Carlos graphs two quadratic equations:

1. $y = -\frac{1}{2}x^2 + 4$
2. $y = -\frac{1}{2}x^2 - 2x + 2$

The graph shows two parabolas, and the task is to identify the conclusion supported by the graph.

Step 1: Analyze the equations

Equation 1: $y = -\frac{1}{2}x^2 + 4$

• This is a downward-opening parabola (since the coefficient of $x^2$ is negative).
• The vertex form of this parabola is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex. For this equation:
  • $a = -\frac{1}{2}$
  • $h = 0$, $k = 4$, so the vertex is $(0, 4)$.

Equation 2: $y = -\frac{1}{2}x^2 - 2x + 2$

• This is also a downward-opening parabola.
• To find its vertex, we rewrite it in vertex form by completing the square.

$y = -\frac{1}{2}x^2 - 2x + 2$ Factor out $-\frac{1}{2}$ from the $x^2$ and $x$ terms: $y = -\frac{1}{2}\left(x^2 + 4x\right) + 2$ Complete the square inside the parentheses: $x^2 + 4x \quad \text{becomes} \quad (x+2)^2 - 4$ Substitute back: $y = -\frac{1}{2}\left((x+2)^2 - 4\right) + 2$ Distribute $-\frac{1}{2}$: $y = -\frac{1}{2}(x+2)^2 + 2 + 2$ $y = -\frac{1}{2}(x+2)^2 + 4$ The vertex of this parabola is $(-2, 4)$.

Step 2: Compare the graphs

From the analysis:

1. The first parabola has its vertex at $(0, 4)$.
2. The second parabola has its vertex at $(-2, 4)$.

Both parabolas share the same maximum $y$-value of 4, but their symmetry axes differ. The first is symmetric about $x = 0$, while the second is symmetric about $x = -2$.

Step 3: Supported conclusion

The graph supports the conclusion that both parabolas have the same maximum value (at $y = 4$) but different axes of symmetry.

Would you like a deeper explanation or have further questions?
Here are five related questions to expand your understanding:

1. How do you find the vertex of a quadratic equation in standard form?
2. What does the coefficient of $x^2$ indicate about the shape of a parabola?
3. How does completing the square help rewrite a quadratic equation?
4. Why is symmetry important in understanding the graph of a parabola?
5. What does the shared maximum value of the parabolas signify?

Tip: Always compare the vertex and the axis of symmetry to understand the differences between two quadratic graphs!