Question 813287
<pre>
y = a(x - h)² + k

Substitute (x,y) = (1,-1)

-1 = a(1 - h)² + k
 
Substitute (x,y) = (2,-1)

-5 = a(2 - h)² + k

Substitute (x,y) = (3,-7)

-7 = a(3 - h)² + k

Solve this system:

-1 = a(1 - h)² + k
-5 = a(2 - h)² + k
-7 = a(3 - h)² + k

Solve the first for k

            -1 = a(1 - h)² + k
-1 - a(1 - h)² = k

Substitute in the second and simplify

-5 = a(2 - h)² + k
-5 = a(2 - h)² - 1 - a(1 - h)²
-4 = a(2 - h)² - a(1 - h)²
-4 = a[(2-h)² - (1-h)²]
-4 = a[(2-h) - (1-h)][(2-h) + (1-h)]
-4 = a[2 - h - 1 + h][2 - h + 1 - h]
-4 = a[1][3 - 2h]
-4 = a(3 - 2h)

Substitute in the thirdd and simplify

-7 = a(3 - h)² + k
-7 = a(3 - h)² - 1 - a(1 - h)²
-6 = a(3 - h)² - a(1 - h)²
-6 = a[(3-h)² - (1-h)²]
-6 = a[(3-h) - (1-h)][(3-h) + (1-h)]
-6 = a[3 - h - 1 + h][3 - h + 1 - h]
-6 = a[2][4 - 2h]
-6 = 2a(4 - 2h)
-6 = 2a·2(2 - h)
-6 = 4a(2 - h)

Divide equals by equals:

{{{(-4)/(-6)}}}{{{""=""}}}{{{(a(3 - 2h))/(4a(2-h))}}}

{{{2/3}}}{{{""=""}}}{{{(cross(a)(3 - 2h))/(4cross(a)(2-h))}}}

{{{2/3}}}{{{""=""}}}{{{(3 - 2h)/(4(2-h))}}}

Cross-multiply:

8(2-h) = 3(3-2h)
16-8h = 9-6h
-2h = -7
  h = {{{7/2}}}

Sustitute in

-1 = a(1 - h)² + k
-5 = a(2 - h)² + k

-1 = a(1 - {{{7/2}}})² + k
-5 = a(2 - {{{7/2}}})² + k

-1 = a{{{(-5/2)^2}|} + k
-5 = a{{{(-3/2)^2}}} + k

-1 = a{{{(25/4)}|} + k
-5 = a{{{(9/4)}}} + k

Multiply both equations through by 4

 -4 = 25a + 4k
-20 =  9a + 4k

Multiply the 1st eq. by -1 and add term by term

  4 = -25a - 4k 
-20 =   9a + 4k
---------------
-16 = -16a
  1 = a

Substitute in  

-4 = 25a + 4k
-4 = 25(1) + 4k
-4 = 25 + 4k
-29 = 4k
{{{-29/4}}} = k

So the equation of the parabola in standard form is

y = a(x - h)² + k
y = 1(x - {{{7/2}}})²  - {{{29/4}}}
y = (x - {{{7/2}}})²  - {{{29/4}}}


Edwin</pre>