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Note that, every second, the cheetah closes the gap between himself (or herself) and the antelope by (90 - 60) =

30 ft

So....the number of seconds to catch up is  [ 120 / 30 ]  = 4 seconds

OK....we can use this "formula" to find "C"

12.3 + (3/4) (25/2 - 12.3) 

12.3 + (3/4) (12.5 - 12.3)  =

12.3 + (.75) (.2) =

12.3 + 0.15 =

12.45   =  "C" 

Welll....I don't know about my "smartness"....some guys WAY smarter than me invented all this stuff

However....I AM an expert in pie-eating....!!!! ....LOL!!!!

Remember that we have

y =  2sin (x)  + 3

We can write this as

y = 2 sin (1x) + 3

So "B"  = 1.....this means that there is no horizontal stretch or compression

It is  "A"  [ the number in front of sin (x) ] that affects the vertical stretching or compression

Since the abs value of this  = 2...and this is > 1.......we have a "stretch"

If the abs value of "A"  is between 0 and 1, we would have a vertical compression...

Thank you again ! You have been so much help, I think I am really starting to get this now ! We should both go get some "pi" lol celebrate the weekend and your smartness!

Note that every hour, it must rise 4 times = 4(1.5) = 6 degrees

So...in two hours...it must rise 2(6 ) =  12 degrees

ohhh that all makes sense now. However such as the problem I had , how did you know "x" was a vertical stretch since x does not have a value ?

First one

2.5   ...Correct !!!

Second one

The amplitude is the absolute value   of - 2   =  l -2 l =   2

[ The "minus" just flips the graph over the x axis .....the amplitude is not affected ]

Third one

"Pi"  is correct...!!!

Speaking of "pi".....that just made me hungry for dessert...!!!!

In the form

y = A * trig function (Bx  + C ) + D

The number on "B"  only affects the horizontal compression or horizontal expansion .....

If this  number  is between 0 and 1, we have a horizontal " stretch"

If this number is > 1, then we have a horizontal "compression"

It is "A"  that affects the vertcal stretching or vertical compression

If   absolute value of A  is  > 0  but < 1,  we have a vertical  compression

If absolute value of A > 1, we have a vertical stretch

"D"  affects the shift of the graph upward   (if D is positive)  or downward ( if D is negative)

Hope that helps, Jenny   !!!!

So you know it is vertically stretched when there is on an (x) and the "B" (period) is not included and it is compressed when there is a value for B ? such as "4x" ?

Looking at the question I didnt realize it was simialr to the last question but now I do it was just the oppostite .

I want to say thank you. These are great explantaions and it has helped me a lot. I am very thankful that you helped with each one. They were more easy then it looked.

We have 34 pieces of fruit....and 10 of them are apples

P (apple)  = 10 / 34   =   5 / 17

4)

The graph of  f(x)=sin(x)  is transformed to a new function, g(x) , by stretching it vertically by a factor of 2 and shifting it 3 units up.

What is the equation of the new function  g(x) ?

This is just the counterpart of the last question

g(x)  =   2sin (x) + 3

3)

The functionf(x) is multiplied by a factor of 2 and then 3 is added to the function. 

f(x)=sin(x)

What effect does this have on the graph of the function?

The "2"  vertically stretches the parent function by a factor of 2

The "3"  shifts the function up 3 units

So.... "A" is correct

2)

What is the frequency of the function f(x)?

f(x)=1/4 cos(2x)+5

Express the answer in fraction form.

In the form

A * trig function ( Bx + C) + D

Only  "B" affects the period

Here...B  = 2      [ and  C = 0 ]

This means that there are 2 periods in 2pi

So....to find the period  take    2pi / 2    =   pi = the period

And the frequency  =  1 / period  =   1 /pi

1) What is the equation of the midline for the function f(x) ?

f(x)= 1/ 2 cos (x) + 5

Notice that   the min for the cos  = -1

So....the lowest point on the graph  is (1/2)(-1) + 5  = -1/2 + 5 =  4.5

The max for the cos  = 1

So...the highest point on the graph  = (1/2)(1) + 5 = 5.5

Midline    ( Highest point + Lowest point) / 2   =    (5.5 + 4.5) / 2  =  10 / 2  =  5

So....the equation for the midline is  y = 5

Here are a few keys

1. Any even is not a prime

2. Any  number ending in 5 is not a prime

3. If we can sum the digits and that sum is divisible by 3, the number is not prime

4. If we double the last digit and subtract the remaining part and the result is divisible by 7, the number is divisible by 7

Example :   154

4(2) - 15  =  8 - 15  =  - 7

And -7/7  =  -1

So.....154 is divisible by 7

5.   Alternate the signs on the digits of a number....add these.....if the result is 0 or a number divisible by 11, the number is divisible by 11

Ex :   1331   =  +1 - 3 + 3 - 1 =  0           ....so 1331 is divisible by 11

There are probably numerous other "tricks" to test for primeness

However, there is no sure-fire algorithm for testing for primes

41, 42, 45, 50, 57, 66, 77, 82, 83, 85, 88, 90, 92, 97   give non-primes  = 14

So

100 - 14  =  86

Which of the following equations is the result of using the factoring method to solve the problem?

(n - 5)(n - 4) = 0

(n - 5)(n + 4) = 0

(n + 5)(n - 4) = 0

(n + 5)(n + 4) = 0

(n - 5)(n - 4)  = 0,     This comes from this quadratic:  n^2 - 9 n + 20 = 0
(n - 5)(n + 4) = 0,     This comes from this quadratic:  n^2 - n - 20 = 0
(n + 5)(n - 4) = 0,     This comes from this quadratic:  n^2 + n - 20 = 0
(n + 5)(n + 4) =0,     This comes from this quadratic:  n^2 + 9 n + 20 = 0

First one....we have that

n^2 + ( n + 1)^2  =  41

n^2 + n^2 + 2n + 1 = 41

2n^2 + 2n - 40  = 0

n^2 + n  - 20 =  0

(n  + 5) ( n - 4) = 0

Second one

Call the first factor, F

So...the second is   F/2 - 11

So.....this implies that

(F)  [ (F/2) - 11]   = -36       simplify

F^2/2 - 11F = - 36

F^2/2 - 11F + 36 = 0         multiply through by 2

F^2 - 22F + 72 = 0

(F - 18) ( F - 4)  = 0

F = 18      or F = 4

When F = 18    the other factor is :  F/2 - 11   =  18/2 - 11  =  9 - 11 =   -2

When F = 4    the other factor is   4/2 - 11 =  2 - 11  =  -9

So....the factors are

(18, -2)   and ( 4, - 9)

0 ≤  t  ≤ 10   ⇒  the domain

The divisors of 2008  are

1, 2, 4, 8, 251, 502, 1004, 2008

"b"  will be smallest  when   we have  251 and 8

So

(x + 251) ( x + 8)  =   x^2 + 259x + 2008

So....the smallest value of "b"  =   259