The standard form of the parabola is \(\boxed{y =- 2{x^2} + 4x - 4}.\)
Further explanation:
The standard form of the parabola can be expressed as follows,
\(\boxed{y = a{x^2} + bx + c}\)
Here, a represents the coefficient of \({x^2},b\) is the coefficient of \(x\) and \(c\) is the constant term.
The vertex form of the quadratic equation can be expressed as follows,
\(\boxed{y = a{{\left( {x - h} \right)}^2} + k}.\)
Here, \(\boxed{\left( {h,k} \right)}\) is the vertex point, h is the \(x-\) coordinate of the equation and k is the y-coordinate.
Given:
The points are \(A\left( { - 2, - 20} \right), B\left( { 0, - 4} \right)\) and \(C\left( { 4, - 20} \right).\)
Explanation:
Substitute \(-2\) for \(x\) and \(-20\) for \(y\) in equation \(y = a{x^2} + bx + c.\)
\(\begin{aligned}- 20&= a{\left({ - 2} \right)^2} + b\left({ - 2} \right) + c\\- 20&= 4a - 2b + c\\- 20 - c&= 4a - 2b\\\end{aligned}\)
Substitute \(0\) for \(x\) and \(-4\) for \(y\) in equation \(y = a{x^2} + bx + c.\)
\(\begin{aligned}- 4&= a{\left( 0 \right)^2} + b\left( 0 \right)+ c\\- 4&= 0 + 0 + c\\- 4&= c\\\end{aligned}\)
Substitute \(4\) for \(x\) and \(-20\) for \(y\) in equation \(y = a{x^2} + bx + c.\)
\(\begin{aligned}- 20&= a{\left(4\right)^2} + b\left( 4\right) + c \hfill\\- 20&= 16a + 4b + c \hfill\\- 20 - c&= 16a + 4b \hfill\\\end{aligned}\)
Substitute \(-4\) for \(c\) in equation \(- 20 - c = 4a - 2b.\)
\(\begin{aligned}- 20 - c&= 4a - 2b\\- 20 + 4 &= 4a - 2b\\- 16&= 4a - 2b\\- 8&= 2a - b\\\end{aligned}\)
Substitute \(-4\) for \(c\) in equation \(- 20 - c = 16a + 4b.\)
\(\begin{aligned}- 20 - c&= 16a + 4b\\- 20 + 4&= 16a + 4b\\- 16 &= 16a + 4b\\ - 4&= 4a + b\\\end{aligned}\)
Add the equations \(- 4 = 4a + b\) and \(- 8 = 2a - b.\)
\(\begin{aligned}- 8 - 4 &= 2a - b + 4a + b\\ - 12&= 6a\\\frac{{ - 12}}{6}&= a\\- 2&= a\\\end{aligned}\)
The value of b can be obtained as follows,
\(\begin{aligned}4\left({ - 2}\right) + b& =- 4\\- 8 + b&=- 4\\b&= - 4 + 8\\\end{aligned}\)
The standard form of the parabola is \(\boxed{y= - 2{x^2} + 4x - 4}.\)
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Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Quadratic equation
Keywords: quadratic equation, standard form, parabola, contain points, (-2,-20), (0,-4), (4,-20), vertex form of the equation, biased, equation, formula, parabola, general equation, explained better.
