For the quadratic function \( f(x)=x^{2}+6 x+9 \), answer parts (a) through ( \( f \) ). (a) Find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down. The vertex is ( \( -3,0 \) ). (Type an ordered pair, using integers or fractions.) What is the equation of the axis of symmetry? The axis of symmetry is \( x=-3 \). (Use integers or fractions for any numbers in the equation.) Is the graph concave up or concave down? Concave up Concave down (b) Find the \( y \)-intercept and the \( x \)-intercepts, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O. The \( x \)-intercept(s) is/are \( \square \). (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts. O

Let's solve the problem step by step. ### Part (a) 1. **Finding the Vertex:** The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] For the function \( f(x) = x^2 + 6x + 9 \): - \( a = 1 \) - \( b = 6 \) Plugging in the values: \[ x = -\frac{6}{2 \cdot 1} = -3 \] Now, substitute \( x = -3 \) back into the function to find the \( y \)-coordinate of the vertex: \[ f(-3) = (-3)^2 + 6(-3) + 9 = 9 - 18 + 9 = 0 \] Thus, the vertex is \( (-3, 0) \). 2. **Finding the Axis of Symmetry:** The axis of symmetry is given by the line \( x = -3 \). 3. **Determining Concavity:** Since the coefficient \( a = 1 \) is positive, the graph is concave up. ### Summary of Part (a): - The vertex is \( (-3, 0) \). - The equation of the axis of symmetry is \( x = -3 \). - The graph is concave up. ### Part (b) 1. **Finding the \( y \)-intercept:** The \( y \)-intercept occurs when \( x = 0 \): \[ f(0) = 0^2 + 6(0) + 9 = 9 \] Thus, the \( y \)-intercept is \( (0, 9) \). 2. **Finding the \( x \)-intercepts:** The \( x \)-intercepts occur when \( f(x) = 0 \): \[ x^2 + 6x + 9 = 0 \] This can be factored as: \[ (x + 3)(x + 3) = 0 \] Therefore, the \( x \)-intercept is: \[ x + 3 = 0 \implies x = -3 \] Since this is a repeated root, there is one \( x \)-intercept at \( (-3, 0) \). ### Summary of Part (b): - The \( y \)-intercept is \( 9 \). - The \( x \)-intercept(s) is/are \( -3 \). ### Final Answers: - (a) Vertex: \( (-3, 0) \), Axis of symmetry: \( x = -3 \), Concave up. - (b) The \( y \)-intercept is \( 9 \), and the \( x \)-intercept is \( -3 \).