Finding the \( x \)-intercept(s) and the vertex of a parabola Find the \( x \)-intercept(s). and the coordinates of the vertex for the parabola \( y=-x^{2}-12 x-36 \). If there is more than one wintercept(s): vertex:

To find the \( x \)-intercept(s) and the vertex of the parabola given by the equation: \[ y = -x^{2} - 12x - 36 \] we can follow these steps: ### 1. Finding the \( x \)-Intercept(s) The \( x \)-intercept(s) occur where \( y = 0 \). Set the equation to zero and solve for \( x \): \[ 0 = -x^{2} - 12x - 36 \] Multiply the entire equation by \(-1\) to simplify: \[ 0 = x^{2} + 12x + 36 \] This is a quadratic equation of the form \( ax^{2} + bx + c = 0 \), where \( a = 1 \), \( b = 12 \), and \( c = 36 \). Calculate the discriminant (\( \Delta \)) to determine the number of real solutions: \[ \Delta = b^{2} - 4ac = 12^{2} - 4(1)(36) = 144 - 144 = 0 \] Since the discriminant is zero, there is exactly one real solution (a repeated root): \[ x = \frac{-b}{2a} = \frac{-12}{2 \times 1} = -6 \] So, the \( x \)-intercept is at: \[ (-6, 0) \] ### 2. Finding the Vertex The vertex of a parabola in the form \( y = ax^{2} + bx + c \) can be found using the formula for the \( x \)-coordinate of the vertex: \[ x_v = \frac{-b}{2a} \] Plugging in the values: \[ x_v = \frac{-12}{2 \times (-1)} = \frac{-12}{-2} = 6 \] **Correction:** There's a sign error here. Given the original equation \( y = -x^{2} - 12x - 36 \), \( a = -1 \), \( b = -12 \). The correct calculation is: \[ x_v = \frac{-(-12)}{2 \times (-1)} = \frac{12}{-2} = -6 \] Now, substitute \( x_v = -6 \) back into the original equation to find the \( y \)-coordinate: \[ y_v = -(-6)^{2} - 12(-6) - 36 = -36 + 72 - 36 = 0 \] So, the vertex is at: \[ (-6, 0) \] ### **Summary** - **\( x \)-Intercept(s):** \((-6, 0)\) - **Vertex:** \((-6, 0)\) This means the parabola touches the \( x \)-axis at the vertex, indicating that there is exactly one \( x \)-intercept, which coincides with the vertex. **Final Answer:** - **\( x \)-intercept:** \((-6,\ 0)\) - **Vertex:** \((-6,\ 0)\)