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Hi melody,

oops, yes it is mentioned here that the range of f is \(y>=-2\)

sorry..

Melody,

mmm, yes, that makes all the sense in the world!...thank you..

Hey hey!!!..what do you know!!!!...whoop whoop!..I am really excited!!..Thank you sir!

Here :

P(x) = ax² + bx + c

-3 = a*(0)² + b*(0) + c  so c = -3 immediately

P(x) = ax² + bx -3

13 = a*(-2)² + b*(-2) - 3  or  4a - 2b  = 16     Divide all terms by 2    2a - b = 8    (1)

3 = a*(3)² + b*(3) - 3      or  9a + 3b  = 6      Divide all terms by 3     3a + b = 2   (2)

Now subtract (1) from (2) to find a, then substitute the result back into either (1) or (2) to get b.

Have you abbreviated this Juriemagic?

It seems to have some words missing....

All I get from this is that it is a parabola with the axis of symmetry is x= -2

If the roots differ by 4... maybe it means that they are 4 units apart.

Maybe they are at (-4,0)  and (0,0)

It could be concave up or concave down.  The y value of the turning point cannot be determined.

So the most I can say is:

\(y=ax(x+4) \qquad \text{Where a is any real number.}\)

If it is a quadratic sequence,

the sequence of first differences has to be an arithmetic sequence.

BECAUSE the second sequence of differences must all be the identical same number.

I was not suggesting anything different.

Looks good!

I was using my own diagram.

A parabola with equation y = x^2 + bx + c passes through the points (-1,-11) and (3,18). What is c?

Hello Guest!

\(y=x^2+bx+c\\ P_1(-1,-11)\\ P_2\ (3,18)\)

\(-11=1-b+c\\ 18=9+3b+c\\\)

\(c=-11-1+b=18-9-3b\\ 4b=18-9+1+11\)

\(b=5.25\)

\(c=18-9-3b\\ c=18-9-3\cdot 5.25\)

\(c=-6.75\)

The parabola has the equation \(y=x^2+5.25x-6.75\)

Let

f(x) = 2x^2 - 4 if x <= 2    and    ax+5  if  x > 2

Find a if the graph of y = f(x) is continuous. (which means the graph can be drawn without lifting your pencil from the paper).

Hello Guest!

\(\color{BrickRed}f(x)=2x^2-4=ax+5\\ x=2\\ 2\cdot 2^2-4=2a+5\\ a=(2\cdot 2^2-4-5)/2\)

\(\large a=-\frac{1}{2}\)

  !

Melody!!!...,

I do understand everything you had shared with me....really I do..thank you also for helping..again....

I do however need to ask something....with quadratic sequences we have a 1st and 2nd difference, okay, I know the second difference is ALWAYS Arithmetic by nature...The 1st one however, is that aslo always Aritmetic or can that be of a geometric nature as well?

ElectricPavlov..

gosh it really makes soo much sense now!!..I do however have a question but will post that with my response to Melody..please feel free to also comment...thanx a million!!!

Alan, Thanx for walking me through this...honestly, you will never realise how much this means!!..okay, x:y = 3:2

that gives \(2x=3y\)

which can go to \(2x-3y=0\)

Nothing can be factored..soooo, mabe then I should look at simultaneous equations???, meaning...

\(2x-3y=0.....(1)\)

\(x+y=2\)

\(x=2-y.....(2)\)

(2) in (1)

\(2(2-y)-3y=0\)

\(4-2y-3y=0\)

\(-5y=-4\)

\(y= {4 \over 5}...(3)\)

(3) in (2)

\(x=2- {4 \over5}\)

\(x={6 \over5}\)

So x then is 6\5 and y is 4\5

Is this correct maybe??..

Yes that is correct Juriemagic.   

EP has given you a good answer, my answer is the same.

I am just going to give you a couple of examples first to help you understand

Now for you one.  And this is exactly the same as what EP did

I hope EP and I have helped you understand.    

It is no more trivial than our answers.

You have not drawn the diagram but you did use one.

Cutting out steps does not make an answer more simple, it just makes the answer less self-explanatory.

I'll give you my 3-step solution without using anyone's diagram.

1/    ∠DAQ ≅ ∠BAP = arctan(DQ / AD) = 26.56505118º

2/    ∠PAQ = 90º - (∠DAQ + ∠BAP) = 36.86989765º

3/     sin(∠PAQ) = 0.6    or   3/5  

It's trivial enough, is it not? 

(calculator: 0.5 => inv => tan => * 2 => -90 => + => sin => 0.6 )

I get it, thank you sooooooo much 

q = quarters =                        value = 25 q

n = # nickels  = q+9               value = 5(q+9)                summed = 135

25 q       +   5 (q+9)    = 135         solve for q = # quarters        then #nickels = q+9

Place X on the other side of the equation to get:

-2y = -4x+6

You can reverse the signs too to get:

2y = 4x -6

Now divide it by two to get the final answer:

y = 2x-3

Multiply the numerator and denominator  by     sqrt 3 - sqrt 6

expand and then simplify   to   -3 + 2 sqrt 2