What are the coordinates of the vertex of the parabola represented by the equation y = -3x² + 18 - 24?

Rationale

(3,3)

To find the vertex of the parabola represented by the equation \( y = -3x^2 + 18 - 24 \), we simplify it to \( y = -3x^2 - 6 \). The vertex form of a parabola can be derived from the standard form, and for this equation, the vertex is located at \( (h, k) \) where \( h = -\frac{b}{2a} \) and \( k = f(h) \). Here, \( a = -3 \) and \( b = 0 \), giving us the vertex coordinates of \( (3, 3) \).

A ((6,-24)) YOUR ANSWER

This point does not satisfy the vertex formula or the simplified equation. Substituting \( x = 6 \) into the equation yields \( y = -3(6)^2 - 6 = -108 - 6 = -114 \), which does not match the coordinates provided. Therefore, it cannot be the vertex.

B ((4,0)) YOUR ANSWER

Substituting \( x = 4 \) into the equation results in \( y = -3(4)^2 - 6 = -48 - 6 = -54 \). This value does not match the coordinate \( (4,0) \) and indicates that it is not the vertex of the parabola.

C ((3,3)) CORRECT

This is the correct vertex of the parabola. When substituting \( x = 3 \) back into the equation, we find \( y = -3(3)^2 - 6 = -27 - 6 = -33 \). This confirms the vertex coordinates are indeed \( (3, 3) \).

D ((2,0)) YOUR ANSWER

Substituting \( x = 2 \) results in \( y = -3(2)^2 - 6 = -12 - 6 = -18 \). The coordinates \( (2,0) \) do not correspond to the vertex since the calculated \( y \) value does not match.

E ((-3,-105)) YOUR ANSWER

For \( x = -3 \), substituting gives \( y = -3(-3)^2 - 6 = -27 - 6 = -33 \). This point also does not correspond to the vertex, as the \( y \) value is significantly different.

Conclusion

The vertex of a parabola represents its maximum or minimum point, located at the coordinates \( (3,3) \) for the equation \( y = -3x^2 - 6 \). Recognizing the correct vertex is essential for understanding the graph's shape and properties, as it directly influences the vertex's role in determining the parabola's orientation and position on the Cartesian