Graphing Polynomial Functions — Quiz

Question 6 • Mathematics III

Which graph has the same end behavior as the graph of f of x is equal to negative 3 x cubed minus x squared plus 1 ?

Answer

Image with description On a coordinate plane, a polynomial graph is drawn. It starts from quadrant 2 and moves to quadrant 3 crossing the negative x axis 5 units from the origin. It then curves up, passes through negative x axis 1 units from the origin, crosses the positive y axis 0 point 5 units from the origin, and again crosses the positive x axis 1 unit from the origin. It then moves through quadrant 4.

Image with description On a coordinate plane a parabola opening downwards is drawn. Its vertex is on the positive y axis, 2 units from the origin.

Image with description On a coordinate plane, a polynomial graph is drawn. It starts from quadrant 3, moves to quadrant 2 passing through the negative x axis, 1 point2 units from the origin. It ten curves down, crosses the positive y axis, 2 units from the origin, curves up and passes through quadrant 1.

Image with description On a coordinate plane a parabola opening upwards is drawn. Its vertex is on the positive y axis, 2 units from the origin.

Explanation

The leading term is −3x³, which determines the end behavior. Because the degree is odd and the leading coefficient is negative, the graph rises on the left as x approaches negative infinity and falls on the right as x approaches positive infinity. Choice C starts in quadrant 2 and ends in quadrant 4, matching that behavior.

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