Let’s solve this system of equations and identify the correct graph.

System of Equations:

1. $y = x^2 - 6$ (a parabola opening upwards)
2. $x^2 + 6y^2 = 36$ (an ellipse)

Step 1: Analyze $y = x^2 - 6$

This equation is a parabola:

• The vertex is at $(0, -6)$.
• It opens upwards.

Step 2: Analyze $x^2 + 6y^2 = 36$

This is an ellipse:

• Rewrite it in standard form: $\frac{x^2}{36} + \frac{y^2}{6} = 1$ Here, $a^2 = 36$ and $b^2 = 6$:
  • Semi-major axis (horizontal direction) = $\sqrt{36} = 6$.
  • Semi-minor axis (vertical direction) = $\sqrt{6} \approx 2.45$.

The ellipse is centered at $(0, 0)$, stretched horizontally.

Step 3: Combine the Two

We need a graph that:

• Contains a parabola with its vertex at $(0, -6)$, opening upwards.
• Contains an ellipse centered at $(0, 0)$, elongated horizontally.

Step 4: Match with the Options

Looking at the graphs:

• Option C matches the criteria:
  • The parabola has its vertex at $(0, -6)$ and opens upwards.
  • The ellipse is centered at $(0, 0)$ and is horizontally stretched.

Answer: C

Would you like to see the equations verified or step-by-step intersections?

Questions to Expand:

1. How do you find the intersection points of these two graphs algebraically?
2. What are the properties of an ellipse and how are they derived?
3. How can you determine the focus points of this specific ellipse?
4. Why does the parabola open upwards in this case?
5. How would the graph change if the ellipse had a different center?

Tip:

When analyzing graphs of systems, always check the symmetry and key points to verify consistency with the equations!