y

x

y

+

=

+

£

£

³

)

,

(

1

2

6

1

We could write this as y= 1304.19x + -2560335.07

or f(x) = 1304.19x + -2560335.07

This equation describes the relationship between our data points.

What is x in our data?

What is f(x) or y in our data?

If we hit the button, we should see a line go through our points.

We can use the equation to make predictions!

For example, what will the average income of US families be in 2010?

First, we need to adjust our window, to make sure we our predicted year is in view.

Press

. We will be changing our Xmax value.

Hit the buttons

INCLUDEPICTURE "../../../../AppData/Documents%20and%20Settings/CB1580/Local%20Settings/Temp/Trace.png" \* MERGEFORMAT to get the following menu.

We will select #1, the value option. We should see the following screen.

x

y

We can input any year and hit enter.

We want to find the income in 2010.Enter 2010 and hit enter.

x

y

We get the following screen. We are interested in what y = at the bottom of the screen.

Average income in 2010 ________

Summary: data is used to make an equation.

Equations can be used to make predictions.

*Our information is only as good as the fit of our line! Meaning, our line should go through the points on our graphs.

Homework Problems p.88#1-15

CDAS: Relations and Functions

Utah State Standards.

2.1.a Model real-world relationships with functions

ACT Warm up:

Linear Function Review

**Exploration – 1) Lines in Motion & 2) Activity 1 – Calorie Consumption

We used our calculators to do linear regression- find the equation of a ____________.

Determine whether each of the following equations is linear or not. Explain your reasoning.

2

22

)24)68)421

)384)3412)7

)49)80)234

11

)2)124)38

24

)6430)28)628

1

)121)30)2

4

axybycxy

dxyyexfyx

gyxhxixy

jxykyxlxyy

mxynyxoabb

pxyqxxrxxy

=+=-=-

+=-==+

-=+=-+=

+==--=

+-=-=-=+

-=++==

What similarities do you see with the linear equations?

What differences do you see?

Linear equation

Can only contain __________, __________, ___________, _____________

No powers on the variable other than ________ .

Variables cannot appear in a _______________.

p. 65 #4-5 Are the equations linear?

4. x2 + y2=4

5. h(x)=1.1 - 2x

Slope-Intercept form-

Formula

m

b

coordinates of y-intercept are actually (____,____)

*that’s a point on the graph of the line!

Point Slope form-

Need coordinates of any point on the line and the slope!

Standard Form

No fractions!

No common factors!

Intercepts:

x-intercept:

y-intercept:

Ex. 2x – 3y = 6

Find the y-intercept. (0,b)

If the equation is in slope-intercept form, the y-intercept is ______.

If not, plug in 0 for x, and solve for y.

Why can we do this?

Find the x-intercept. (#,0)

Plug in 0 for y, and solve for x.

What type of equation is in our example?

Graph the equation.

Find the x and y- intercepts for the following equations:

a. x – 5y = 10

b. 8x – y = 16

Constant Function

f(x) = ____________

ex. f(x) = 12

Slope = ______

Graph: __________ line

ex. f(x) = 12 or y = 12

Homework Problems. p. 66 #15-23 odd, 39-47 odds 56, 71, 72


1.1.c Solve absolute value and compound inequalities of a single variable

ACT Warm up:

Inequalities

**Exploration – Quality Control

Properties of Inequality

Addition and Subtraction Properties

Multiplication and Division Properties

Solving an Inequality

· Isolate the variable using inequality properties

Ex. Solving an Inequality

1. Solve

2

5

3

4

+

<

-

y

y

. Report your answer and then graph the solution set on a number line.

*Three different ways to report your solutions!

A graph on a number line

Open dot-

Closed dot-

Set-builder notation-

Interval notation

Use of

¥

±

Use of ( vs. [

Ex. Solve

p

10

3

12

-

³

. Report your answers and graph the solution set on a number line.

Ex. 3.Solve

7

2

-

>

-

x

x

. Report your answers and graph the solution set on a number line.

Keep your eyes open for these tricks!

*When you are a solving an inequality, the variable disappears, and you arrive at a false statement, like 2>8, then the solution set for that inequality is the empty set

f

* When you are a solving an inequality, the variable disappears, and you arrive at a true statement, such as 5<9, then the solution set for that inequality is the set of all real numbers.

Key phrases

Less than, fewer than

Greater than, more than

At most, no more than, less than or equal to, maximum

At least, no less than, greater than or equal to, minimum

Ex. Writing and solving an equality

p.36ex4. Craig is delivering boxes of paper to each floor of an office building. Each box weighs 64 pounds, and Craig weighs 160 pounds. If the maximum capacity of the elevator is 2000 pounds, how many boxes can Craig safely take on each elevator trip?

2. Alida has at most $15.50 to spend at a convenience store. She buys a bag of potato chips and a can of soda for $1.55. If gasoline at this store costs $2.85 per gallon, how many gallons of gasoline can Alida buy for her car, to the nearest tenth of a gallon?

Homework problems: p. 37 #17, 27, 31, 35, 37, 39, 41, 43, 48, 49, 59, 69


1.1.c Solve absolute value and compound inequalities of a single variable

ACT Warm up:

Compound Inequality-

*You must solve each part of the inequality!

Intersection of the solution sets of the two inequalities-

*A compound inequality containing the word and is true if and only if both inequalities are true. (where they overlap)

Ex. Solving an “And” compound inequality.

Solve

19

2

3

10

<

-

£

y

. Graph the solution set on a number line. Report solutions.

* this doesn’t have the word “and”. What should we do?

Union of the solution sets of the two inequalities-

*A compound inequality containing the word or is true if one or more of the inequalities is true. (any covered areas).

Ex. Solving an “or” compound inequality

Solve

2

3

<

+

x

or

4

-

£

-

x

. Graph the solution set on a number line.

Report your answer in interval notation as well.

Homework Problems: p.44 #8, 9, 27 – 32 all, 45 – 47 all, 51, 52


1.2.a Solve systems of linear, absolute value equations algebraically and graphically

ACT Warm up:

**Exploration – Systems of Linear Inequalities

**Reading – Linear Inequalities

The graph of a linear inequality is a ______________ region.

This region is made up of a gazillion ____________.

Every point in the shaded region makes the inequality “true”.

To graph the inequality, follow these steps:

1. Determine whether you will use a dashed or solid line.

If we have or , use a dashed line.

If we have or , use a solid line.

2. Change the inequality sign to an = sign, and graph the line.

3. Choose ANY POINT that does not lie on the line you just graphed.

Write down the coordinates.

Plug the coordinates into the original inequality.

4. Determine where to shade your graph.

If the test point made a true inequality, shade the region of the graph that contains your point.

If the test point made a false inequality, shade the region that does NOT contain your point.

Ex. Graphing Linear Inequality

Graph x – 2y < 4

Ex. Graph 4x – y > 8

Homework: p.98 #13 – 23 odd, 35 – 37 all



ACT Warm up:

**Exploration – Linear Systems

**Reading – System of Linear Equations

Warm Up! Graph the two equations on the same coordinate system.

2x + y = 5

x – y = 1

A _____________ of equations is two or more equations with the same variables.

A _____________ to a system of equations is the point or points (x, y) where the lines _______________. (The point or points whose coordinates “work” in all equations.)

In the above warm-up exercise, what is the solution to the system of equations?

What are the possible ways for lines to intersect?

1. Consistent Independent System

2. Consistent Dependent System

3. Inconsistent System

There are different ways to find the solution to a system of equations:

1. By graphing and finding the point(s) of ______________.

2. Algebraically

3. Using matrices (next year!)

Solving systems by graphing

Graph each equation individually on the same coordinate system.

Try to find the coordinates of the point where the lines cross.

*This will not always be easy, accurate, or possible.

Ex. Solving systems by graphing.

Solve the system of equations and determine what kind of system you have.

1. x - 2y = 0

x + y = 6

2. 9x – 6y = -6

6x – 4y = -4

3.

0

6

15

=

-

y

x

5x – 2y = 10

Solving systems algebraically (using the equations, not the graphs).

Substitution Method

Take one equation and solve for x or y. *Whatever is easier!

Substitute your new expression into the second equation.

You will only have one variable, so…You can solve!

Find the second unknown by plugging your answer into an equation.

Ex. Using the Substitution Method

Solve the system using the substitution method.

x + 4y = 26

x – 5y = -10

Practice: Solve the system using the substitution method.

2x – y = 13

3x + y = 2

Elimination Method

We will add or subtract our equation together to eliminate one variable.

Why is this legal?

Solve for the remaining variable

Using the value you found, plug into an equation and find the other variable.

Ex. Using the Elimination Method

Solve the system using the elimination method.

1. x + 2y = 10

x + y = 6

2. 2x + 3y = 12

5x – 2y = 11

*Weird situations when solving algebraically…

If all your variables cancel out and you get a true equation

Ex. 3 =3

Then your system is Consistent Dependent.

What does that mean?

If all your variables cancel out and you get a false equation

Ex. 0 = 19

Then your system is Inconsistent.

What does that mean?

Ex. Weird system of equations

Solve the system of equations algebraically.

-3x + 5y = 12

6x – 10y = -21

**Exploration – Substitution & Elimination

Homework Problems p. 113 #13, 15, 25, 27, 44

p. 120 #13, 19, 25, 29, 34, 37, 41, 60, 65



ACT Warm up:

Systems of Inequalities-

**Exploration – Under the Big Top but Above the Floor

Solutions to systems of inequalities- all the points that make the inequalities true.

(the area where the shaded solution regions _____________.)

If the shaded regions do NOT ______________, then there is ________________.

You must solve these systems on a graph.

Graph each inequality separately. Determine where the shading overlaps.

Ex. Solving systems of inequalities

1.

2

3

2

+

-

<

-

³

x

y

x

y

2.

2

4

3

1

4

3

-

-

£

+

-

³

x

y

x

y

3.

1

2

-

³

-

y

x

4

£

+

y

x

4

4

³

+

y

x

In example 3, we had a shaded region that was a polygon. It had corners or __________.

*These corners are important points!!!

How can we find the coordinates of those points?

We solve little _______________ of equations.

Let’s find the corners/vertices in example 3.

Homework Problems. Worksheet p.133-134


1.2.c Solve application problems involving systems of equations and inequalities

ACT Warm up:

Linear programming

**Reading – Feasible Region

**Exploration – Maximizing Profit

We can use a system of linear inequalities to find maximum and minimum values of real-world situations.

Steps to solve linear programming problems

1. Define the variables

2. Write a system of inequalities.

3. Graph the system of inequalities.

4. Find the coordinates of the vertices of the feasible region.

5. Write (use) a function to be maximized or minimized.

6. Substitute the coordinates of the vertices into the function.

7. Select the greatest or least result. Answer the problem.

*Important terms:

Constraints-

Feasible region

Bounded

Unbounded

Vertices

*the maximum or minimum value of a related function always occurs at one of the vertices.

Ex. Linear Programming

1.

8

6

4

2

-2

-4

-6

-8

-15-10-551015

Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region.

4

3

2

1

-1

-2

-3

-4

-6-4-2246

A

2. Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region.

x

y

±

=

x

y

y

x

f

x

y

x

y

y

x

+

=

+

£

-

³

³

+

3

)

,

(

8

4

6

3

2

4

2

2

3. A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the company receives per day for one of its crews.

4. As a receptionist for a vet, one of Dolores Alvarez’s tasks is to schedule appointments. She allots 20 minutes for a routine visit and 40 minutes for a surgery. The vet cannot do more than 6 surgeries per day. The office has 7 hours available for appointments. If an office visit costs $55 and most surgeries cost $125, find a combination of office visits and surgeries that will maximize the income the vet practice receives per day.

Homework Problems: Worksheet p.139-140

CDAS Systems of Equations


1.1.a Solve and graph first-degree absolute value equations of a single variable

ACT Warm up:

Absolute Value-

**Reading – p. 28 in text

*always nonnegative

Symbol for absolute value of a number

Ex.

3

-

=

Ex.

=

7

.

5

If a number _____ is positive or 0, the absolute value of that number is ______.

If a number _____ is negative, the absolute value of that number is _______.

*The absolute value bars act like parentheses. Remember order of operations!

Ex. Evaluating an expression with absolute value

1. Evaluate

7

8

5

+

-

2. Evaluate

x

2

6

3

-

+

if x = 4.

Look at

4

=

x

. What are two possible values for x?

Look at |x – 5| = 8. What are two possible values for x?

Solving absolute value equations

1. Isolate the Absolute Value.

2. check that the an absolute value is equal to a number greater than or equal to zero. Why?

*If it is equal to a number less than zero, there is NO SOLUTION!

The solution set is the empty set

Symbolized by

3. absolute value has two different cases to worry about.

If

b

a

=

then

b

a

=

or

b

a

=

-

. Two cases to solve!

Notice that the two new cases do not have absolute value anymore!

4. check answers. Sometimes we get an answer that is not an actual solution.

5. Report our solutions in a solution set. Use { } and list your solution(s) inside.

Ex. Solving an absolute value equation.

1. Solve for y and check your answers.

8

3

=

+

y

2. Solve for t and check your answers.

0

5

4

6

=

+

-

t

3. Solve for z and check your answers.

3

2

8

-

=

+

z

z

Homework Problems: p. 30-31 #21-27 odd, 33-43odd, 53


1.1c Solve absolute value and compound inequalities of a single variable

ACT Warm up:

Absolute Value Inequalities

**Reading – p. 42 in text, examples 3 & 4

There will be two cases to look at for each problem.

*Remember: Absolute value means distance from zero on a number line.

Let’s say

b

a

<

(the absolute value is less than a positive number)

Graph :

For this inequality,

ab

>-

_________

ab

<

.

< is an ______________________.

Ex. Solving a “less than” Absolute Value inequality

Solve. Graph the solution set on a number line.

5

1

3

<

+

x

Let’s say

b

a

>

(the absolute value is greater than a number)

Graph:

For this inequality

ab

<-

________

ab

>

> is a __________________.

Ex. Solving a “Greater Than” Absolute Value Inequality

Solve. Graph the solution set on a number line.

4

2

2

³

-

x

Practice: p.44 #16, 18

Ex. Solve and graph the solution set on a number line.

2137

x

-+>

*Things to watch out for

The solution of an inequality like

5

-

<

a

is the empty set. Why?

The solution of an inequality like

5

-

>

a

is the set of all real numbers. Why?

Homework Problems. P.44 #15-19 odd, 33 – 39 odd, 40, 43

CDAS: Absolute Value


3.1.a Identify the domain and range of the absolute value function

3.1.b Graph the absolute value function

3.1.c Graph functions using transformations of parent functions

ACT Warm up:

The Absolute Value Function

f(x)= ________

Let’s make a t-chart

Basic Shape of the Absolute Value Function _______________

Domain:

Range:

Transformations of the absolute value function.

Investigation

Make a sketch of each function using T-charts.

2

)

(

+

=

x

x

f

3

)

(

-

=

x

x

f

What happens to the v-shape when we add or subtract a number outside the absolute value bars?

4

)

(

-

=

x

x

f

1

)

(

+

=

x

x

f

What happens to the v-shape when we add or subtract a number inside the absolute value bars?

x

x

f

5

)

(

=

x

x

f

8

1

)

(

=

What happens to the v-shape when we multiply x by a number?

x

x

f

-

=

)

(

What happens when we multiply by a negative number outside

of the absolute value signs?

What do you think the inverse of the Absolute Value would look like? Why? Is the inverse a function?

Homework Problems. p. 94 #30-36 all, #56-57


1.2a Solve systems of linear, absolute value equations algebraically and graphically.

ACT Warm up:

Graphing an Absolute Value Inequality

Follow the same steps as for linear inequalities.

1. Determine whether you will have a dashed or solid line.

2. Replace the inequality with an = sign and graph the equation.

3. Choose a test point that is NOT an the graph you just drew, and plug into the original inequality.

4. Determine where to shade your graph.

Ex. Graphing an Absolute Value Inequality

Graph

2

-

³

x

y

Dashed or solid line?

Rewrite the inequality with an = sign.

And graph!

Choose a test point NOT on the line.

Plug into the inequality

2

-

³

x

y

Determine shading: is the above true or false?

Practice: p.98 #27, 28

Homework Problems: Worksheets p. 95-96


1.2a Solve systems of linear, absolute value equations algebraically and graphically

ACT Warm up:

Systems of Absolute Value equations

Graph each equation

The intersection of the v-shapes represents the solution(s).

Ex. Find the solution(s) to the system of equations.

4

2

+

-

=

+

=

x

y

x

y


2

2

=

-

=

y

x

y


2

3

+

=

+

=

x

y

x

y

Systems of Inequalies

Solve the following systems graphically:

1.

2

2

>

+

+

>

y

x

x

y

Where do the maximum and minimum

for this system occur?

2.

x

y

x

y

>

+

<

3

Homework: p. 833, Lesson 3-3 #1-12all

Project – Pictures


1.1.e Simplify algebraic expressions involving negative and rational exponents

1.3.a Simplify numerical expressions, including those with rational exponents

ACT Warm up:

**Exploration – Power Equations

Monomial

*Cannot have variables in denominators

*Cannot have variables with negative exponents

*Cannot have variables under radicals

Examples

Coefficient

Degree

Ex.

Constants

Ex.

Degree

**Exploration – Properties of Exponents

Negative exponents

Ex.1. x-3

Ex.2.

4

1

-

x

To simplify an expression: rewrite the expression without parentheses or negative exponents.

Product of Powers

For any real number a and integers m and n, am∙an = _______.

Ex. x5∙x3

Quotient of Powers

For any real number a (not equal to 0) and integers m and n,

=

n

m

a

a

______.

Ex.

3

7

r

r

=

Practice Problems. Simplifying expressions.

1. (-2a3b)(-5ab4)

2.

10

2

d

d

*Remember: any number (except 0) raised to the power 0 is equal to 1

a0 = 1

00 = undefined

Properties of Powers (exponents)

1. Power of a Power (am)n = ______

Ex.

2. Power of a Product (ab)m = ______

Ex.

3. Power of a Quotient

=

n

b

a

)

(

______

Ex.

Practice Problems. Simplifying expressions

1. (b2)4

2. (-3c2d5)3

3.

5

2

)

2

(

b

a

-

4.

4

)

3

(

-

x

Homework Problems: p. 226 #19-41 odd, 60




ACT Warm up:

Square Root-

*The square root is the inverse operation of squaring a number

Ex.

Nth Root-

Ex.

Symbol

Index

Radical sign

Radicand

Notation

=

-

=

-

=

±

=

-

=

4

3

81

8

16

16

16

Summary of Real nth Roots

N

nth root if b > 0

nth root if b < 0

b = 0

Even number

One positive root, one negative root

No real roots

One real root = 0

Odd number

One positive root, no negative roots

No positive roots, one negative root

One real root = 0

Practice Problems. Finding roots.

1.

64

2.

3

3

5

**Reading – p. 250 in text – Simplify Radical Expressions

Product Property of Radicals

For any integer n > 1,

n

n

n

b

a

ab

×

=

*If n is even, both a and b must be nonnegative.

Ex.

=

2

2

2

9

d

a

Ex.

=

3

3

8

b

More Practice. Finding Roots.

1.

4

25

x

±

2.

8

2

)

2

(

+

-

y

3.

5

20

15

32

y

x

4.

9

-

To get an approximate answer, you may use a calculator to simplify radicals.

To take the square root of a number, press

and type in your number. CLOSE THE PARENTHESES! Hit enter.

To take the nth root of a number, type in the index number.

Hit the

button and select #5.

Then type in the radicand. Hit enter.

Homework Problems: p.248 #17-53 odd, 58,59,62,75




ACT Warm up:

**Reading – p. 251 in text (top of page)

Quotient Property of Radicals

For any integer n > 1, and for any real numbers a and

0

¹

b

,

n

n

n

b

a

b

a

=

Ex.

=

3

27

Summary of Simplifying Radicals

1. The index n is as small as possible

2. The radicand contains no factors (other than 1) that are the nth powers of an integer or polynomial

3. The radicand contains no fractions

4. No radicals appear in denominator

To get radicals out of denominator… you “rationalize” the denominator.

*multiply the numerator and denominator by a quantity so that the radicand has an exact root.

Ex.

x

1

Ex.

5

4

y

x

Ex.

5

4

5

a

Ex.

3

9

2

x

Operations with Radicals

*We use Multiplication and Quotient properties to work with and combine radicals.

Ex.

3

3

2

24

3

9

6

n

n

×

Ex.

3

3

2

10

100

5

a

a

×

*Does addition work the same way? Does

?

25

9

16

=

+

Addition and Subtraction only work with like radical expressions.

(indices AND radicands are alike)

Ex.

4

3

2

a

and

4

3

5

a

are like expressions and can be added.

=

4

3

7

a

Ex. of UNLIKE expressions… cannot be added.

5

3

,

3

4

4

5

,

5

x

Practice Simplifying.

1.

20

4

80

5

45

3

+

-

2.

)

3

2

)(

3

2

5

3

(

+

-

3.

)

7

2

4

)(

7

2

4

(

-

+

Conjugates- pairs of binomials of the form

d

c

b

a

+

and

d

c

b

a

-

.

Ex.

What happens when you multiply a pair of conjugates?

(what happened in example 3 on the previous page?)

(What happens with example above?)

*The product of a conjugate pair is always a ___________________.

*We can use conjugates to rationalize ugly denominators.

Ex.

3

4

3

2

-

+

Ex.

5

3

5

1

-

+

Homework Problems: p. 254 #15-45 odd, 57, 61, 79




ACT Warm up:

**Exploration – Rational Exponents and Roots

Rational Exponents- exponents that are fractions

Think about this…

2

1

2

1

2

2

1

)

(

x

x

x

×

=

=

For any real number b and for any positive integer n,

n

n

b

b

=

1

Except when b<0 and n is even.

Ex.

=

3

1

8

Ex.

=

6

1

m

Ex.

=

5

b

Ex.

w

Ex.

=

2

1

49

Ex.

2

1

25

-

For any nonzero real number b and any integers m and n, with n>1,

m

n

n

m

n

m

b

b

b

)

(

=

=

except when b < 0 and n is even.

Ex.

5

3

243

=

Ex.

5

2

32

*We can use these properties of powers to help us with any kind of rational exponent expressions.

Ex.

7

4

7

1

y

y

×

Ex.

3

2

-

x

*If a problem is originally written with radicals, write your final answer with radicals.

*If a problem is originally written with rational exponents, write your final answer with rational exponents.

Practice Problems

1.

3

6

2

16

2.

6

4

4

x

3.

1

1

2

1

2

1

-

+

y

y

Summary

No negative exponents!

No fractional exponents in the denominator!

The index of a remaining radical must be as small as possible.

No complex fractions!

Homework Problems: p. 261 #21-61 odd, 76, 81-82

CDAS: Square Roots and Rational Exponents

**Project – Powers of 10


1.1.b Solve radical equations of a single variable, including those with extraneous roots

ACT Warm up:

**Reading – p. 263 in text – How do radical equations apply to manufacturing?

Radical equations- equations that have variables under a radical!

To solve:

Isolate the more complicated radical

Raise each side of the equation to a power equal to the index of the radical (to eliminate the radical!)

Check your solutions: there may be extraneous solutions (a number that doesn’t work in the original equation)

Ex.

9

7

=

+

x

Ex.

4

2

1

=

+

+

x

Ex.

x

x

-

=

-

3

15

Ex.

0

5

1

3

3

=

+

+

y

Ex.

0

2

)

1

5

(

3

3

1

=

-

-

n

**Explore solutions on the calculator – See p. 264 in text

Homework Problems p. 266 #13-27 odd, 45, 59

CDAS: Radical Equations

Utah State Standard

1.3.b Simplify expressions involving complex numbers and express them in standard form, a+bi

ACT Warm up:

** Reading – History of Complex Numbers & Applications of Complex Numbers

A new number in our lives… i .

When we look at the problem x2= -1, is there any real number that would fit or solve this equation? Why or why not?

__________________________ decided to define a new kind of number that would be a solution to this equation.

He called the number i. i is the principal square root of -1.

i =

It is called the imaginary unit.

i2 =

Pure imaginary numbers- square roots of negative real numbers.

Ex. 3i, -5i, and

2

i

all came from taking the square root of a negative real number.

9

-

25

-

2

-

We can use the commutative and associative properties of multiplication to help us simplify pure imaginary numbers.

Ex.

i

i

7

2

×

-

Ex.

15

10

-

×

-

Take the i’s out first!

To simplify big powers of i, we have to think a bit.

Ex. i8

remember i2 = -1

Ex. i35

Complex numbers- an expression with both a real number and a pure imaginary number (that cannot be combined).

Ex. 7 + 4i and 2 – 6i and 5 + (-3)i

Standard Form of complex numbers: a + bi

a is the real part and

b is the imaginary part.

What happens if b = 0 ? For example, 5 + 0i

Two complex numbers are equal

If and only if their real parts are equal and their imaginary parts are equal.

Ex. Find the values of g and k that would make this equation true.

i

ki

g

17

5

+

=

+

Ex. Find the values of x and y that would make this equation true.

i

i

y

x

2

3

)

4

(

3

2

+

=

-

+

-

Add and Subtract Complex Numbers

Combine the real parts and combine the imaginary parts.

Ex. (3+5i) + (2-4i)

Ex. (4-6i) – (3-7i)

Multiply Complex Numbers

Use FOIL. (distribute)

Ex. (1+3i)(7-5i)

*(electrical engineers use this kind of number when calculating current and voltage. They use j instead of i to avoid confusion with I used for current)

Complex Conjugates: a + bi and a - bi

The product of complex conjugates is always ____________________

Ex. (5 + 3i)(5 – 3i)

Divide Complex Numbers

We use the idea of the product of complex conjugates to help!

Make sure to put answers in standard complex form.

Ex.

i

i

4

2

3

+

Ex.

i

i

2

5

+

Homework Problems p. 274 #19-41 odd, 49, 55, 62, 73

CDAS: Complex Numbers


1. 4.a Model real-world situations using quadratic equations

ACT Warm up:

Quadratic Regression- Modeling Real-World Data

Practice problems p.300

Refresher on Calculator steps for regression.

Things to do first!

and and turn on Stat Plot 1.

Hit and clear any equations typed in.

Hit and clear list. You must then type

,

to name List 1 and List 2. Hit enter until your screen says “done”.

To enter data:

Hit and select Edit. You will see empty lists. List 1 will be your x-values. List 2 will be your y-values.

To see the graph of your data:

You must change the window values (remember to look at your lowest and highest x-values, and lowest and highest y-values).

Hit graph.

To find the linear regression equation:

Hit and move to the 2nd column “Calc”.

Select #4 “LinReg(ax+b)”.

You need to type in L1, L2, and

to select the Y1 variable.

When your screen looks like this, Hit enter.

Write down your linear regression equation.

Hit graph, to see how the new line fits your data.

To find the quadratic regression equation:

Hit and move to the 2nd column “Calc”.

Select #5 “Quad Reg”.

You need to type in L1, L2, and

and select the Y2 variable. Hit .

Write down your quadratic regression equation.

Hit graph, to see how the new curve fits your data. (You should still see your linear regression equation on the screen).

Compare which line/curve fits your data better.

To make predictions, remove (clear) the equation that does NOT fit best.

(We will keep the quadratic regressions for these exercises).

Hit the and select #1, Value. Type in the number for which you are making a prediction. (if you have a problem, make sure your window x-values include the number you are typing)

Hit enter to receive the prediction value.

*Before beginning the next problem, make sure to clear your lists and clear out your equations in y = .

Homework Problems: Quadratic Regression Worksheet


1.4.b Approximate the real solutions of quadratic equations graphically

1.4.c Solve quadratic equation of a single variable over the set of complex numbers by factoring, completing the square, and using the quadratic formula

1.4.e Write a quadratic equation when given the solutions of the equation

ACT Warm up:

Solutions to a quadratic equation- when a quadratic function is set equal to __________.

Ex.

Solutions

Roots

Zeros

Ex.

Real Solutions of a Quadratic Equation

One real solution

Two real solutions

No real solutions

Factoring- (the reverse of multiplying or distributing or FOILing.) Writing the expression as a product.

Step 1. Pull out the Greatest Common Factor of all terms.

Ex. 8x2+4

Ex. 12x2 + 9x – 3

Step 2. Look at how many terms are in the expression.

*If there are two terms, see if you have the _______________________

a2 – b2 =

Ex. x2- 144

Ex. 16x2 – 9

Ex. x2 -

4

1

Practice with grouping

Ex.

22

3926 5x-50-6x60

xxxx

++++

*If there are three terms (trinomials)

*Sometimes, it is not possible to factor!

Ex. x2 + 10x + 16

Ex. x2 – 12x + 35

Ex. 3x2 – x – 4

Ex. 2x2 + 7x + 3

Special trinomials: Perfect Square Trinomials

Ex. x2 + 10x + 25

Ex. x2 – 4x + 4

Zero Product Property

We use the zero product property to help us solve quadratic equations.

First, make sure everything is to one side of the equation (leave 0 on one side).

Factor.

Set each factor = 0 and solve.

Ex. Solve by factoring.

x2 = 6x

2x2 + 7x= 15

x2 – 16x + 64 = 0

Given the roots, write a quadratic equation in the form ax2 + bx + c = 0

Remember:

x- root = 0 x – root = 0

Put them together as a product!

And distribute!

Ex. Given the roots -4 and 7, write a quadratic equation (with a, b, and c as integers!)

Ex. Given the roots

2

1

and

3

4

, write a quadratic equation (with a, b, and c as integers!)

*make sure a,b, and c are integers.

**Application Problems

Homework Problems p. 297 # 14-19

Homework Problems. Factor Worksheet, p. 304 #15-31 odd, 35-41 odd, 57, 59


1.4.c Solve quadratic equation of a single variable over the set of complex numbers by factoring, completing the square, and using the quadratic formula

ACT Warm up:

Square Root Property

For any real number n, if x2 = n, then x =

n

±

.

Examples. Solve using the square root property.

1. x2 = 36

2. (x+2)2 = 242

*This Square Root Property only works when the side with the quadratic expression is a perfect square trinomial. Rare!

We can make any quadratic expression a perfect square, by a process called ________________________________.

Let’s look at a perfect square trinomial.

(x + 7)2 =

**Exploration – Complete the Square

To complete the square for any quadratic expression of the form x2 + bx

*The coefficient of x2 must be a 1!!!

1. Find one half of b. (the coefficient of x)

2. Square the result of step 1.

3. Add the result of step 2 to the original expression.

Ex. Complete the square to make a perfect square trinomial.

x2 + 12x

How does this help solve an equation?

First, create the form x2 + bx on one side of the equation.

Complete the square.

Remember: whatever you add to one side, you must add to the other.

Write the perfect square.

Square root each side.

Solve for x.

*We can use completing the square to solve any quadratic equation.

Ex. Solve by completing the square. x2 + 8x – 20 = 0

Ex. Solve by completing the square. 3x2 -2x – 1 = 0

Ex. Solve by completing the square. x2 +2x + 3 = 0.

Complete the square for the equation ax2 + bx + c = 0.

We have derived the ____________________________.

Formula:

*this formula can be used to solve any quadratic equation.

Ex. Solve using the quadratic formula. x2 - 8x -33 = 0

a= b= c=

Sketch.

Ex. Solve using the quadratic formula. x2 + 22x +121 = 0

a= b= c=

Sketch.

Ex. Solve using the quadratic formula. 2x2 + 4x -5 = 0

a= b= c=

Sketch.

Ex. Solve using the quadratic formula. x2 - 4x = -13

a= b= c=

Sketch.

Remember: We can solve quadratic equation a number of ways.

Graphing (finding the x-intercepts) and factoring only work SOME of the time.

Completing the Square and the Quadratic Formula work ALL of the time.

**Exploration – How High Can You Go?

Homework Problems. p. 311 #33-47 odd, 57, 58

Homework Problems. p. 318 #17-23 parts b and c odd, 29-35 odd


3.1.b Graph the quadratic function.

3.1.d Write an equation of a parabola in the form

k

h

x

a

y

+

-

=

2

)

(

when given a graph or an equation

ACT Warm up:

Vertex Form of a Parabola

y = a(x – h)2 + k

Just a different form of the equation y = ax2 + bx + c

(h, k) are the coordinates of the __________________

If a > 0, the parabola

If a < 0, the parabola

Given an equation in vertex form y = a(x – h)2 + k, we can get a rough graph pretty easily.

Example. y = -2(x-2)2 + 3

How do we write an equation in vertex form, given y = ax2 + bx + c.

We will use the process of Completing the Square.

Ex. y = 3x2 + 24x + 50

What is the vertex, axis of symmetry, and direction of opening of the parabola?

Ex. y = -x2 – 2x + 3

Put in Vertex Form. What is the vertex, axis of symmetry, and direction of opening of the parabola?

How can we find an equation in vertex form, when given a graph?

Identify the vertex:

(h, k)

Identify one other point on the parabola. (x,y)

Plug in all the values into y = a(x – h)2 + k.

Find a.

Rewrite y = a(x – h)2 + k, plugging in only a, h, and k.

Ex.

Homework problems. p. 424 # 12, 22, 26. For each problem, write in vertex form, identify the vertex, axis of symmetry, and direction of opening for the parabola.

p.326 #39, 41, 43

Worksheet. Using graphs to find equation in vertex form.


3.1.c Graph functions using transformations of parent functions.

ACT Warm up:

Transformations of a quadratic:

Think of an equation of a parabola in vertex form. y = a(x – h)2 + k.

Analyzing a change in k.

Graph each set of equations:

y = x2

y = x2 + 3 y = x2 - 5

Sketch:

What does a change in k do to the graph of a parabola?

Analyzing a change in h.


y = x2

y = (x+3)2 y = (x-5)2

Sketch:

What does a change in h do to the graph of a parabola?

Analyzing a change in a.


y = x2

y = -x2

Sketch:


y = x2

y = 4x2 y =

x

4

1

2

Sketch:

What does a change in a do to the graph of a parabola?

Homework Problems. p. 321 #1-15

CDAS: Solving Quadratic Equations


3.1.a Identify the domain and range of the absolute value, quadratic, radical, sine, and cosine functions

3.1.b Graph the quadratic function.

ACT Warm up:

Quadratic Function

**Reading – p. 286 in text – How can income from a rock concert be maximized?

Graph of a quadratic function:

Basic shape:

*We can graph a quadratic function by making a t-chart! Or we can learn more and save ourselves some time!

Axis of symmetry: x =

*this is a vertical line! ** Reading p. 287 in text

Vertex:

x-coordinate of vertex:

How can we find the y-coordinate of the vertex?

y-intercept-

In a quadratic function, the y-intercept is simply ______.

All these pieces of information can help us graph a parabola.

And we can make a small t-chart or use symmetry to help us make a smooth curve.

Ex. Graphing a quadratic function.Consider the function f(x)=2 - 4x + x2

1. Find the y-intercept, the equation of the axis of symmetry, and the coordinates of the vertex.

2. Use this information to graph the function.

Maximum and minimum values of a quadratic function.

The _______________ of the vertex is the max or min value.

The graph of f(x) = ax2 + bx + c

Opens up when

Diagram

The minimum value is the y-coordinate of the vertex.

Opens down when

Diagram

The maximum value is the y-coordinate of the vertex.

*A quadratic function does not have both a maximum and minimum value.

Example 4, p.289 – 290 in text

Ex. Find max or min of a quadratic function.

Consider the function f(x) = -x2+2x+3

Determine whether the function has a maximum or minimum value and find that value!

With the information of max and min, we can discuss domain and range of quadratic functions.

Domain (possible x-values)-

Range (possible y-values)-

What do the max/min values and y-intercept mean in our real-world data?

Homework Problems p. 291 #19-27 odd (find the domain and range of each function too!),

33-43 odd, 44, 45, 64, 68


1.2.a Solve systems of quadratic equations algebraically and graphically

1.2.b Graph the solutions of systems of quadratic inequalities on the coordinate plane

1.4.d Solve quadratic inequalities of a single variable.

ACT Warm up:

Systems of quadratic equations

To solve, find the point(s) where the parabolas or lines intersect.

Algebraically

Use the substitution method to solve.

Ex.

2

2

5

2

2

+

=

-

=

x

y

x

y

Ex.

1

3

2

-

=

-

=

x

y

x

y

Quadratic inequalities of one variable

To solve means to find all the real numbers that make the inequality true.

How to solve.

Ex. x2 + x > 6

1. First solve the related equation.

2. Plot the solutions to #1 on a number line.

3. Test a value in each interval in the original inequality.

4. Summarize results.

Ex. Solve x2 + x

£

2

Quadratic inequalities of two variables.

To solve means to find all the points (x,y) that make the inequality true.

How to solve.

Ex. y > -x2 -