To determine which parabola $y = ax^2 + bx + c$ passes through the points $(0, -2)$, $(1, -1)$, and $(-1, 1)$, we need to substitute these points into the general form of the parabola and solve for $a$, $b$, and $c$. Let's solve this step-by-step:

Step 1: Substituting Points into the General Form

For a parabola $y = ax^2 + bx + c$, substitute each point:

1. Point $(0, -2)$: $-2 = a(0)^2 + b(0) + c \implies c = -2$

2. Point $(1, -1)$: $-1 = a(1)^2 + b(1) + c \implies -1 = a + b - 2$ a + b = 1 \tag{1}

3. Point $(-1, 1)$: $1 = a(-1)^2 + b(-1) + c \implies 1 = a - b - 2$ a - b = 3 \tag{2}

Step 2: Solving the System of Equations

We now solve the system of equations:

1. $a + b = 1$
2. $a - b = 3$

Add the two equations: $(a + b) + (a - b) = 1 + 3 \implies 2a = 4 \implies a = 2$

Substitute $a = 2$ into $a + b = 1$: $2 + b = 1 \implies b = -1$

From earlier, $c = -2$.

Step 3: Writing the Parabola Equation

The equation of the parabola is: $y = 2x^2 - x - 2$

Step 4: Comparing with the Options

From the options provided, the equation matches (c) $y = 2x^2 - x - 2$.

Final Answer: (c)

Would you like further clarification or more detailed steps? Here are five related questions for you to consider:

1. How do you determine the general equation of a parabola given points?
2. What happens if more than three points are given for a quadratic equation?
3. How can you verify the solution graphically?
4. How do different coefficients $a$, $b$, and $c$ affect the shape and position of the parabola?
5. Can you derive a quadratic equation if only two points are provided?

Tip: Always check your solution by substituting the points into the final equation to ensure consistency!