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One method

If x + 4 is a factor then - 4 is a root

So

f(-4) should = 0

So we have    4(-4)^4 - 2(-4)^3  -52(-4)^2  - 15 (-4) + 8 =

1024  + 128 - 832 + 60 + 8  =  388...so .... not a root

Another method....synthetic division

-4    [  4     -2     - 52     - 15     8   ]

                -16      72      -80  380

       ________________________

          4     -18    20      -95   388            same remainder as above  !!!

If it was a root....we sould get 0  as a result

The period of a pendulum is the amount of time it takes a pendulum to complete one full cycle. One cycle would be the travel from one side to another, and then back to its starting position. Looking at the data we see that it completes a half cycle in 1 second. It goes from what we can imagine as the left side peak of 5.4 cm to a right side peak of 5.4 cm in 1 second. It would take 2 seconds to complete the movement back to its starting position. The period of the function is therefore 2 seconds. The frequency of a function is the inverse of the period, i.e.: period = seconds/cycle, frequency = cycles/second. If the period is 2 seconds, then the frequecy is 1/2 seconds^-1. The only correct answer is the amplitude is 0.4 cm as the distance between the peak and rest positions is 5.4-5 = 0.4 cm.

Assuming that t = 0   = 12 AM....the correct function is

h(t) = 0.6cos ( pi * t /6 ) + 1.6

Mathematician

c is correct

3pi * 4  =

12 pi

pi is irrational.....and a real non-zero multiple of an irrational is still an irrational

BTW.... b =  .0444... =   4/90 = 2/45....so 4 * 2 /45 = 8 /45  which is rational

MathWizard likes to play a fun number trick on her friends. She tells them to think of a number. She then tells them to subtract their number from 2 and multiply the result by 5. To this product she tells them to add half the difference when 24 is subtracted from 12 times their original number.

(a) If I use the number 8 and follow MathWizard's steps, what number will I get?

Mmmmmm....that's a lot of stuff..LOL!!!

2 - 8  = - 6

-6 * 5 = -30

"  half the difference when 24 is subtracted from 12 times their original number."   translates to

(1/2) ( 8 * 12 - 24 ]  =    (1/2) [ 96 - 24 ]  = (1/2)(72) = 36

So

-30 + 36 =

6 

The sum of the roots = -b / a =  -21/18  =  -7/6

The product of the roots = c / a = -400/18 = -200/9

Let the roots be  p, q

p + q =   -7/6    

pq = -200/9    ⇒   2pq = -400/9

(p + q)^2 =   49/36

p^2 + 2pq + q^2 = 49/36

p^2 +  (-400/9) + q^2  = 49/36

p^2 + q^2 =  49/36 + 400/9

p^2 + q^2 =  [ 49 + 1600] / 36  = 1649 / 36   ≈   45.8....just as EP found !!!!

I assumed this was    18x^2 +21x -400 = 0      Yah?

(d)   y = 2 ( x + 1)^2 - 5             add 5 to both sides

y + 5 =  2 ( x + 1)^2             multiply both sides by 1/2

(1/2) ( y + 5) = ( x + 1)^2

The vertex is ( -1, -5)       and this opens upward

4p = 1/2

p = 1/8

The focus is ( -1, -5 + 1/8) = (-1, -39/8 )

The directrix is y = -5 - p =  - 5 - 1/8 =   -41/8  ⇒  y = -41/8

Sorry but that was wrong

(c)   y = (1/16) ( x - 3)^2       easier!!!

16 y = (x - 3)^2

16 ( y - 0) = (x - 3)^2

The vertex is (3, 0)       and this turns upward

4p = 16

p = 4

The focus is  ( 3, 0 + p)  = ( 3, 0 + 4) = ( 3, 4)

The directrix is    y = 0 - p  = 0 - 4  =  -4  ⇒    y = -4

Quadratic formula....

roots   -5.333  and  4.166     squared and added together = 45.79

(b)   y = (-1/4) ( x + 2)^2 + 1

y - 1  =  -(1/4)(x + 2)^2

4 ( y - 1) = - ( x + 2)^2        the parabola turns downward

The vertex is ( -2, 1)

4p = 4      so   p = 1

The focus is ( -2, 1 - p) = (-2, 1 - 1 ) = (-2, 0)

The directrix is    y = 1 + p  =  1 + 1   = 2   ⇒    y = 2

a)   y = (1/8) ( x - 6)^2 + 4

We are working backwards, here

(y - 4) = (1/8) ( x - 6)^2

8(y - 4) = ( x - 6)^2           the vertex is ( 6, 4)

4p = 8   ⇒    p = 2

The parabola turns upward  so  the focus is  ( 6 , 4 + p)  =  ( 6 , 4 + 2) = (6, 6)

And the directrix is   y = 4 - p  =  4 - 2   = 2   ⇒  y = 2

(x-y)/2    would be half the difference between  x and y

half the difference between 9 and 3    would be     (9-3)/2 = 3

When the object hits the ground   height (h) will = 0   obviously....substitute this in for h and then solve for t

0 = -16t^2+64t

0 = -16t(t-4)         so t = 0   or   4 seconds       (0 is the time before it was launched)

I am so so sorry but I am sure someone else will know!

I can’t view that! 

I mean I am not to sure but I think it is (The period of the function is 5.4)

I see C-Phill is answering to so he will probally give you a acurate answer XD!

This is the reflection of the "floor" function across the x axis  and shifted to the left by 2 units

Here is the graph:

The correct graph is at the top left

See the answer here :

https://web2.0calc.com/questions/amount-of-real-solutions