Select the graph which correctly displays the solutions of y – x2 + 4 ≥ 0.



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.

Select the graph which correctly displays the solutions of y – x2 + 4 ≥ 0.

Learn how to graph the solution set for the quadratic inequality y ≥ x^2 - 4. This problem involves understanding parabolas, their properties, and how to represent inequalities graphically. Perfect for students in Grades 9-10.

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