Respuesta :
In order to complete the square, the leading coefficient has to be a positive 1, which it is. Now we will set the polynomial equal to 0 and move the 2 over by subtraction to isolate the x terms. [tex]x^2-6x=-2[/tex]. The rule now is to take half the linear term, square it, and then add it in to both sides. Our linear term is 6. Half of 6 is 3, and 3 squared is 9, so we add 9 to both sides. [tex]x^2-6x+9=-2+9[/tex]. Simplifying we have [tex]x^2-6x+9=7[/tex]. During this process, and the reason for it, was to create a perfect square binomial on the left which will give us the x coordinate (or the h) for our vertex. That perfect square binomial is [tex](x-3)^2=7[/tex]. Now we will move the 7 over by subtraction and set the polynomial back equal to y to get [tex](x-3)^2-7=y[/tex]. Our vertex, then, is (3, -7) and this is a min value since our parabola is positive and opens up like a cup that has a bottom instead of mountain that has a top. And there you go! Your answer is C
Vertex of the equation will be denoted by the point given in option (C).
Quadratic equation given in the question is,
- y = x² - 6x + 2
To convert this equation into vertex form we will rewrite the equation into square form.
y = x² - 6x + 2
= x² - 2(3x) + 2
= x² - 2(3x) + (3)² - (3)² + 2
= [x² - 2(3x) + 3²] - 3² + 2
= (x - 3)² - 9 + 2
y = (x - 3)² - 7
By comparing this equation with the vertex form of the quadratic equation,
y = a(x - h)² + k
Vertex of the parabola will be (3, - 7).
And the leading coefficient 'a' = 1
Since, leading coefficient is positive, parabola will open upwards.
Therefore, vertex will be the minimum or lowest point of the curve.
Hence, Option (C) will be the answer.
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