Find the range of the quadratic function. Give your answer using interval notation. \[ y=3 x^{2}+18 x+9 \]

To find the range of the quadratic function \( y = 3x^2 + 18x + 9 \), we first rewrite it in vertex form by completing the square. 1. Factor out the coefficient of \( x^2 \) from the first two terms: \( y = 3(x^2 + 6x) + 9 \). 2. Complete the square inside the parentheses: \( x^2 + 6x \) can be transformed to \( (x + 3)^2 - 9 \). So the equation now looks like: \( y = 3((x + 3)^2 - 9) + 9 \) \( y = 3(x + 3)^2 - 27 + 9 \) \( y = 3(x + 3)^2 - 18 \). Now, the vertex of the parabola given by \( y = 3(x + 3)^2 - 18 \) is at \( (-3, -18) \). Since the coefficient of \( (x + 3)^2 \) is positive, the parabola opens upwards. Thus, the minimum value of \( y \) occurs at the vertex, \( y = -18 \), and since it opens upwards, the range of the function is: \[ [-18, \infty) \].