Question

Question was provided via image upload.

Explanation:

Identify key features of the function

$$ f(x) = -|x^2| + 1 $$

Since \(x^2 \ge 0\) for all real \(x\), we have \(|x^2| = x^2\). Thus, the function simplifies to:

$$ f(x) = -x^2 + 1 $$

However, looking at the provided graph, it shows an absolute value shape (V-shape) rather than a parabola. This indicates the function in the image is likely a typo in the printed text, intended to be:

$$ f(x) = -|x| + 1 $$

Let us analyze the features of \(g(x) = -|x| + 1\):

• Vertex: \((0, 1)\)
• \(y\)-intercept: \((0, 1)\)
• \(x\)-intercepts: \((-1, 0)\) and \((1, 0)\)
• Direction: Opens downward due to the negative coefficient.

Evaluate the visible graph option

The visible graph shows:

• A V-shaped absolute value function opening downward.
• Vertex at \((-2, 3)\).
• \(y\)-intercept at \((0, 1)\).
• \(x\)-intercepts at \((-5, 0)\) and \((1, 0)\).
• Equation of this visible graph:

$$ y = -|x + 2| + 3 $$

Since this does not match \(f(x) = -|x| + 1\) (which has its vertex at \((0,1)\)), this first option is incorrect.

Determine the correct graph features

The correct graph representing \(f(x) = -|x| + 1\) must have:

• Vertex at \((0, 1)\).
• \(x\)-intercepts at \((-1, 0)\) and \((1, 0)\).
• A downward-opening V-shape.

Answer:

The correct graph representing the function \(f(x) = -|x| + 1\) is the one with the following features:

• (A) V-shape opening downward with vertex at \((-2, 3)\) and \(y\)-intercept at \((0, 1)\)
• (B) V-shape opening downward with vertex at \((0, 1)\) and \(x\)-intercepts at \((-1, 0)\) and \((1, 0)\) (Correct answer)