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Education is Power!

Objective: Prerequisite ReviewDate: 9/9/14 Bell Ringer:

•What is Calculus and why study it? •Http://www.youtube.com/watch?v=ismnD_QHKkQ• Homework Requests: •Rational expressions - Check solutions•Absolute Values

• Memory Sheet• In class: Rational Expressions (Tricks)• Complex Fractions• Homework:• Finish Worksheets Rational Expressions (Tricks), Complete Complex Fractions WS

Syllabus online Parent sign in sheet due Tues. 9/9If needed, bring in calculator loan contract 9/9 (online)Bring Calculator Register on morganparkcps.org due Tues. 9/9Bring in $15.00 fee by FRIDAY

• Announcements: • Monday Day for Mandatory Session

• Periodic Functions (Unit Circle)• Domain and Range

Get Email AddressesFind the domain of each function.Use interval and set builder notation

1.

2.

Objective

• Simplify complex fractions• Lets Review fraction rules first…………..

Complex fractions

Tips for finding the domain algebraically.

1. Before any restrictions are considered, the domain is (-∞,∞). Start narrowing the domain from this point with any restrictions.

2. If there is a radical ( ) in your equation, then the argument under the radical must be greater than zero. arg 4x – 8 x ∞)

3. If you have a rational expression (fraction) , then bFind out what makes the denominator, b= 0, and restrictthis from the domain.

+ 2 D = (-∞,∞)

Simplifying Complex Fractions

A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator.

454

3xx

3x

ba9-a

ba

Example:

Adding/Subtracting Fractions

0c ,c

ba

c

b

c

a 0c ,

c

ba

c

b

c

a

712

= 512

212

+

Add . 512

212

+

Common Denominators

1. Add or subtract the numerators.2. Place the sum or difference of the

numerators found in step 1 over the common denominator.

3. Simplify the fraction if possible.

Subtract .5

6

5

7-2x

5

13-2x

5

6-7-2x

5

6

5

7-2x

Common Denominators

a.) Add .12ww

4-2w-

12ww

53w22

Example:

12ww

4-2w-53w

12ww

4-2w-

12ww

53w222

1)(w

1

1)(w

1w2

12ww

4-2w-53w2

Common Denominators

b.) Subtract

.649x

29x-x

649x

54x2

2

2

2

649x

29)x-(x-54x

649x

29x-x

649x

54x2

22

2

2

2

2

649x

24x3x

649x

29xx-54x2

2

2

22

8)(3x

3)(x

8)8)(3x(3x

8)3)(3x(x

Example:

Unlike Denominators

1. Determine the LCD.2. Rewrite each fraction as an

equivalent fraction with the LCD.3. Add or subtract the numerators

while maintaining the LCD.4. When possible, factor the

remaining numerator and simplify the fraction.

Unlike Denominators

a.)w

5

2w

3

2w

2w

w

5

w

w

2w

3

The LCD is w(w+2).

2)w(w

2)5(w

2)w(w

3w

2)w(w

105w

2)w(w

3w

2)w(w

108w

answers. acceptable also are and 2ww

108w

2)w(w

5)2(4w2

Example:

Unlike Denominators

b.)3x

1

4-4x

x The LCD is 12x(x – 1).

3x

1

1)-4(x

x

1)-4(x

1)-4(x

3x

1

3x

3x

1)-4(x

x

1)-12x(x

44x3x

1)-12x(x

1)-4(x

1)-12x(x

3x 22

This cannot be factored any further.

Example:

Simplifying Complex Fractions

A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator.

454

3xx

3x

ba9-a

ba

Example:

So how can we simplify them?

• Remember, fractions are just division problems.• We can rewrite the complex fraction as a division

problem with two fractions.• This division problem then changes to multiplication

by the reciprocal.

5

62

3

5

6

2

3

5

6

3

2

5

4

Simplifying Complex Fractions Rule

• Any complex fraction

dcba

Where b ≠ 0, c ≠ 0, and d ≠ 0, may be expressed as:

bc

ad

What if we have mixed numbers in the complex fraction?

• If we have mixed numbers, we treat it as an addition problem with unlike denominators.

• We want to be working with two fractions, so make sure the numerator is one fraction, and the denominator is one fraction

• Now we can rewrite the complex fraction as a division of two fractions

Example

21

25

Try on your own…

4

113

What about complex rational expression?

• Treat the complex rational expression as a division problem

• Add any rational expressions to form rational expressions in the numerator and denominator

• Factor• Simplify• “Bad” values - Extraneous Roots

Ex. 2: Simplify .11

11

yx

yx

xyx

xyy

xyx

xyy

yx

yx

11

11

xyxy

xyxy

← The LCD is xy for both the numerator and the denominator.

← Add to simplify the numerator and subtract to simplify the denominator.

xy

xy

xy

xy

← Multiply the numerator by the reciprocal of the

denominator.

Ex. 2: Simplify .11

11

yx

yx

xy

xy

xy

xy

← Eliminate common factors.

xy

xy

Example

x 1

xx 1

(x 1

x) (x 1)

(x 2 1

x)

1

x 1

(x 1)(x 1)

x

1

x 1

x 1

x, x 0, 1

Example

x 2

1 5

x 6

x 2

Try on your own

2

3x1

x

One more for you

x 16

xx 2 8x 16

Ex. 3: Simplify

348

11

41

4

xx

xx

348)3)(11(

41)4)(4(

xxxxxx ← The LCD of the numerator is x +

4, and the LCD of the denominator is x – 3.

Ex. 3: Simplify

348

11

41

4

xx

xx

348338

41168

2

2

xxxxxx

← FOIL the top and don’t forget to subtract the 1 and add the 48 on the bottom.

Ex. 3: Simplify

348

11

41

4

xx

xx

3158

4158

2

2

xxx

xxx

← Simplify by subtracting the 1 in the numerator and adding the 48 in the denominator.

Ex. 3: Simplify

348

11

41

4

xx

xx

158

3

4

1582

2

xx

x

x

xx

← Multiply by the reciprocal.

x2 + 8x +15 is a common factor that can be eliminated.

Ex. 3: Simplify

348

11

41

4

xx

xx

4

3

x

x ← Simplify

Factor the sum or difference of two cubesThe product of the same three factors is called a ___________________________.KNOW THESE…..a3 + b3 = (a + b)(a2 – ab + b2)

a3 – b3 = (a – b)(a2 + ab + b2) Factor x3 – 8 This is x3 – 23, so:x3 – 8 = x3 – 23 = (x – 2)(x2 + 2x + 22) = (x – 2)(x2 + 2x + 4) Factor 27x3 + 1 Remember that 1 can be regarded as having been raised to any power you like, so this is really (3x)3 + 13. 27x3 + 1 = (3x)3 + 13 = (3x + 1)((3x)2 – (3x)(1) + 12) = (3x + 1)(9x2 – 3x + 1)

Ex: Answer the following questions1. How many separate pieces does this function have?2. List the three equations3. For equation, list the x intervals for which it is valid. (use inequality notation)4. How do we write this as a piecewise function?

5. What is the domain of the function? D = (-∞, 6] R = [0, ∞)

Model Problems

5

31)1

2

x

x

12)

1y

yy

24

33)36

3

x

x

4)1 1

x yx

x y

2 65)

2 3

k k

k k

71

26)

31

2

y

y

Rule of 4The Rule of 4 refers to representing mathematical functions with graphs, tables, equations, and words. As learners discover how to represent functions in each of these ways, the mathematics becomes more meaningful. For example, consider the following cell phone plan offered by T-Mobile in 2011, represented using the Rule of 4.

1. Words Representation (from website) 3. Graph RepresentationEven More 1000 Talk + Unlimited Text$59.99 includes 1000 whenever minutesAdditional minutes $0.45 per minute

2. Table Representation

4. Equation Representation

Despite the fact that each of these representations of the cell phone cost function looks different, the same function is represented in each representation. All learners should practice to increase their ability to “see” the other forms mentally even when only one form is given.

Absolute Value

| x | = 5

Absolute Value Equations

x = 5 x = – 5Same Opposite

| x | = –2

No Solution

Two Solutions

Absolute Value PropertyIf |x| = a, where x is a variable or an expression and a 0, then x = a or x = a.

Solving Absolute Value Equations

1. Isolate the absolute value so that the equation is in the form |ax + b| = c. If c < 0, the equation has no solution.

2. Separate the absolute value into two equations, ax + b = c and ax + b = c.

3. Solve both equations.

3. Check your answers. Make sure they are not extraneous.

Absolute Value Equations with 2 Absolute Values

2342 ww

Same Opposite

2342 ww 2342 ww

6

52

,

ww 33 24 w44

6 w

6w

2342 ww

245 w

ww 33

44 25 w55

52w

Check your work!

Absolute Value Equations

9523 x

423 x

55

Same Opposite

423 xDrop the absolute value bars!

Keep the absolute value bars!

423 x33 12 x22

21x

33

22 72 x

27x

27

21

,

1. Isolate2. Two Cases3. Solve

Check your work!

Absolute Value Equations with 2 Absolute Values

332 kk

Same Opposite

332 kk 332 kk

21

45

,

kk 322 k33

k21

21k

332 kk

324 k

kk 33

22 54 k44

45k

22

Check your work!

Absolute Value Equations

6413 k

1013 k

44

Same Opposite

1013 kDrop the absolute value bars!

Keep the absolute value bars!

1013 k11

113 k33

311k

11

3393 k

3k

311

3,

1. Isolate2. Two Cases3. Solve

Check your work!

Absolute Value Equations

7127 x

57 x

1212

1. Isolate2. Two Cases3. Solve

No Solution

Absolute Value Equations

052 y

Same Opposite

052 y55 52 y

22

25y

25

1. Isolate2. Two Cases3. Solve

Check your work!

Absolute Value Equations with 2 Absolute Values

xx 2392

Same Opposite

xx 2392 xx 2392

3

xx 22

394 x99

124 x

3x

xx 2392

39

xx 22

44No Solution

Factoring

Factoring a polynomial means expressing it as a product of other polynomials.

Factoring polynomials with a common monomial factor (using

GCF).

**Always look for a GCF before using any other factoring method.

Factoring Method #1

Steps:

1. Find the greatest common factor (GCF).

2. Divide the polynomial by the GCF. The quotient is the other factor.

3. Express the polynomial as the product of the quotient and the GCF.

3 2 2: 6 12 3Example c d c d cd

3GCF cdStep 1:

Step 2: Divide by GCF

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

3cd(2c2 4cd 1)

The answer should look like this:

Ex: 6c3d 12c2d 2 3cd

Factor these on your own looking for a GCF.

1. 6x3 3x2 12x

2. 5x2 10x 35

3. 16x3y4z 8x2y2z3 12xy3z 2

23 2 4x x x

25 2 7x x

2 2 2 24 4 2 3xy z x y xz yz

Factoring polynomials that are a difference of squares.

Factoring Method #2

A “Difference of Squares” is a binomial (*2 terms only*) and it factors like this:

a2 b2 (a b)(a b)

To factor, express each term as a square of a monomial then apply the rule... a2 b2 (a b)(a b)

Ex: x2 16 x2 42

(x 4)(x 4)

Here is another example:1

49x2 81

1

7x

2

92 1

7x 9

1

7x 9

Try these on your own:

1. x 2 121

2. 9y2 169x2

3. x4 16

Be careful!

11 11x x

3 13 3 13y x y x

22 2 4x x x

End of Day 1

Sum and Difference of Cubes:

a3 b3 a b a2 ab b2 a3 b3 a b a2 ab b2

3: 64Example x (x3 43 )

Rewrite as cubes

Write each monomial as a cube and apply either of the rules.

Apply the rule for sum of cubes:

a3 b3 a b a2 ab b2

(x 4)(x2 4x 16)

(x 4)(x2 x 4 42 )

Ex: 8y3 125

Rewrite as cubes

((2y)3 53)

2y 5 4y2 10y 25

Apply the rule for difference of cubes:

a3 b3 a b a2 ab b2 2y 5 2y 2 2y5 5 2

Factoring Method #3

Factoring a trinomial in the form:

where a = 1 ax 2 bx c

Next

Factoring a trinomial:

ax 2 bx c

2. Find the factors of the c term that add to the b term. For instance, let

c = d·e and d+e = bthen the factors are

(x+d)(x +e ).

.

1. Write two sets of parenthesis, (x )(x ). These will be the factors of the trinomial.

xx

2: 6 8Example x x

x x

Factors of +8: 1 & 8

2 & 4

-1 & -8

-2 & -4

Factors of +8 that add to -6

2 + 4 = 6

1 + 8 = 9

-2 - 4 = -6

-1 - 8 = -9

-2-4

x2 6x 8

Check your answer by using FOIL

(x 2)(x 4)

(x 2)(x 4) x2 4x 2x 8

F O I L

x2 6x 8

Lets do another example:

6x2 12x 18

6(x2 2x 3) Find a GCF

6(x 3)(x 1) Factor trinomial

Don’t Forget Method #1.

Always check for GCF before you do anything else.

When a>1, let’s do something different!

2: 6 13 5Example x x

Step 1:

Multiply a · c

Step 2: Find the factors of a·c (-30) that add to the b term

= - 30

Factors of 6 · (-5) : 1, -30 1+-30 = -29

-1, 30 -1+30 = 29

2, -15 2+-15 =-13

-2, 15 -2+15 =13

3, -10 3+ -10 =-7

-3, 10 -3+ 10 =7

5, -6 5+ -6 = -2

-5, 6 -5+6 =1

2: 6 13 5Example x x

Step 2: Find the factors of a·c that add to the b term

Let a·c = d and d = e·fthen e+f = b

d = -30e = -2f = 15

-2, 15 -2+15 =13

2: 6 13 5Example x x

Step 3: Rewrite the expression separating the b term using the factors e and f

-2x+15x -5 13x

-2x + 15x -5

Step 4: Group the firsttwo and last two terms.

2: 6 13 5Example x x

Step 4: Group the firstTwo and last two terms.

Step 5: Factor GCF from each groupCheck!!!! If you cannotfind two common factors,Then this method does not work.

Step 6: Factor out GCF

- 2x + 15x - 5

3x - 1) + 5(3x - 1)

3x - 1) (2x + 5)

Common factors

Step 3: Place the factors inside the parenthesis until O + I = bx.

6x2 30x x 5F O I L

O + I = 30 x - x = 29xThis doesn’t work!!

2: 6 13 5Example x x

6x 1 x 5 Try:

I am not afan of guessand check!

6x2 6x 5x 5F O I L

O + I = -6x + 5x = -xThis doesn’t work!!

2: 6 13 5Example x x

6x 5 x 1

Switch the order of the second termsand try again.

Try another combination:

(3x 1)(2x 5)

6x2 15x 2x 5F O I L

O+I = 15x - 2x = 13x IT WORKS!!

(3x 1)(2x 5)6x2 13x 5

Switch to 3x and 2x

Factoring Technique #3continued

Factoring a perfect square trinomial in the form:

a2 2ab b2 (a b)2

a2 2ab b2 (a b)2

Perfect Square Trinomials can be factored just like other trinomials (guess and check), but if you recognize the perfect squares pattern, follow the formula!

a2 2ab b2 (a b)2

a2 2ab b2 (a b)2

Ex: x2 8x 16

x2 8x 16 x 4 2

2

x 2 4 2

Does the middle term fit the pattern, 2ab?

Yes, the factors are (a + b)2 :

b

4

a

x 8x

Ex: 4x2 12x 9

4x 2 12x 9 2x 3 2

2

2x 23 2

Does the middle term fit the pattern, 2ab?

Yes, the factors are (a - b)2 :

b

3

a

2x 12x

Factoring Technique #4

Factoring By Groupingfor polynomials

with 4 or more terms

Factoring By Grouping1. Group the first set of terms and

last set of terms with parentheses.2. Factor out the GCF from each group

so that both sets of parentheses contain the same factors.

3. Factor out the GCF again (the GCF is the factor from step 2).

Step 1: Group

3 2

3 4 12b b b Example 1:

b3 3b2 4b 12 Step 2: Factor out GCF from each group

b2 b 3 4 b 3 Step 3: Factor out GCF again

b 3 b2 4

3 22 16 8 64x x x

2 x3 8x2 4x 32 2 x3 8x2 4x 32 2 x 2 x 8 4 x 8 2 x 8 x2 4 2 x 8 x 2 x 2

Example 2:

Try these on your own:

1. x 2 5x 6

2. 3x2 11x 20

3. x3 216

4. 8x3 8

5. 3x3 6x2 24x

Answers:

1. (x 6)(x 1)

2. (3x 4)(x 5)

3. (x 6)(x2 6x 36)

4. 8(x 1)(x2 x 1)

5. 3x(x 4)(x 2)

t (months) 1 2 3 4 5 6 7

S(t) (100’s) 1.54 1.88 2.32 3.12 3.78 4.90 6.12

The number of new software products sold is given by S(t), where S is measured inhundreds of units and t is measured in months from the initial release date of January 1. 2012. The software company recorded these sales data:

a) Estimate the number of units sold between April 1, 2012 through June 30, 2012.b) Assuming the data is linear, determine a the prediction equation estimating the number of units sold during months April and June.c) Using your prediction equation, predict the number of units sold in March and July?d) Graph the data and draw a quick sketch.e) Is your prediction equation accurate, why might it not be accurate?

(2x+5) = 4x2+16x+15

t (months 1 2 3 4 5 6 7St (100’s) 1.54 1.88 2.32 3.12 3.78 4.90 6.12

The number of new software products sold is given by S(t), where S is measured inhundreds of units and t is measured in months from the initial release date of January 1. 2012. The software company recorded these sales data:

a) Estimate the number of units sold between April 1, 2012 through June 30, 2012.Using the table data (4, 5, 6) , there were 1180 units sold

b) Assuming the data is linear, determine a the prediction equation estimating the number of product sold during months April and June. Using the point (4, 3.12) and (6, 4.9), the equation of the line is: y - 3.12 = .9(x-4) +2 ptsc) Using your prediction equation, predict the number of units sold in March and July? Evaluating the equation found in step (b), the number of units sold are for March x = 3 y = 2.22 units and for July x = 7, y = 4.92 unitsd) Graph the data and draw a quick sketch of your prediction equation. The red line is the graph of the original table data. The green line is the graph of the prediction equation found in (b). The graph was generated using Excel.e) Explain whether your prediction equation is accurate? What might make it be more accurate? Our initial assumption that the data was linear was incorrect. We can look at other regressions that might match the data more accurately. Challenge: Using your calculator, which regression yields the best fit?f) 1 pt b) 2 pts c)1 pts d) 2 pts 6) 5 pts

1 2 3 4 5 6 70

1

2

3

4

5

6

7

Series1Series2

Rule of 4The Rule of 4 refers to representing mathematical functions with graphs, tables, equations, and words. As learners discover how to represent functions in each of these ways, the mathematics becomes more meaningful. For example, consider the following cell phone plan offered by T-Mobile in 2011, represented using the Rule of 4.

1. Words Representation (from website) 3. Graph RepresentationEven More 1000 Talk + Unlimited Text$59.99 includes 1000 whenever minutesAdditional minutes $0.45 per minute

2. Table Representation

4. Equation Representation

Despite the fact that each of these representations of the cell phone cost function looks different, the same function is represented in each representation. All learners should practice to increase their ability to “see” the other forms mentally even when only one form is given.

Studying ProcessWhat does it mean to study in Mrs. Harton’s Class? Studying begins At Home• Before coming to class:• Read the section and start your notes. • Write down the definitions of words and

copy theorems. • Work through each example.• Make note of what you don’t understand.

Studying continues In Class• Listen to Teacher led instruction• Take Notes• Ask Questions• Do in class work

Studying continues with HomeworkTo do Assigned Homework• Write Header -Name, Date, Section number,

page numbers, problem numbers• Copy Question and figures• Look at notes and the text to see how to

apply definitions and theorems• Mark up the figure and solve the problem. Studying continues In Class: • Request to see problems you don’t

understand and make corrections to Homework

Quizzes and Test• Show your work. This means I would like to

see the set up, the method used and the solution. Sometimes, no work is needed. However, explain your reasoning, why are you able to do what you are doing? If a formula is used, mention it.