Respuesta :
The quadratic equation has the general formula:
y = ax^2 + bx + c
The vertex form has the general formula:
y = a(x-h)^2 + k
To get the vertex formula, we will need to get the values of a,h and k as follows:
1- The value of a:
The value of "a" in the vertex form is the same as the value of "a" in the quadratic form
Therefore: a = -1
2- The value of h:
the value of "h" can be computed using the formula:
h = -b / 2a
From the given quadratic equation: b = 12 and a = -1
Therefore: h = -12 / 2(-1) = 12/2 = 6
3- The value of k:
The value of k can be computed easily by evaluating y at the calculated h as follows:
The given equation is: y = -x^2 + 12x - 4
Compute the result at x=h=6 to get k as follows:
k = y = -(6)^2 + 12(6) - 4 = 32
Therefore, based on the above calculations, the vertex form would be:
y = a(x-h)^2 + k
y = -(x-6)^2 + 32
y = ax^2 + bx + c
The vertex form has the general formula:
y = a(x-h)^2 + k
To get the vertex formula, we will need to get the values of a,h and k as follows:
1- The value of a:
The value of "a" in the vertex form is the same as the value of "a" in the quadratic form
Therefore: a = -1
2- The value of h:
the value of "h" can be computed using the formula:
h = -b / 2a
From the given quadratic equation: b = 12 and a = -1
Therefore: h = -12 / 2(-1) = 12/2 = 6
3- The value of k:
The value of k can be computed easily by evaluating y at the calculated h as follows:
The given equation is: y = -x^2 + 12x - 4
Compute the result at x=h=6 to get k as follows:
k = y = -(6)^2 + 12(6) - 4 = 32
Therefore, based on the above calculations, the vertex form would be:
y = a(x-h)^2 + k
y = -(x-6)^2 + 32
The vertex form of the equation [tex]y =-x^{2}+12x-4[/tex] is [tex]\boxed{y=- {{\left({x - 6}\right)}^2}+32}.[/tex]
Further explanation:
The general form of the quadratic equation can be expressed as follows,
[tex]\boxed{y = a{x^2} + bx + c}[/tex]
Here, a represents the coefficient of [tex]{x^2}[/tex] , b is the coefficient of x and c 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 x-coordinate of the equation and k is the y-coordinate.
Given:
The quadratic equation is [tex]y = - {x^2} + 12x -4.[/tex]
Explanation:
Compare the quadratic equation is [tex]y = - {x^2} + 12x - 4[/tex] with the general quadratic equation.
The value of a is -1.
[tex]\boxed{a = - 1}.[/tex]
The value of h can be obtained as follows,
[tex]\begin{aligned}h&= -\frac{b}{{2a}}\\&=-\frac{{12}}{{2\left( { - 1}\right)}}\\&=\frac{{12}}{2}\\&=6\\\end{aligned}[/tex]
Substitute 6 for x in equation [tex]y = - {x^2} + 12x - 4[/tex] to obtain the value of y.
[tex]\begin{aligned}y &= - {\left(6\right)^2}+ 12\left(6\right)- 4\\&= - 36 + 72 - 4\\&=32\\\end{aligned}[/tex]
Therefore, the value of k is 32.
Substitute -1 for a, 6 for h and 32 for k in equation [tex]y = a{{\left( {x - h} \right)}^2} + k[/tex] to obtain the vertex equation.
[tex]\begin{aligned}y&= - 1{\left({x - 6}\right)^2}+32\\&= - {\left( {x - 6}\right)^2}+32\\\end{aligned}[/tex]
Hence, thevertex form of the equation [tex]y = - {x^2} + 12x - 4[/tex] is [tex]\boxed{y = - {{\left( {x - 6}\right)}^2} + 32}.[/tex]
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Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Quadratic equation
Keywords: quadratic equation, vertex form of the equation, biased, equation, formula, parabola, general equation, [tex]y = - {x^2} + 12x - 4[/tex], explained better.
