\begin{tabular}{l|l} N list & \( \begin{array}{l}\text { For the quadratic function } \mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+6 \mathrm{x}+9 \text {, answer parts (a) through (c). } \\ \text { (a) Graph the quadratic function by determining whether its graph opens up or down } \\ \text { and by finding its vertex, axis of symmetry, } \mathrm{y} \text {-intercept, and } \mathrm{x} \text {-intercepts, if any. } \\ \text { Does the graph of } \mathrm{f} \text { open up or down? } \\ \text { tion } 10\end{array} \) \\ \( \begin{array}{ll}\text { What are the coordinates of the vertex? }\end{array} \) \\ The vertex of the parabola is \( \square \).\end{tabular}

1. **Determine the direction the parabola opens:** The quadratic function is given by \[ f(x) = x^2 + 6x + 9. \] Since the coefficient of \( x^2 \) is positive (\(1 > 0\)), the parabola opens upward. 2. **Find the vertex:** The vertex of a parabola given by \[ f(x) = ax^2 + bx + c \] can be found using the formula \[ x = -\frac{b}{2a}. \] Here, \( a = 1 \) and \( b = 6 \), so: \[ x = -\frac{6}{2 \cdot 1} = -3. \] To find the \( y \)-coordinate, substitute \( x = -3 \) into \( f(x) \): \[ f(-3) = (-3)^2 + 6(-3) + 9 = 9 - 18 + 9 = 0. \] Therefore, the vertex is at \( \boxed{(-3, 0)} \). 3. **Determine the axis of symmetry:** The axis of symmetry is vertical through the vertex, i.e., \[ x = -3. \] 4. **Find the \( y \)-intercept:** The \( y \)-intercept occurs when \( x = 0 \): \[ f(0) = 0^2 + 6(0) + 9 = 9. \] Thus, the \( y \)-intercept is \( (0, 9) \). 5. **Find the \( x \)-intercepts:** The \( x \)-intercepts occur when \( f(x) = 0 \): \[ x^2 + 6x + 9 = 0. \] Factor the quadratic: \[ (x+3)^2 = 0. \] Hence, \[ x + 3 = 0 \quad \Longrightarrow \quad x = -3. \] This means there is one \( x \)-intercept at \( (-3, 0) \) (a repeated root). **Summary of Findings:** - **Direction:** Opens upward. - **Vertex:** \( (-3, 0) \). - **Axis of symmetry:** \( x = -3 \). - **\( y \)-intercept:** \( (0, 9) \). - **\( x \)-intercept:** \( (-3, 0) \). Thus, the coordinates of the vertex are \(\boxed{(-3, 0)}\).