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Happy 2018, Rosala......!!!

x =  (4 - x) / (y^2 - x)

y^2x  - x^2  =   4 - x

y^2x  =   x^2 - x  +   4

y^2  =   [ x^2  -  x  +  4 ]  /  x

y^2  =  x  -  1   +  4/x

Notice  that  the first two terms always result in an integer for any integer value of x

But   4/x     is only an integer  when x  =  ±1, ±2 or ±4

But we can reject  the negative values  because  they make the right side negative.....so y is not a real number  for these values

And when x = 2,  y  = ±√3    which isn't an integer pair

So.....when  x  =  1,  y   = ±2

And when  x  =   4,  y   = ±2

So......the ordered pairs of positive integers that make this true  are  (1,2), (-1,2), (4, 2)  and (4 , -2)

So....the correct answer is  (e)

We can find the height of this trapezoid as follows.....

h =  sqrt [ 10^2   -  6^2]  =  sqrt [100 - 36 ]  =  sqrt [ 64]  =   8

And the length  of the diagonal  will be the hypotenuse of a right triangle with legs of 8  and 16

So....the diagonal   is

sqrt  [ 8^2  +  16^2]  =    sqrt  [  8^2  ( 1  + 2^2) ]  =   8 sqrt (5)  units

Hi Rosala, it is always a pleasure to see you here!

Happy New Year to you too :)

13.5 m/s

It will do a lot more than that in an hour!

13.5*3600 m/h

It will do a lot less km than metres in an hour

13.5*3600 / 1000  km/hour    =   48.6km/hour

That is definitely less than 50km/h 

4 = 10log₁₀(S1/40)

4 = log₁₀((S1/40)¹⁰)

Now raise both sides to the power of 10

10^4=10^(log₁₀((S1/40)¹⁰)

10000=(S1/40)^10

(10000)^(0.5)=((S1/40)^10)^0.5

100=(S1/40)^5

100*40^5 = S1^5

100*10^5*4^5=S1^5

10 000 000*4^5=S1^5

4*10^(1.4)=S1

S1=100.47  approx

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That was really messy, I will try and check it.

4 = 10log₁₀(S1/40)

0.4=log₁₀(S1/40)

10^0.4=S1/40

40*10^0.4=S1

S1 = 100.47  approx

The second method is much better !!

8x^2  +  15x   + c  =   0

This will nave two real  roots if

15^2  -  4 * 8 * c  >  0

225  -  32c  >  0

225  >  32c

225/ 32  > c

7.03125  > c

So......c will have two real roots  [and  be positive ]   for  all  integer values  from 1 to 7, inclusive

And the procuct of these possible values for c   =  7!   =  5040

-2x^2 -  20x -  53   complete the square

-2   ( x^2   +  10x  +  25   +  53/2  - 25)

-2 [  (x + 5)^2  +  53/2  -  50/2 )

-2 [ (x + 5)^2  +  3/2  ] 

-2( x + 5)^2  -  3

a  =  -2         d  =  5           e   =  -3

And their sum  =  0

#11 slope = 2/3; y intercept is 9

#12 slope = 7; y intercept is -3

#13 passes through (2, -1) and (0,1)

#14 passes through (2, 3) and (4,4)

I'll  do   11  and 13......try to do the other two.......12 is similar to 11  and  14 is similar to 13

11.....if the y intercept  is  9,  the point   (0,9)  is on the graph

So.....we have.....

y  =  (2/3)(x  -  0)  +  9

y  =  (2/3)x  +  9

13.....first.....find the slope  = 

 [subtract the y values in some order] /  [ subtract the x values in the same order ]

[ 1  -  -1  ]  /  [ 0  - 2 ]  =    2 / -2   = -1

So.....using the slope  and either of the points  - I'll use (0, 1) - we have

y  =  -1  (x  - 1)  +  0

y = -x + 1

(x, y)  =  (5, -12)

And sin   =   y / r

So....we need to find r

r  =  sqrt [ x^2 + y^2]  =  sqrt  [ 5^2  + (-12)^2 ]   =  sqrt  [ 25  +  144 ]  =  sqrt [ 169] = 13

So....sin  =   y / r  =  -12 / 13

See if you can work out the first two from this:

Thank you !!!

1) What is  -pi/4 in degrees?  What quadrant is it in?

2) The reference angle is the one from the poisitve, or negative x axix. 

    Willl you use the positve x axis OR the negative x axis for this quadrant?

3) What reference (acute) angle will give the same trig ratios?

Think about my questions, if you get stuck let me know where  :)

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Here are a couple of general videos that might help with this topic.

Hi Lexi,

1) To answer this question you first need to graph (-5,12) on the number plane.

2) What quadrant are you in?

3) Is sine positive or negative in this quadrant?

Now sine is opposite/hypotenuse  Hopefully you already know that.

The angle you must use is the one that has one arm on the positive or negative x axis. (whichever will give you an acute angle.

If it is in the 1st or 2nd quad you go up and if it is in the 3rd or 4th quad you go down.

4) Do you know acute angle to use?

5) Now you have a rise, that will definitely be 12 (the y vaue), and you ave a run, that will be 5 (the x vaue) but you will nee to use pythagoras's theorem to work out the hypotenuse.

6) Now you know what sine is.

Follow these instructions and tell me the answer or tell me the answers to the questions I have asked and tell me where you get stuck and I will help you more. :)

You should begin by deducing what the 'parent function' looks like, and then working from left to right, rather than right to left as you have tried to do.

For this example, it should be clear that the 'pf ' will be a quadratic, so start with the assumption that

\(\displaystyle f(x) = ax^{2}+bx +c\).

In that case,

\(\displaystyle f(x+2)=a(x+2)^{2}+b(x+2)+c\\ =ax^{2}+(4a+b)x+(4a+2b+c)\\\equiv 4x^{2}-2 .\)

Equating coefficients across the identity leads to, (in turn), a = 4, b = -16 and c = 14.

If though, for example, you were given \(\displaystyle f(x^{2}+2)=4x^{2}-2\), then your presumed 'pf ' would be \(\displaystyle f(x)=ax+b\), leading to \(\displaystyle f(x^{2}+2)=a(x^{2}+2)+b\) , etc. .

Tiggsy

Notice that  as y  changes by  +1,  x changes  by   -2

So....this is a linear equation  where the slope, m,   = 

change in y  / change in  x   =     

 +1  / -2    =    -1/2

So.....using   y  =   m  ( x  -  x1)  +  y1         where  (x1, y1)  is any point on the graph, we have

y  =  (-1/2) ( x - 4)   +  0      simplify

y  =  (-1/2)x  +  2

f(x + 2)  =  4x^2  - 2

Notice that  this must be the parent function, f(x),  shifted to the left by 2 units.....thus....we can shift the function 2 units to the right to find the parent function....so we have

4(x - 2)^2  -  2  =

4(x^2 - 4x  + 4)  -  2  =

4x^2 - 16x  +  16  - 2  =

4x^2 -  16x  +  14  =     f(x)

Happy New Year, Melody.....[  and everyone else on the site  !!!  ]

x=age

h(x)=height

f(h(x))=y=food consumption

x=time spent in a gym

h(x)=muscle bulk

f(h(x))=strength

I didn't think that question would get answered on that thread, so made a new one. Either ways, thank you for your help CPhill, I really appreciate it! :)

HAPPY NEW YEAR, WEB2.0CALC! WISH EVERYONE A GREAT YEAR AHEAD!

Thanks, azsun......I  hope we're correct....LOL!!!

k(n)  =  32  - 0.05n

Note that

k(n)  + 0.05n  =  32

So

Volume of Water  + Volume of marbles  =  Total Volume

So...this implies that  the volume of  each marble must be 0.05  L

Note  ...  (a  + b)^2  =  a^2  + 2ab  +  b^2

Let  a  =  3x      and b  =  (2y + 1)     ....   so we have

(3x  + (2y + 1) )^2  =

(3x)^2   +  2 (3x)(2y+1)  + (2y + 1)^2

We only  care about the middle term....so....

2(3x)(2y + 1)   =

6x (2y + 1)  =

12xy  +  6x

So.....the coefficient is  12