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

Rationale

The coordinates of the vertex of the parabola represented by the equation y = -3x² + 18x - 24 are (3, 3).

To find the vertex of the parabola given in standard form \(y = ax^2 + bx + c\), we can use the formula for the x-coordinate of the vertex, \(x = -\frac{b}{2a}\). In this case, \(a = -3\) and \(b = 18\), leading to the vertex coordinates of (3, 3).

A ((6, -24)) YOUR ANSWER

This point does not satisfy the vertex formula. When substituting \(x = 6\) into the equation, the corresponding y-value is \(y = -3(6^2) + 18(6) - 24\), which does not yield -24. Hence, this is not the vertex.

B ((4, 0)) YOUR ANSWER

For \(x = 4\), substituting into the equation gives \(y = -3(4^2) + 18(4) - 24\), resulting in \(y = 0\). While this is a valid point on the parabola, it is not the vertex since the vertex occurs at \(x = 3\).

C ((3, 3)) CORRECT

This is the correct choice. Substituting \(x = 3\) into the equation results in \(y = -3(3^2) + 18(3) - 24\), which simplifies to 3. Therefore, (3, 3) represents the vertex of the parabola.

D ((2, 0)) YOUR ANSWER

When substituting \(x = 2\) into the equation, \(y = -3(2^2) + 18(2) - 24\) results in \(y = 0\). Although this point lies on the parabola, it does not indicate the vertex, as the vertex is located at (3, 3).

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

Calculating \(y\) for \(x = -3\) gives \(y = -3(-3^2) + 18(-3) - 24\), which results in a much lower value than -105. This point does not represent the vertex of the parabola.

Conclusion

The vertex of a parabola provides critical information regarding its maximum or minimum point. For the equation \(y = -3x^2 + 18x - 24\), the vertex is accurately calculated to be at (3, 3), confirming it as the peak of the parabolic curve. All other options represent points on the parabola but do not serve as the vertex, underscoring the