The given function is $y = -(x - 1)^2 + 3$. Let's break it down and sketch its graph:

1. Basic Form: The function is a transformation of the standard parabola $y = x^2$.
2. Vertex: The vertex form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. Here, $h = 1$ and $k = 3$, so the vertex is $(1, 3)$.
3. Direction: The negative sign before the squared term (the $-$) means the parabola opens downward.
4. Stretch/Compression: Since there is no coefficient other than -1, the parabola maintains its standard width.

The correct graph would:

• Have a vertex at $(1, 3)$,
• Open downward,
• Be symmetric around $x = 1$.

The first option in your image shows a downward-facing parabola with its vertex at $(1, 3)$, which is the correct graph.

Would you like a detailed explanation of this transformation or have any questions? Here are 5 related questions to deepen your understanding:

1. How do we identify the vertex of a parabola in vertex form?
2. What effect does a negative sign in front of the quadratic term have on the graph?
3. How would the graph change if the equation had a positive coefficient?
4. What are the transformations if the function is $y = (x + 2)^2 - 1$?
5. How does adding a constant outside the squared term affect the graph?

Tip: Always check the sign of the leading coefficient to determine whether a parabola opens upwards or downwards.