geno3141

1) Since the center is at (3, -1) and a vertex is at (5, -1), it opens horizontally.

The general equation will be:  (x - h)²/a² - (y - k)²/b² = 1

with       center = (h, k),          a = length of transverse radius,           b = length of conjugate radius

For this problem:  h = 3,  k = -1,  a = 2

The asymptote is 2y = 3x - 11   --->  y = (3/2)x - (11/2)   --->  m = 3/2

Since m = 3/2, for every run of 2, it has a rise of 3   --->  Since a = 3, b = 2.

--->   (x - 3)²/2² - (y - -1)²/2² = 1

Finish this, and you can choose the answer.

2) The general equaiton will be:  (x - h)²/a² - (y - k)²/b² = 1

with        center (h, k),          a = x-radius,          b = y-radius          c = focal length

Since foci are at (1, -1) and (1, 5), the center will be at (1, 2)     ---> h = 1, k = 2, c = 3

--->   (x - 1)²/a² - (y - 2)²/b² = 1

Since the point (4, 2) is on the graph   --->   (4 - 1)²/a² - (2 - 2)²/b² = 1,

solve this equation to find the value of a.

To find the value of b, use the equation:  b²  =  a² + c².

(Use that equation if the major axis is vertical; if the major axis is horizontal, use a²  =  b + c².)

Place those values into the general equation to get your answer.

Compound Interest Forumla:  A  =  P(1 + r/n)^(n·t)

A = Final Amount          P = Initial Amount = 2000         r = rate (as a decimal) = 0.08

n = number of times compounded per year = 1         t = number of years = 14

A  =  2000(1 + 0.08/1)^(1·14)  =  $5874.39

The formula for future value is:  PV  =  FV / (1 + r)^n

PV = Present Value         FV = Future Value = 45000            n = number of compounding periods = 7

r = rate (as a decimal divided by the number of compounding periods per year) = .025  

PV  =  45000 / (1 + .025)^7  =  $37,856.94

(3/4)(100x) = 75x

25x + 50x = 75x

When does  75x  =  75x ?

Always!  This is an "identity", meaning that it is always true for all values of x.

At the end of 1 hour, Tony has collected 3 bags, while Ralph and Della combined has collected 3 bags.

At this rate, for any amount of time Tony collects as much trash as Ralph and Della combined.

I will replace the symbol "f(x)" with "y", so that my typing is easier.

Problem:  y  =  (3x - 7) / (x + 1)

Two steps:

Step 1:  interchange x and y:

--->   x  =  (3y - 7)/(y + 1)

Step 2:  solve for y:

--->  x(y + 1)  =  3y - 7

--->  xy + x  =  3y - 7

--->  xy - 3y  =  -x - 7

--->  y(x - 3)  =  -(x + 7)

--->  y  =  -(x + 7) / (x - 3)

That is the inverse.

Slope is defined to be rise / run

or:  the change in y / change in x

or:  m  =  (y2 - y1) / (x2 - x1)

For two points like (3, 5) and (12, 8) the slope is:  m = (8 - 5) / (12 - 3)  =  3 / 9  =  1 / 3,

so, for every run of 3, there will be a rise of 1, for every change in x of 3, there is a change in y of 1.

Now, let's try two points that lie on a vertical line:  say,  (7, 2)  and  (7, 12)

     --->  m  =  (12 - 2) / (7 - 7)  =  10/0  <--- this is undefined because you can't divide by zero

     --->  every vertical line has "no slope" or an "undefined slope".

Let's compare this to a horizontal line:  say,  (4, 5)  and  (21, 5)

     --->   m  =  (5 - 5) /(21 - 4)  =  0/17  =  0      

     --->  although there was a run of 17 (from 4 to 21), there was no rise (it started at 5 and ended at 5, it is              perfectly flat)

     --->  since zero is a number, every horizontal line has a slope, and its value is 0.

In this case "zero" doesn't mean "none"  

--->  "zero" means a flat line, a horizontal line

--->  "none" (or "no slope") indicates a vertical line

With an equilateral triangle, all three sides are equal.

I'll use KM and KS.

KM = KS   --->              x + 6  =  2x - 3

Subtract x from both sides:  6  =  x - 3

Add 3 to both sides:            9  =  x

So, if  x = 9  and KM = x + 6, how much is KM?

To figure this out:

take the first function:  f(x) = x³+ 8x² + 10x - 1    and the first point: (1, 6)

     find f(1)  =  (1)³+ 8(1)² + 10(1) - 1  =  ?

     If this answer isn't 6, this can't be the correct answer, so eliminate it and go to the next function.

     If this answer is 6, then check out (2,11) and then (4, 39) and, finally, (7, -54).

If all four points work, that is the correct answer; if any one doesn't work, it can't be the right answer, go to the next function.