Question

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Explanation:

Analyze the roots and their multiplicities

Using the Polynomial Multiplicity and Graph Behavior at Roots knowledge points

• At \(x = -6\), the root has multiplicity 3 (odd multiplicity \(\ge 3\)), meaning the graph crosses the x-axis with an inflection point (it flattens out as it crosses).
• At \(x = 2\), the root has multiplicity 4 (even multiplicity), meaning the graph touches the x-axis and turns around (tangent to the x-axis, U-shaped behavior).

Analyze the end behavior of the polynomial

Using the Polynomial End Behavior knowledge point

• The polynomial has an odd degree and a negative leading coefficient.
• As \(x \to \infty\), \(y \to -\infty\).
• As \(x \to -\infty\), \(y \to \infty\).
• Therefore, the graph starts in the upper-left quadrant (Quadrant II) and ends in the lower-right quadrant (Quadrant IV).

Evaluate the visible graph options

• Looking at the first graph shown:
• It starts in the upper-left quadrant and goes down to the lower-right quadrant, matching the end behavior.
• At \(x = -6\), the graph crosses the x-axis while flattening out (inflection point).
• At \(x = 2\), the graph touches the x-axis and turns around (tangent, staying below or on the x-axis).
• This perfectly matches all given criteria.

• Looking at the second graph (partially visible at the bottom):
• It goes upwards as \(x \to \infty\), which violates the negative leading coefficient condition for an odd-degree polynomial.

Answer:

• (A) The first graph, which starts in the upper-left, crosses the x-axis with an inflection point at \(x = -6\), touches and turns around at \(x = 2\), and goes down to the lower-right. (Correct answer)
• (B) The second graph, which goes upwards as \(x\) increases to the right.