Answer:
[tex]y = (x - 3)^2 - 12[/tex]
[tex](3,-12)[/tex]
Step-by-step explanation:
Given
[tex]y = x^2 - 6x - 3[/tex]
Solving (a): In vertex form
The vertex form of an equation is:
[tex]y = a(x - h)^2 + k[/tex]
To do this, we make use of completing the square method.
We have:
[tex]y = x^2 - 6x - 3[/tex]
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Take the coefficient of x (i.e. -6)
Divide by 2; -6/2 = -3
Square it: (-3)^2 = 9
Add and subtract the result to the equation
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[tex]y = x^2 - 6x - 3[/tex]
[tex]y = x^2 - 6x + 9 - 9 - 3[/tex]
[tex]y = x^2 - 6x + 9 - 12[/tex]
Factorize [tex]x^2 - 6x + 9[/tex]
[tex]y = x^2 - 3x-3x + 9 - 12[/tex]
[tex]y = x(x - 3)-3(x - 3) - 12[/tex]
Factor out x - 3
[tex]y = (x - 3)(x - 3) - 12[/tex]
Express as squares
[tex]y = (x - 3)^2 - 12[/tex]
Hence, the vertex form of [tex]y = x^2 - 6x - 3[/tex] is: [tex]y = (x - 3)^2 - 12[/tex]
Solving (b): State the coordinates of the vertex.
In [tex]y = a(x - h)^2 + k[/tex]; the vertex is: (h,k)
The vertex of [tex]y = (x - 3)^2 - 12[/tex] will be [tex](3,-12)[/tex]
