Question 32

The coordinates of the vertex of the parabola are (3, 3).

The vertex of the parabola represented by the equation y = -3x² + 18x - 24 can be found using the vertex formula. For a quadratic equation in the form y = ax² + bx + c, the x-coordinate of the vertex is given by -b/(2a). Substituting the values, we find the vertex at the coordinates (3, 3).

A) (6, -24)

This option is incorrect as it suggests a vertex at (6, -24), which does not satisfy the vertex formula derived from the given quadratic equation. The vertex must be calculated using the values of a and b from the equation and does not align with these coordinates.

B) (4, 0)

This option is also incorrect. While (4, 0) may be a point on the parabola, it does not represent the vertex. The vertex is found at a specific point that results from the maximum or minimum value of the parabola, which in this case is not at (4, 0).

C) (3, 3)

This option is correct as it represents the vertex of the parabola. By applying the vertex formula, we find that the x-coordinate is 3, and substituting this value back into the equation gives the corresponding y-coordinate as 3, confirming that the vertex is indeed at (3, 3).

D) (2, 0)

This option is incorrect as well. The coordinates (2, 0) do not correspond to the vertex of the parabola represented by the equation. The vertex is determined through the specific calculations of the vertex formula, which does not yield this point.

E) (-3, -105)

This option is incorrect. The coordinates (-3, -105) do not reflect the vertex of the parabola defined by the given equation. The calculations for the vertex yield different values, making this option invalid.

Conclusion

The correct answer, (3, 3), accurately identifies the vertex of the parabola based on the application of the vertex formula. All other options fail to represent the vertex as they do not conform to the calculations derived from the quadratic equation provided. Thus, (3, 3) is definitively the correct choice.