Question

Question was provided via image upload.

Explanation:

Identify the function's key features

Using the End Behavior of Polynomials and Zeros of a Function from Graph knowledge points

$$
LATEXBLOCK0
$$

Analyze the first graph option

We observe the behavior of the first graph:

• As \(x \to -\infty\), the curve goes downwards to \(-\infty\).
• As \(x \to \infty\), the curve goes upwards to \(\infty\).
• The graph crosses the \(x\)-axis at \(x = -1\), \(x = 1\), and \(x = 5\).
• It has a local maximum at \((0, 5)\) and a local minimum at \((3, -17)\).
• This matches a standard cubic polynomial \(f(x) = (x+1)(x-1)(x-5) = x^3 - 5x^2 - x + 5\).

Analyze the second graph option

We observe the behavior of the second graph:

• As \(x \to -\infty\), the curve goes upwards to \(\infty\).
• As \(x \to \infty\), the curve goes downwards to \(-\infty\).
• This represents a cubic polynomial with a negative leading coefficient, which is the vertical reflection of the first graph.

Match with the correct choice

• The first graph correctly represents the cubic function \(f(x) = x^3 - 5x^2 - x + 5\) with positive end behavior.
• Therefore, the first option is the correct graph.

Answer:

• (A) The first graph, which starts in the third quadrant, goes up through \((-1,0)\) to a local maximum at \((0,5)\), goes down through \((1,0)\) to a local minimum at \((3,-17)\), and then goes up through \((5,0)\) into the first quadrant. (Correct answer)
• (B) The second graph, which is a vertical reflection of the first graph, starting in the second quadrant and ending in the fourth quadrant.
• (C) The third graph, which shows a narrow parabola-like shape opening upwards.