To solve the inequality $y - x^2 + 4 \geq 0$, we first rewrite it to make it easier to understand:

$y \geq x^2 - 4$

This inequality states that $y$ is greater than or equal to the parabola $y = x^2 - 4$.

Steps to determine the correct graph:

1. Identify the parabola: The equation $y = x^2 - 4$ is a parabola that opens upwards (since the coefficient of $x^2$ is positive) and is shifted down by 4 units compared to the standard parabola $y = x^2$.

2. Graph the parabola: The vertex of the parabola is at $(0, -4)$. The parabola intersects the y-axis at $(0, -4)$ and the x-axis at points where $y = 0$, which happens when $x^2 = 4$, giving $x = \pm 2$. So, the parabola intersects the x-axis at $(-2, 0)$ and $(2, 0)$.

3. Shading the region: Since the inequality is $y \geq x^2 - 4$, the solution region will include the parabola itself and all the area above it.

The correct graph should have:

• A parabola opening upwards with a vertex at $(0, -4)$.
• The area above the parabola shaded or filled in, including the parabola.

If you have different graphs to choose from, select the one that matches this description.

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Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How does the graph change if the inequality were $y > x^2 - 4$ instead of $y \geq x^2 - 4$?
2. What is the significance of the vertex in the graph of a parabola?
3. How would the graph look if the equation were $y \leq x^2 - 4$ instead?
4. What transformations occur to the graph $y = x^2$ to get $y = x^2 - 4$?
5. Can you find the x-intercepts of any parabola without graphing?

Tip: Remember, inequalities involving $\geq$ or $\leq$ include the boundary line, which is why the parabola is part of the solution in this case.