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$y=x^2-6x+6$ Which of the following is an equivalent form of the equation of the graph shown in the $xy$ -plane, from which the coordinates of vertex $V$ can be identified as constants in the equation?

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Write a quadratic function in vertex form

Full solution

Q.

y=x^2-6x+6

Which of the following is an equivalent form of the equation of the graph shown in the

xy

-plane, from which the coordinates of vertex

V

can be identified as constants in the equation?

Identify Quadratic Equation: Identify the given quadratic equation. $\newline$ The given quadratic equation is $y = x^2 - 6x + 6$ . We need to rewrite this equation in vertex form, which is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.
Factor Coefficient: Factor out the coefficient of $x^2$ if it is not $1$ . In this case, the coefficient of $x^2$ is $1$ , so we do not need to factor anything out. We can proceed to complete the square.
Complete the Square: Complete the square to transform the equation into vertex form. $\newline$ To complete the square, we take the coefficient of $x$ , which is $-6$ , divide it by $2$ , and square it. This gives us $(-6/2)^2 = (-3)^2 = 9$ .
Add/Subtract Value: Add and subtract the value obtained in Step $3$ inside the parentheses. We add $9$ and subtract $9$ inside the parentheses to maintain the equality of the equation. This gives us $y = (x^2 - 6x + 9) - 9 + 6$ .
Rewrite with Perfect Square: Rewrite the equation with a perfect square trinomial. $\newline$ The equation now becomes $y = (x - 3)^2 - 3$ . This is the vertex form of the equation, and from this form, we can identify the vertex $V$ of the parabola as $(3, -3)$ .

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