To determine if a polynomial graph belongs to a third-degree polynomial (also known as a cubic polynomial), we can look for certain characteristics that are typical of cubic functions:
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End Behavior: A third-degree polynomial has the end behavior that both ends of the graph go in opposite directions. Specifically, if the leading coefficient is positive, the graph will rise to the right and fall to the left; if the leading coefficient is negative, it will fall to the right and rise to the left.
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Number of Turning Points: A cubic polynomial can have up to 2 turning points (local minima and maxima). If the graph has more than two turning points, it cannot be a cubic polynomial.
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Intercepts: A cubic polynomial can have up to three x-intercepts (real roots), and it must have at least one real root due to the Fundamental Theorem of Algebra.
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Symmetry: Cubic polynomials are generally not symmetric, but they may have some points of rotational symmetry.
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Shape: The general shape of a cubic graph resembles the letter "S." It typically starts from one end of the graph going up or down, crosses the x-axis up to three times, and then finishes on the opposite side.
By examining the actual graph and these characteristics, you can conclude whether it appears to be a cubic polynomial (third-degree) or not. If you see that the graph exhibits the behaviors mentioned above, then Celina may be correct in thinking that the polynomial graph belongs to a third-degree polynomial. If it shows behaviors that are inconsistent with cubic functions (such as more than two turning points, or a different end behavior), then she may not be correct.
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