Lesson # 52-55

Practice tests for  for lessons # 52-54.  Test during Lesson# 55!

Lesson # 50& 51

Identity Package assigned. Due on Tuesday, Nov. 30 (day of TEST)

Lesson # 29: y = aSin(x-p)+C

A - Amplitude     P.S. - Phase Shift     P - Period

2Cos(x+1)
- vertical stretch x2
- up 1 on the center l
- period = 1
- 1/4 period = 1/4

Lesson # 18: Infinite Sum & Sigma Sum

Infinite Sum:

Add the infinite sum of 6+2+2/3+.....

S =     6     
     (1-(-1/3))
   =      6    
         (2/3)
   = 6(3/2)
   =9

What is the formula:  S=   a   
                                       l-r

Sigma:

5

∑ 3i+2
i=1

5 - the terms
i =1  - input #
3i+2 - formula

Lesson # 17: Geometric Sequences & Series

Formula:

tn = ar^n-1
tn - value
n - the position
r - multiplier (2nd/1st)

Example:             2, 6, 18, 54,......
1) t10
tn = 2(3)^9
tn = 39 366

2) How many terms are in the following sequence?   8, 12, 18, 27,...... 205.03125
205.03125 = 8(1.5)^n-1
        8                  8
log(25.6289) = (n-1) log(1.5)
   log(1.5)             log(1.5)
8 = n-1     
9 = n


Series:
 A series is the sum of a sequence
S= add   t= find number

Formula:
 
Sn = a(1 - r^n)   or   Sn = a- rl                           
             1-r                       1-r      
 
Sn: series (sum)       
r: ratio     
l: last term> tn
a: first #         
n: how many terms there is

Example:
1) Find the sum of the series          5 + 15+ 45 +.......... + 885 735
Sn =  a - rl
          1-r               
Sn= 5 - 3(885 735)
             1-3                                                                                      
     =  1, 328, 600

Lesson # 16: Richter Scale, pH Scale & Decibels

Earth Quake: Richter Scale




4.0 ------x10------ 5.0 ------x10 ------ 6.0 ------x10 ------ 7.0


R = 10    P = 1

pH Scale:

Sounds- decibels dbs:

40 dbs  > whisper      90 dbs > shout

           |  100000 times louder   |

R= 10      P= 10


Example 1:
How much more intense is an Earthquake of 6.8 to a 3.5?
F = IR ^ t/p                     M= magnitude   M=F       I= 1            t= change   
M= 10^ 3.3/1
M= 1995.26

Example 2:
A chainsaw of 105dbs to a whisper of 40dbs
M = R^t/p
M= 10^ 65/10                     
M= 10^ 6.5
M= 3162277.66

Example 3:
Find the pH that is 80x more basic than water?
water= pH 7
M = R ^t/p
80= 10 ^t
log 80 = t log10                    *note log10 = 1*
1.90= t
7+ 1.9= 8.9
t= 8.9

Lesson # 15: Exponential Growth and Decay

There are two different formulas:

The most common formula to use is:

F= IR^ t/p                    F: final amount       I: initial amount     R: rate of growth ( 5% = 0.05 +1 R>1)   growth decay ( 5%  R= 0.95)     t: time       p: time for R to occur (days, weeks, months,etc)  

* note: t and p must be in same units

The second formula is for continuous interest rates:

 P= Po e^ Kt          P: final amount       Po: initial amount      e: calculator function   K: growth/decay (no 1)  

t: time 

Example 1:

What will $3500 grow to, if invested at 6% interest for 10 years, compounded annually?

F= IR ^ t/p               F=?     I= 3500   R= 1.06  t= 10    p=1 

(*note: if p is a fraction take the reciprocal and multiply it to the top number) 

F= 3500(1.06)^10/1

F= 3500(1.06)^10

F= $6267.97

Example 2:

What amount of money would grow to $ 4000 if invested at 91/4%, compounded annually for 4 years?

F= IR ^ t/p              F= 4000   I= ?   R= 1.0925   t= 4     p=1

4000   =     I  (1.0925) ^ 4/ 1

(1.0925)^4     (1.0925)^4

I= $2807.85

Example 3:

In April. the atmosphere at Chernobyl was contaminated with radioactive iodine-131, which has a half-life of 8.1 days.  How long did it take for the level of radiation to reduce to 1% of the level immediately after the accident?

*Half-life  R= 0.5     P= half-life
1/100 = 100(0.5)^t/8.1
0.01 = 0.5^t/8.1
log0.01 = t/8.1 log0.5
8.1 log0.01/log0.5 = t

Lesson # 14: Logarithm Equations & Identities

Today's lesson had two methods for solving equations, log= # and log=log

Method 1: Log = #                                                                                                                                             

Example:
log(2)x = 3                  1) Put into (b)^E = A form

2^3 = x

x = 8 

Example:

log(5)(x-3)+ log(5)x = log(5)10

log(5)x(x-3) = log(5)10               1) Put into B^e = A form

         x(x-3) = 10

       x^2- 3x = 10          2) Multiply both sides by x

  x^2-3x-10 = 0

   (x-5)(x+2) = 0        3) Simplify

           x = 5, -2     *-2 rejects*

           x = 5

*Note: negative answers are tricky, they must become positive in original equation or are rejected.*

       

Method 2: log = log                                                                                                                       

log5(x-3) + log5x = log5 10     1) Create only one log

log5(x-3)x = log5 10          2) Drop logs

(x-3)x = 10

x^2 - 3x = 10

x^2 -3x -10 = 0

(x-5)(x+2)

x= 5, -2            *-2 rejects*

x=5

Identities:

Equation        vs.         Identity

2x+3 = 5                    x + x = 2x

x=1                            x = all real numbers

x= specific #                rejects allowed

Prove the identity and state the value(s) of x for which it is true: (make both sides equal)

  

logx + log(x +3) = log(x^2 +3x)

  LEFT SIDE        RIGHT SIDE

Left side:  logx(x+3)

               logx^2 + 3x = Right side

*x must be great the 0     x>0*

Lesson # 13: Exponential Equations

In this lesson there are 2 different methods to find x: Common Bases method, or the Log method

Method 1- Common Bases:

3^2x = 27^3x-1              1) Find Common Base
3^2x = (3^3)^ 3x-1
3^2x = 3^9x-3                2) Drop the Base
2x = 9x-3
3 = 7x                           3) Solve
3/7 = x

Check:
3^3/7 = 27^3(3/7) -1
2.55 = 2.55  
correct!

Method 2- Logs:

3^2x = 27^3x-1                                  1) Log both sides
log3^2x = log27^3x-1
2xlog3 = (3x-1)log27                          2) Factor out brackets
2xlog3= 3xlog27 - log27
2xlog3- 3xlog27= -log27                     3) Factor Out x
x ( 2log3 - 3log27)  =      -log27          4) Solve for x
    (2log3- 3log27)     (2log3- 3log27)
x= 0.42857
x= 3/7

*Note: when using calculator, remember to use brackets or the answer may be wrong*

Lesson # 12: Laws of Logarithms

The Laws of Logarithms:

1) log(a)(xy) = log(a)x + log(a)y
2) log(a)(x/y) = log(a)x - log(a)y
3) log(a)(x^n) = nlog(a)x
4) log(a)n*root*(x)^1 = log(a)x^1/n or 1/nlog(a)x
5) log(a)b = log(b)/log(a)

Example: Express log8 in otherways
- log2^3 = 3log(2)
- log(2x2x2) = log2 + log(2) + log(2)

Example: Express 5 as a power of 2     *Log Both Sides!*
2^x = 5
log2^x = log5
xlog2 = log5
  log2    log2
x = 2.32

Lesson # 11: Exponentials & Logarithms

Exponentials vs. Logarithms:

Exponentials:      3^x                                      log(b) A=E <==> (b)^E=A
Logarithms:        log^x                            
                                                                 b=base     A=answer   E=exponent 

Example of Exponential to Logarithm Form:
1) 4^3 = 64   <==>  log(4)64 = 3
2) 2^-4 = 1/16  <==>  log(2)1/16 = -4

A logarithm is.... the exponent with a base of 10.

*To get the answer of a logarithm with a base of 10 is to count the zeros!*
Example:
log(10)1,000 = 3

Converting to base of x:

Example:     *base of 2*                     
1) 8^3x+1
(2^3)^3x+1
2^9x+3

2) 1/32^x-1
(2^-5)^x-1
2^-5x+5

Lesson # 7+8: Practice Tests

On the first day we wrote an in class practice test for chapter one.  We reviewed the test the next day, and got our marks back.  We were also given another test to take home to practice with over the weekend.

Lesson # 6: Absolute Value Tool

How to use the Absolute Value Tool:                    *y = f(x) to y = |f(x)|*

For the positive y values do not do anything. But for the negative values, they reflect over the x-axis.

Example:
base graph --- > y = 2x+3                   *All negative values become positive values*
absolute value graph ---> y = |2x+3|

How to use the Reciprocal Tool:                           *y = f(x) to y = 1/f(x)*

Steps:
1) Base
2) A.V. (zeros)
3) invarient points (ones)
4) reciprocal height

Lesson # 5: Combinations

Combinations:                       *Reflections, Shifts and Dilations*

If you have a dilation or reflection with a translation => FACTOR IT!

Example:                   *Factor only if needed*
y = *root*(2x) which makes it... horizontally compress 1/2x
y = *root*(2x+6) but you would factor it out, which makes it... y = *root*(2(x+6))

*Do stretch and reflections first then do the slides*

List the transformations:

y = 1/2f(-3(x+1))-4          
   
      Vert.                      Hor.
comp by 1/2x      reflect over y-axis
    down 4              comp by 1/3x
                                    left 1

Lesson # 4: Dilations

Stretching or Compressing:

y = f(x) goes to ky = f(x)        *This will make the equation, vertically stretch or compress by 1/k*

y = f(x) goes to y = f(kx)        *This will make the equation, horizontally stretch or compress by 1/k*
The way I remember these is when... if it's a whole number it will be compressed.
                                                      if it's a fraction it will be stretched.

Example:
1) y = f(1/2x) goes to x = 1/2x   which makes a... horizontal stretch x2
2) 2/3y = f(x) goes to y = 2/3x   which makes a... vertical stretch x3/2 or x1.5

Lesson # 3: Relections

Rules for x and y-axis:

For y = f(x) to go to y = f(-x), the graph of f(x) is reflected over the y-axis. In other words, x is replaced by -x.
For y = f(x) to go to -y = f(x), the graph of y = f(x) is reflected over the x-axis. In other words, y is replaced by -y.

Examples of basic equations:

1) y = (-x)^2 is the same as y = x^2, but it reflects over the y-axis.
2) y = -(x^2) is the same as y = x^2, but it refelcts over the x-axis.
3) y = 1/x is the same as y = 1/-x, but it reflects over the y-axis.
4) -y = *root*(16-x^2) is the same as y = *root*(16-x^2), but it reflects over the x-axis.

Rules for Inverses:     *An inverse is the reflection across the line y = x.*

1) Swap x and y                 y = f(x) goes to y = f^-1(x)
2) Isolate y
3) rewrite f^-1(x)

Example:               *Find f^-1(x)*
f(x) = -1/2x+3
x = -1/2x+3
(Multiply everything by 2)
2x = -y+6
y = -2x+6
f^-1(x) = -2x+6     *Remember that y = f(x) and f^-1(x) and NOT 1/f(x)*

Lesson # 2: Translations

Horizontal Translations   

For y = f(x) to go to y = f(x-k), the graph of f(x) shifts to the right by k if k>0, and it shifts to the left by k if k<0.
*A trick I use to remember this is, if the equation is (x-k) it will move to the right by k because it will become possitive.  And if the the equation is (x+k) it will move to the left by k. It will become negative.  So since the the graph is horizontal, the points will only move left to right.*

Example:        y = |x|                                                                        y = |x-4|

Vertical Transaltions

For y = f(x) to go to y-k = f(x), the graph of f(x) shifts up by k if k>0, and it shifts down by k if k<0.
*A trick I use to remember this is, if the equation is y-3 = f(x) it with move up by 3 because it will become possitive.  And if the equation is y+4 = f(x) it will move down because it will become possitive. It would be the same for y = f(x)+3 it would move up 3, and for y = f(x)-5 it would would move down by 5.*

Example:         y = x^2 +2

Lesson # 1: Functions You Should Know

1) Linear   y = mx+ b   (m = slope, b = y-int)                             

Example: y = 1/2x+2
    rise = +1
    run = +2                                                             
    y-int = +2

                                                                                      

2) Quadratic   y = (x+2)^2 -1                                                    Parabola = *root* (-2,-1)
                             

3) Cubics   *Deg 3* 

          

 y = (x+2)(x-3)(x-4)                                  Leading Term = x^3....+24

 zeros =-2, +3, +4

 y = -(x+1)^2(x-2)                                    Leading Term = -x^3....+2
 zeros =-1, -1, +2

*The negative makes it flip vertically*

4) Quartics   *Deg 4*                                        

                         
y = (x+1)^2 (x-2)(x-3)                                    Leading Term = x^4....+6

                        

5) Reciprocal   y = x  >> y = 1/x                      

a) draw base

b) vertical asymptote (zeros)

c) imvarient

d) reciprocal heights

                                                        

                                                       

6) Circles   r = 2                                              

 x^2+y^2 = 4

             y = 4- x^2

             y = *root*(4- x^2)       *top half*

             y = -*root*(4- x^2)     *bottem half*

                                                      

                                                         

7) Absolute Value   y = |x|                                                                         



|5| = 5

|0| = 0

|-2| = 2                                    

                                                    

8) Sqaure Root   y = *root* (x)        

             

 X   Y

 0    0

 1    1

 4    2

 9    3

-4   n/a            

                                              

9) Exponential   y = 2x

           

 X   Y

 0    1

 1    2

 2    4

 4    8 

-1  1/2

-2  1/4