Question

Can someone explain to me how to find the vertex form of this equation?

Answer 1

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

Answer 2

The vertex form of the equation $y =-x^{2}+12x-4$ is $\boxed{y=- {{\left({x - 6}\right)}^2}+32}.$

Further  explanation:

The general form of the quadratic equation 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 quadratic equation is  $y = - {x^2} + 12x -4.$

Explanation:

Compare the quadratic equation is $y = - {x^2} + 12x - 4$ with the general quadratic equation.

The value of a is -1.

$\boxed{a = - 1}.$

The value of h can be obtained as follows,

$\begin{aligned}h&= -\frac{b}{{2a}}\\&=-\frac{{12}}{{2\left( { - 1}\right)}}\\&=\frac{{12}}{2}\\&=6\\\end{aligned}$

Substitute 6 for x in equation $y = - {x^2} + 12x - 4$ to obtain the value of y.

$\begin{aligned}y &= - {\left(6\right)^2}+ 12\left(6\right)- 4\\&= - 36 + 72 - 4\\&=32\\\end{aligned}$

Therefore, the value of k is 32.

Substitute -1 for a, 6 for h and 32 for k in equation $y = a{{\left( {x - h} \right)}^2} + k$ to obtain the vertex equation.

$\begin{aligned}y&= - 1{\left({x - 6}\right)^2}+32\\&= - {\left( {x - 6}\right)^2}+32\\\end{aligned}$

Hence, thevertex form of the equation $y = - {x^2} + 12x - 4$ is $\boxed{y = - {{\left( {x - 6}\right)}^2} + 32}.$

Learn more:

1. Learn more about inverse of the function brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: Middle School

Subject: Mathematics

Chapter: Quadratic equation

Keywords: quadratic equation, vertex form of the equation, biased, equation, formula, parabola, general equation, $y = - {x^2} + 12x - 4$, explained better.