The equation of a parabola is a quadratic equation.
The equation of the parabola is: [tex]\mathbf{y = 4x^2 + 32x -32}[/tex]
The points are given as:
[tex]\mathbf{(x,y) = (-5,-224),(-3,-92),(1,4)}[/tex]
Substitute these values in:
[tex]\mathbf{y = ax^2 + bx +c}[/tex]
So, we have:
[tex]\mathbf{-224 = a(-5)^2 + b(-5) +c}[/tex]
[tex]\mathbf{-224 = 25a - 5b +c}[/tex] ------ (1)
[tex]\mathbf{-92 = a(-3)^2 + b(-3) +c}[/tex]
[tex]\mathbf{-92 = 9a -3b +c}[/tex] ---- (2)
[tex]\mathbf{4 = a(1)^2 +b(1) +c}[/tex]
[tex]\mathbf{4 = a +b +c}[/tex] ---- (3)
Subtract (3) from (2)
[tex]\mathbf{9a - a - 3b - b + c - c =-92 - 4}[/tex]
[tex]\mathbf{8a - 4b =-96}[/tex]
Multiply by 3
[tex]\mathbf{24a - 12b = -288}[/tex]
Subtract (3) from (1)
[tex]\mathbf{25a -a - 5b -b +c - c = -224 - 4}[/tex]
[tex]\mathbf{24a - 6b = -228}[/tex]
Subtract [tex]\mathbf{24a - 6b = -228}[/tex] from [tex]\mathbf{24a - 12a = -288}[/tex]
[tex]\mathbf{24a- 24a - 12ba +6b = -288 + 96}[/tex]
[tex]\mathbf{- 6b = -192}[/tex]
Divide through by -6
[tex]\mathbf{b = 32}[/tex]
Substitute [tex]\mathbf{b = 32}[/tex] in [tex]\mathbf{8a - 4b =-96}[/tex]
[tex]\mathbf{8a - 4 \times 32 = -96}[/tex]
[tex]\mathbf{8a - 128 = -96}[/tex]
Collect like terms
[tex]\mathbf{8a = 128 -96}[/tex]
[tex]\mathbf{8a = 32}[/tex]
Divide both sides by 8
[tex]\mathbf{a = 4}[/tex]
Substitute [tex]\mathbf{a = 4}[/tex] and [tex]\mathbf{b = 32}[/tex] in [tex]\mathbf{4 = a +b +c}[/tex]
[tex]\mathbf{4 + 32 + c = 4}[/tex]
[tex]\mathbf{36 + c = 4}[/tex]
Subtract 36 from both sides
[tex]\mathbf{c = -32}[/tex]
Substitute values of a, b and c in: [tex]\mathbf{y = ax^2 + bx +c}[/tex]
[tex]\mathbf{y = 4x^2 + 32x -32}[/tex]
Hence, the equation of the parabola is: [tex]\mathbf{y = 4x^2 + 32x -32}[/tex]
Read more about quadratics and parabolas at:
https://brainly.com/question/10738281
