Question 1114193

Find the quadratic equation that fits the following set of data points.
(−2​,−8​),(−4​,−12​),(1​,13​)
​f(x)=______.
​(Use integers or fractions for any numbers in the expression. Do not​ factor.)
<pre>Standard equation of a quadratic function: {{{matrix(1,11, f(x), "=", ax^2 + bx + c, or, y, "=", ax^2 + bx + c, or, ax^2 + bx + c, "=", y))}}}

(&#8722; 2&#8203;,&#8722; 8&#8203;)
{{{matrix(1,3, ax^2 + bx + c, "=", 0)}}}
{{{matrix(1,3, a(- 2)^2 + b(- 2) + c, "=", - 8)}}} ------ Substituting (- 2, - 8) for (x, y)
4a - 2b + c = - 8 ------- eq (i)


(&#8722; 4&#8203;, &#8722; 12&#8203;)
{{{matrix(1,3, ax^2 + bx + c, "=", 0)}}}
{{{matrix(1,3, a(- 4)^2 + b(- 4) + c, "=", - 12)}}} ------ Substituting (- 4, - 12) for (x, y)
16a - 4b + c = - 12 ------- eq (ii)


(1&#8203;, 13&#8203;)
{{{matrix(1,3, ax^2 + bx + c, "=", 0)}}}
{{{matrix(1,3, a(1)^2 + b(1) + c, "=", 13)}}} ------ Substituting (1, 13) for (x, y)
a + b + c = 13 ------- eq (iii)

 4a - 2b + c = -  8 ------- eq (i)
16a - 4b + c = - 12 ------- eq (ii)
  a +  b + c =   13 ------- eq (iii)

12a - 2b = - 4 ------ Subtracting eq (i) from eq (ii) 
2(6a - b) = 2(- 2) 
6a - b = - 2 ------ eq (iv)

3a - 3b = - 21 ------ Subtracting eq (iii) from eq (i) 
3(a - b) = 3(- 7) 
a - b = - 7 ------ eq (v)

5a = 5 ------- Subtracting eq (v) from eq (iv)
{{{matrix(1,5, a, "=", 5/5, "=", 1)}}}

1 - b = - 7 ------- Substituting 1 for a in eq (v)
- b = - 7 - 1
- b = - 8
{{{matrix(1,5, b, "=", (- 8)/(- 1), "=", 8)}}}


1 + 8 + c = 13 ------- Substituting 1 for a, and 8 for b in eq (iii)
9 + c = 13
{{{matrix(1,5, c, "=", 13 - 9, "=", 4)}}}


{{{matrix(1,3, ax^2 + bx + c, "=", 0)}}}
{{{highlight_green(matrix(1,7, x^2 + 8x + 4, "=", 0, or, f(x), "=", x^2 + 8x + 4))}}} ------- Substituting 1 for a, 8 for b, and 4 for c</pre>