To analyze the polynomial \( f(x) = -x(x+2)^2 \), we first need to identify its zeros and the behavior of the graph at those points.
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Finding the Zeros:
- The polynomial is factored as \( f(x) = -x(x+2)^2 \).
- The zeros occur when \( f(x) = 0 \):
- \( x = 0 \) (one zero)
- \( (x + 2)^2 = 0 \) gives \( x = -2 \) (with a multiplicity of 2).
- Thus, the zeros are \( x = 0 \) and \( x = -2 \).
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Determining the Behavior Around the Zeros:
- Since the leading coefficient of the polynomial is negative, the graph opens downward.
- At \( x = 0 \) (odd multiplicity), the graph will cross the x-axis.
- At \( x = -2 \) (even multiplicity), the graph will touch the x-axis but will not cross it.
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Finding the End Behavior:
- As \( x \to \infty \), \( f(x) \to -\infty \) (because the leading term is negative).
- As \( x \to -\infty \), \( f(x) \to -\infty \).
Considering this information:
- The graph will start high in quadrant 2, drop down to touch the x-axis at \( x = -2 \), then continue downward, crossing the x-axis at \( x = 0 \) before going down into quadrant 4.
Given these characteristics, we can eliminate options based on their descriptions:
- The second option (downward-opening parabola) does not match our findings about the multiplicities and specific shape around the zeros.
- The graphs with curves passing through quadrants 2 and 4 indicate that the graph likely touches at -2 and crosses at 0, which is consistent with our analysis.
Thus, the correct response would be the one that describes a curve passing through quadrants 2 and 4, indicating it properly touches the x-axis at \( x = -2 \) and crosses it at \( x = 0 \).
Recommended Response: A curve passes through quadrants 2 and 4 on a coordinate plane with the x-axis ranging from negative 2 to 6 in unit increments.