Question

which graph represents the function $f(x) = -(x - 3)^2 + 1$?
a) graph with vertex at (-3, -1)
b) graph with vertex at (3, 1)

Explanation:

Step1: Identify the vertex form

The function \( f(x) = -(x - 3)^2 + 1 \) is in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, \( h = 3 \) and \( k = 1 \), so the vertex is \((3, 1)\).

Step2: Determine the direction of opening

The coefficient \( a = -1 \), which is negative. So the parabola opens downward.

Step3: Analyze the graphs

• Graph a: The vertex is at \((-3, -1)\) (approximately) and opens downward, but the vertex coordinates don't match \((3, 1)\).
• Graph b: The vertex is at \((3, 1)\) and opens downward, which matches the vertex and direction from the function.

Answer:

b) The graph with vertex at (3, 1) and opening downward.