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Starting with this form:  y - k  =  a(x - h)²   where (h, k) is the vertex.

--->     y - 4  =  a(x - 2)²

Now, the problem is to find the value of a.

Use the point (1, 1)     --->     1 - 4  =  a(1 - 2)²     --->     -3  =  a(-1)²     --->   -3  =  a

So, the equation is:  y - 4  =  -3(x - 2)²   

Multiply this out:  y - 4  =  -3(x² - 4x + 4)

                            y - 4  =  -3x² + 12x - 12

                                 y  =  -3x² + 12x - 8

ummmm... why does it say this post is invisible to normal users? 

circumference = 2 pi r  =  16 pi

A fraction of the circumference   6 ft / (16 pi)   x  360^o  = 42.97 ^o

Maybe   6n = 36     n = 6

divisors of 36   =   1   2   3   4   6    9   12    18   36

Profit  or (loss)     =    marginal revenue -  marginal cost of production - fixed cost

Profit  or loss              80x       - 60sqrt(x+1) - 340

 3 units                      (80)*3    - 60(sqrt4)     - 340 =   -220     a loss

8 units                       (80)*8    - 60(sqrt9)     - 340 = + 120     a profit

If Tyler's original height was  x  it is now  x + 0.20x  or  1.20x.

Since he is not 4 feet tall:  1.20x  =  4   --->   x  =  4 / 1.2  =  3.33.

So, both Tyler and Cody were 3.33 feet (or 3 feet 4 inches)

If Cody grew at 10%, we need to add 10% to his original height:  10% of 3 feet is 0.30 feet or 3.6 inches.

Also, 10% of 4 inches is 0.40 inches.

Starting with 3 feet 4 inches and adding 3.6 inches and 0.40 inches, we get 3 feet 8 inches.

An angle whose vertex is on a circle has one-half the number of degrees as the intercepted arc.

Another way of saying this is to say that the arc has twice as many degrees as the inscribed angle.

In number 1:  angle(L) = 38^o so arc(JK) as 76^o.

In problem #2, find the number of derees in angle(L) and then use that answer to find the number of degrees in arc(MN).

Hi Shaezy,

It is good to see you interacting here.

Your substitution is perfect, EP taught you well,  it is your simplifying that is incorrect.

\(If \;\;f(x) = (1 − x)^2 \qquad g(x) = 9 − x,\\  find\;\; f(g(x)) \;\;and ;\;g(f(x)). \\\\~\\ g(f(x))= 9-(1-x)^2\)

So far so good.  BUT

\(\text{This is NOT }(8-x)^2\\ \text{it is  }   \quad9-[(1-x)^2] \)

You have to expand the bracket and then take away ALL the terms from 9.

Have another go at it    

understood i really appreciate it!

The first one:  the series is not geometric:  16, 7.5, -1, -9.5  is arithmetic with a common difference of -8.5.

This series won't converge because its sum gets smaller than any limit who can try.

You can rewrite  x² + 8x + 4  as  4( ¼x² + 2x + 1 )

If you graph   y  =  ¼x² + 2x + 1,   it will have the same roots as  y  =  x² + 8x + 4  (but it will not be otherwise congruent).

So:    A  =  ¼     and     B  =  2     and    A + B  =  9/4.

One way to do this is to find the vertex of this quadratic:  y  =  0.5b² + 5b - 3

Move the 3 to the other side:                                      y + 3  =  0.5b² + 5b

Factor out the 0.5:                                                       y + 3  =  0.5(b² + 10b)

Complete the square:                                       y + 3 + 12.5  =  0.5(b² + 10b + 25)

Factor and simplify:                                                 y + 15.5  =  0.5(b + 5)²

The vertex is (-5, -15.5) which means that the minimum is -15.5 and this occurs at -5.

Some respond to this now please!!!!!!

Question #1:  The base of the ladder is 7m from the building; the length of the ladder is 25m. 

     Pythagorean Theorem:  7² + h²  =  25²     --->     49 + h² =  625     --->     h²  =  576     --->     h  =  24.

     Since the base of the ladder is 2 meters above the ground, it will reach 26 meters.

Question #2:  The length of the ladder now is 'x', the hypotenuse. The base of the triangle is still 7m.

     The ladder will have to reach a height of 20m (we need to subract the distance of the ladder from the ground.)

      7² + 20²  =  x²     --->    49 + 400  =  x²     --->     449  =  x²     --->     x  =  21.2 meters

                                  --->  retract the ladder  25m - 21.2m  =  3.8m

Qustion #3:  The length of the ladder is 25m; the height is 20m, find the base of the triangle.

     b² + 20²  =  25²     --->     b² + 400  =  625     --->     b²  =  225     --->     b  =  15  (or 8m farther than it was)

Since 100 + x is a streight angle and stright angles equal 180 degrees, x = 80 degrees since 180-100 equals x.

80 degrees

Those are supplementary angles which means that their combined total is equal to 180 degrees.

180-100 is 80

so the answer is 80 degrees

Equation of a circle:  (x - h)² + (y - k)²  =  r²  where the center is (h, k) and the radius is r.

Since the center is (2,3)  h = 2  and  k = 3.

To find the radius we need to know the distance from (-1,6) to (2,3).

Using the distance formula:  r  =  sqrt( (2 - -1)² + (3 - 6)² )  =  sqrt( (3)² + (-3)² )  =  sqrt( 9 + 9 )  =  sqrt( 18 )

Since  r  =  sqrt( 18 )  --->   r²  =  18

So, our equation is:  (x - 2)² + (y - 3)²  =  18

Multiplying this out:  (x² - 4x + 4) + (y² - 6y + 9)  =  18

The equation for the circle is:     x² + y² - 4x - 6y - 5  =  0                                                              

From here, pick out A, B, and C, and multiply them together.

A + B = 4/3.

I am answering question 2. I'm assuming this question is from AOPS algebra a, and I also got stuck on this question. i got it wrong about 23 times. the answer is [200, 350)

Tan 60 =  sin 60 / cos 60 = sqrt3/2  / 1/2 = sqrt 3

16  tan 60 = DF  =   16 sqrt 3

sin 60 = sqrt 3 / 2   = DF/DE

              sqrt3 /2 =   16 sqrt3 / DE

               DE = 16 sqrt3 / (sqrt3 /2) = 32

This is a mountain shaped parabola           -25t^2+75t+24,

the maximum will occur at   t = - b/2a    =  -75/(2 (-25)) = .......   seconds 

If you need to know the HEIGHT at that time, t , you will need to plug that value of t into the equation .....

𝐬𝐮𝐦𝐟𝐨𝐫(𝐧, 𝟏, 𝟏𝟎𝟎𝟎𝟎𝟎𝟎, 𝟐 / ( 𝐧^𝟑 + 𝟑* 𝐧^𝟐 + 𝟐* 𝐧))=𝐢𝐭 𝐜𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐬 𝐭𝐨 𝟏 / 𝟐

𝐄𝐱𝐩𝐚𝐧𝐝   (𝟐𝐱^𝟐*𝐲^𝟑  -  𝟏/𝟒𝐲)^𝟗

𝟓𝟏𝟐 𝐱^𝟏𝟖 𝐲^𝟐𝟕 - 𝟓𝟕𝟔 𝐱^𝟏𝟔 𝐲^𝟐𝟓 + 𝟐𝟖𝟖 𝐱^𝟏𝟒 𝐲^𝟐𝟑 - 𝟖𝟒 𝐱^𝟏𝟐 𝐲^𝟐𝟏 + (𝟔𝟑 𝐱^𝟏𝟎 𝐲^𝟏𝟗)/𝟒 - (𝟔𝟑 𝐱^𝟖 𝐲^𝟏𝟕)/𝟑𝟐 + (𝟐𝟏 𝐱^𝟔 𝐲^𝟏𝟓)/𝟏𝟐𝟖 - (𝟗 𝐱^𝟒 𝐲^𝟏𝟑)/𝟏𝟎𝟐𝟒 + (𝟗 𝐱^𝟐 𝐲^𝟏𝟏)/𝟑𝟐𝟕𝟔𝟖 - 𝐲^𝟗/𝟐𝟔𝟐𝟏𝟒𝟒
(𝟏𝟎 𝐭𝐞𝐫𝐦𝐬)

Thanks Alan,

It really is a strange one :)  

It is happening relatively often.

Maybe it is something to do with copyright that Admin is trying to fix? 

I have no idea.