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Sagot :
To determine the function that reveals the vertex of the parabola given by [tex]\(y = x^2 + 6x + 3\)[/tex], we need to complete the square. Here is the detailed, step-by-step solution:
1. Move the constant term to the other side of the equation:
[tex]\[ y - 3 = x^2 + 6x \][/tex]
2. Take half of the coefficient of [tex]\( x \)[/tex], square it, and add it to both sides:
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
Adding 9 to both sides:
[tex]\[ y - 3 + 9 = x^2 + 6x + 9 \][/tex]
Simplify the left side:
[tex]\[ y + 6 = (x + 3)^2 \][/tex]
3. Isolate [tex]\( y \)[/tex]:
[tex]\[ y = (x + 3)^2 - 6 \][/tex]
The function [tex]\( y = (x + 3)^2 - 6 \)[/tex] reveals the vertex of the parabola. Therefore, the correct answer is:
D. [tex]\( y = (x + 3)^2 - 6 \)[/tex]
1. Move the constant term to the other side of the equation:
[tex]\[ y - 3 = x^2 + 6x \][/tex]
2. Take half of the coefficient of [tex]\( x \)[/tex], square it, and add it to both sides:
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
Adding 9 to both sides:
[tex]\[ y - 3 + 9 = x^2 + 6x + 9 \][/tex]
Simplify the left side:
[tex]\[ y + 6 = (x + 3)^2 \][/tex]
3. Isolate [tex]\( y \)[/tex]:
[tex]\[ y = (x + 3)^2 - 6 \][/tex]
The function [tex]\( y = (x + 3)^2 - 6 \)[/tex] reveals the vertex of the parabola. Therefore, the correct answer is:
D. [tex]\( y = (x + 3)^2 - 6 \)[/tex]
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