CPhill

Oops....mis-read the denominator

x^2 +4x - 21    will have 0's  at

(x + 7) (x - 3)   = 0       set each factor to 0   and x = 3  or x = -7

So....the  domain  is  (-inf, -7) U (-7, 3) U (3, inf)

Thanks to hectictar and EP  for catching my error.....!!!!!

(x^2 + 10x + 21) / ( x^2 + 4x + 21)

The domain will be undefined for any "x" that makes the denominator = 0

But

x^2 + 4x  + 21  = 0      has no real 0's

So....this rational function is defined everywhere ....  (-inf, inf)

See the graph, here : https://www.desmos.com/calculator/t3utixb7ot

If you wanted to send money to me....I'll make sure that the other members get a percentage....

One question.....how do you make (63/6)  = 10 + 1/2 pies ????

By definition, 0!  =1  ......so.....

0 / 0 !   =    0 / 1    =  0

Find a+b+c if the graph of the equation   is a parabola with vertex (5,3) , vertical axis of symmetry, and contains the point (2,0)

We have that

y = a(x - h)^2 + k  and we know that (h, k)  = (5,3)  so

y = a(x - 5)^2 + 3       and since the point (2,0) is on the graph, we have that

0 = a (2 - 5)^2 + 3

0 = a(-3)^2 + 3

0 = 9a + 3

-3 = 9a        divide through by 3

-1 = 3a

-1/3  = a

So...our function is

y = (-1/3) (x - 5)^2 + 3     simplify

y = (-1/3) (x^2 - 10k + 25) + 3

y = (-1/3)x^2 + (10/3)x - 25/3 + 3

y = (-1/3)x^2 + (10/3)x - 25/3 + 9/3

y = (-1/3)x^2 + (10/3)x - 16/3

a = (-1/3)  b = (10/3)  and c = (-16/3)    so

a + b + c  =   [ -1 + 10 - 16] / 3  =  -7/3

Here's the graph : https://www.desmos.com/calculator/chosahxdr6

3 + 1/2   =  7/2       and

2 + 1/4  =   9/4

7/2 - 9/4    =

14/4 - 9/4  =

5/4  =

1 + 1/4

Set these equal

-x^2 - x + 1  = 2x^2 - 1   simplify

0 = 3x^2 + x - 2       factor

0 = (3x - 2) ( x + 1)    set both factors = 0   and the x intersection points are x = 2/3  and x = -1

So...we have the intersection points are  (a,b) and (c,d)  = (-1,b) and (2/3, d)

So.....c - a  =  2/3 - (-1)  =  1 + 2/3   =   5/3

We know that

y = a(x - 2)^2 - 4

And the point (4,12)  is on the graph....so...

12 = a(4 - 2)^2 - 4

12 = 4a - 4

16 = 4a

4 = a

So  we have that

y = 4(x - 2)^2 - 4

y = 4(x^2 - 4x + 4) - 4

y = 4x^2 - 16x + 16 - 4

y = 4x^2 - 16x + 12

And to find the roots

0 = 4x^2 - 16x + 12      divide through by 4

0 = x^2 - 4x + 3   factor

0 = (x - 3) ( x - 1)      set each factor to 0  and m = 3  and n = 1

So....m - n  =  3 - 1 = 2

Here's a graph :  https://www.desmos.com/calculator/yvxfjhlbkv

Could you post the original problem???