\( x \)-intercept(s) and the coordinates of the vertex for the parabola \( y=-x^{2}-2 x+3 \). If there is more than one \( x \)-intercept, separate them with comr

To find the \( x \)-intercept(s) and the coordinates of the vertex for the parabola \( y=-x^{2}-2x+3 \), we can follow these steps: 1. Find the \( x \)-intercept(s) by setting \( y = 0 \) and solving for \( x \). 2. Find the vertex of the parabola by using the formula for the vertex of a parabola in the form \( y = ax^2 + bx + c \), which is given by \( x = -\frac{b}{2a} \). Let's start by finding the \( x \)-intercept(s). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(-x^{2}-2x+3=0\) - step1: Factor the expression: \(\left(-x+1\right)\left(x+3\right)=0\) - step2: Separate into possible cases: \(\begin{align}&-x+1=0\\&x+3=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=1\\&x=-3\end{align}\) - step4: Rewrite: \(x_{1}=-3,x_{2}=1\) To find the vertex of the parabola, we can use the formula for the vertex of a parabola in the form \( y = ax^2 + bx + c \), which is given by \( x = -\frac{b}{2a} \). For the given parabola \( y = -x^2 - 2x + 3 \), we have \( a = -1 \) and \( b = -2 \). Substitute these values into the formula to find the \( x \)-coordinate of the vertex: \[ x = -\frac{-2}{2(-1)} = -\frac{-2}{-2} = -1 \] Now, substitute \( x = -1 \) back into the equation of the parabola to find the \( y \)-coordinate of the vertex: \[ y = -(-1)^2 - 2(-1) + 3 = -1 + 2 + 3 = 4 \] Therefore, the coordinates of the vertex are \( (-1, 4) \).