The problem provides the equation of a parabola, $y = 3 - 2x - x^2$. We need to find the coordinates of points A and C where the parabola intersects the x-axis, and the coordinates of the turning point of the graph.

1. Problem Description

The problem provides the equation of a parabola, $y = 3 - 2x - x^2$. We need to find the coordinates of points A and C where the parabola intersects the x-axis, and the coordinates of the turning point of the graph.

2. Solution Steps

i. Finding the coordinates of A and C:

Points A and C are the x-intercepts of the parabola, meaning $y = 0$. We need to solve the equation $3 - 2x - x^2 = 0$ for $x$.

We can rewrite the equation as $x^2 + 2x - 3 = 0$.

This is a quadratic equation, which can be factored:

$(x+3)(x-1) = 0$.

Therefore, the solutions are $x = -3$ and $x = 1$.

The coordinates of A are $(-3, 0)$ and the coordinates of C are $(1, 0)$.

ii. Finding the coordinates of the turning point:

The turning point of a parabola in the form $y = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$. In our case, the equation is $y = -x^2 - 2x + 3$, so $a = -1$ and $b = -2$.

$x = -\frac{-2}{2(-1)} = -\frac{-2}{-2} = -1$.

To find the $y$-coordinate, substitute $x = -1$ into the original equation:

$y = 3 - 2(-1) - (-1)^2 = 3 + 2 - 1 = 4$.

Therefore, the coordinates of the turning point are $(-1, 4)$.

3. Final Answer

i. Coordinates of A and C:

A: $(-3, 0)$

C: $(1, 0)$

ii. Coordinates of the turning point:

$(-1, 4)$