What is the equation, in standard form, of a parabola that contains the following points?(–2, –20), (0, –4), (4, –20)

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23-06-2016
Mathematics

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What is the equation, in standard form, of a parabola that contains the following points?(–2, –20), (0, –4), (4, –20)

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meerkat18 meerkat18 · Answer 1 · 2016-07-06T22:42:45+02:00

From the following points, we can see that for y equals -20, there are two x's. Hence the equation should contain x^2. The graph is going downward too so we are expecting a -4a. The equation is (y-k) = 4a (x-h)^2. h as the x component of the vertex is 1 from the points. substituting, we get (y-k) = 4a (x-1)^2. we substitute the points to get k and a, k is equal to -2 and a is equal to -0.5. Hence the equation becomes (y+2) = -2 (x-1)^2. The standard form is -2x^2 + 4 x - 4 - y = 0.

AkshayG AkshayG · Answer 2 · 2019-11-02T11:00:40+01:00

The standard form of the parabola is [tex]\boxed{y =- 2{x^2} + 4x - 4}.[/tex]

Further explanation:

The standard form of the parabola can be expressed as follows,

[tex]\boxed{y = a{x^2} + bx + c}[/tex]

Here, a represents the coefficient of [tex]{x^2},b[/tex] is the coefficient of [tex]x[/tex] and [tex]c[/tex] is the constant term.

The vertex form of the quadratic equation can be expressed as follows,

[tex]\boxed{y = a{{\left( {x - h} \right)}^2} + k}.[/tex]

Here, [tex]\boxed{\left( {h,k} \right)}[/tex] is the vertex point, h is the [tex]x-[/tex] coordinate of the equation and k is the y-coordinate.

Given:

The points are [tex]A\left( { - 2, - 20} \right), B\left( { 0, - 4} \right)[/tex] and [tex]C\left( { 4, - 20} \right).[/tex]

Explanation:

Substitute [tex]-2[/tex] for [tex]x[/tex] and [tex]-20[/tex] for [tex]y[/tex] in equation [tex]y = a{x^2} + bx + c.[/tex]

[tex]\begin{aligned}- 20&= a{\left({ - 2} \right)^2} + b\left({ - 2} \right) + c\\- 20&= 4a - 2b + c\\- 20 - c&= 4a - 2b\\\end{aligned}[/tex]

Substitute [tex]0[/tex] for [tex]x[/tex] and [tex]-4[/tex] for [tex]y[/tex] in equation [tex]y = a{x^2} + bx + c.[/tex]

[tex]\begin{aligned}- 4&= a{\left( 0 \right)^2} + b\left( 0 \right)+ c\\- 4&= 0 + 0 + c\\- 4&= c\\\end{aligned}[/tex]

Substitute [tex]4[/tex] for [tex]x[/tex] and [tex]-20[/tex] for [tex]y[/tex] in equation [tex]y = a{x^2} + bx + c.[/tex]

[tex]\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}[/tex]

Substitute [tex]-4[/tex] for [tex]c[/tex] in equation [tex]- 20 - c = 4a - 2b.[/tex]

[tex]\begin{aligned}- 20 - c&= 4a - 2b\\- 20 + 4 &= 4a - 2b\\- 16&= 4a - 2b\\- 8&= 2a - b\\\end{aligned}[/tex]

Substitute [tex]-4[/tex] for [tex]c[/tex] in equation [tex]- 20 - c = 16a + 4b.[/tex]

[tex]\begin{aligned}- 20 - c&= 16a + 4b\\- 20 + 4&= 16a + 4b\\- 16 &= 16a + 4b\\ - 4&= 4a + b\\\end{aligned}[/tex]

Add the equations [tex]- 4 = 4a + b[/tex] and [tex]- 8 = 2a - b.[/tex]

[tex]\begin{aligned}- 8 - 4 &= 2a - b + 4a + b\\ - 12&= 6a\\\frac{{ - 12}}{6}&= a\\- 2&= a\\\end{aligned}[/tex]

The value of b can be obtained as follows,

[tex]\begin{aligned}4\left({ - 2}\right) + b& =- 4\\- 8 + b&=- 4\\b&= - 4 + 8\\\end{aligned}[/tex]

The standard form of the parabola is [tex]\boxed{y= - 2{x^2} + 4x - 4}.[/tex]

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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.